some important lie symmetries in both general relativity

Some Important Lie Symmetries in Both General Relativity and

Teleparallel Theory of Gravitation

By

Suhail Khan

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF

PHILOSOPHY IN MATHEMATICS

Supervised by Dr. Ghulam Shabbir

Faculty of Engineering Sciences,

Ghulam Ishaq Khan Institute of Engineering Sciences and Technology Topi, District: Swabi, Khyber Pakhtunkhwa, Pakistan.

2011

ii

Author’s Declaration

I Suhail Khan S/O Haji Nawab declare that the work in this dissertation was carried out in

accordance with the regulations of Ghulam Ishaq Khan Institute of Engineering Sciences and

Technology Topi, Swabi. The work is original except where indicated by special reference in the

text and no part of the dissertation has been submitted for any other degree either in Pakistan or

overseas. Most of the work presented in this dissertation has been published in reputed journals.

Suhail Khan

ES0503

Faculty of Engineering Sciences,

GIK Institute, Topi, Swabi,

Khyber Pakhtunkhwa, Pakistan.

August, 2011.

iii

Certificate

It is certified that the research work presented in this thesis, entitled “Some

Important Lie Symmetries in Both General Relativity and Teleparallel

Theory of Gravitation” was conducted by Mr. Suhail Khan under the supervision

of Dr. Ghulam Shabbir.

(External Examiner) (External Examiner) Prof. Dr. Khalida Inayat Noor, Prof. Dr. Rasheed A. Muneer Professor, Professor, COMSATS Institute Islamabad. NUST, Islamabad. (Supervisor) (Dean) Dr. Ghulam Shabbir, Prof. Dr. Syed Ikram A. Tirmizi Associate Professor, Faculty of Engineering Sciences. Faculty of Engineering Sciences.

iv

Dedication

I DEDICATE THIS THESIS TO

MY LOVING SON

SAAD AHMAD

AND

MY LOVING ANGEL

SILAH SUHAIL

v

Table of Contents Table of Contents v

Abstract viii

Acknowledgments ix

1. Preliminaries 1.1 Introduction................................................................................................................................... 1

1.2 Basics of General Relativity ......................................................................................................... 3

1.2.1 Manifolds .............................................................................................................................. 3

1.2.2 Tangent Spaces ..................................................................................................................... 5

1.2.3 Tensors .................................................................................................................................. 6

1.2.4 Covariant Derivative ............................................................................................................. 8

1.2.5 Parallel Transport and Lie Derivative ................................................................................... 9

1.2.6 Riemann Curvature Tensor ................................................................................................. 10

1.2.7 Symmetries in General Relativity ....................................................................................... 11

1.2.8 Space-Times and Tetrad...................................................................................................... 12

1.2.9 Classification of Tangent Spaces ........................................................................................ 14

1.3 Basics of Teleparallel Theory of Gravitation.............................................................................. 14

1.3.1 Tetrad in Teleparallel Theory ............................................................................................. 14

1.3.2 Space-Time Structure.......................................................................................................... 15

1.3.3 Teleparallel Lie Derivative ................................................................................................. 17

1.4 Literature Review of Conformal, Homothetic, Killing and Self Similar Vector Fields in General Relativity .................................................................................................................................... 18

1.5 Literature Review of Teleparallel Killing and Homothetic vector fields in Teleparallel Theory of Gravitation.............................................................................................................................. 21

2. Teleparallel Killing Vector Fields in Bianchi Types I, II, VIII and

IX Space-Times 2.1 Introduction................................................................................................................................. 23

vi

2.2 Teleparallel Killing Vector Fields in Bianchi Type I Space-Times............................................ 24

2.3 Teleparallel Killing Vector Fields in Bianchi Type II Space-Times .......................................... 53

2.4 Teleparallel Killing Vector Fields in Bianchi Types VIII and IX Space-Times......................... 69

2.5 Summary of the Chapter ............................................................................................................. 85

3. Teleparallel Killing Vector Fields in Kantowski-Sachs, Bianchi

Type III, Static Cylindrically Symmetric and Spatially

Homogeneous Rotating Space-Times 3.1 Introduction................................................................................................................................. 87

3.2 Teleparallel Killing Vector Fields in Kantowski-Sachs and Bianchi Type III Space-Times...... 88

3.3 Teleparallel Killing Vector Fields in Static Cylindrically Symmetric Space-Times ................ 102

3.4 Teleparallel Killing Vector Fields in Spatially Homogeneous Rotating Space-Times............. 133

3.5 Summary of the Chapter ........................................................................................................... 147

4. Teleparallel Proper Homothetic Vector Fields in Bianchi Type I,

Non Static Plane Symmetric and Static Cylindrically Symmetric

Space-Times 4.1 Introduction............................................................................................................................... 149

4.2 Teleparallel Proper Homothetic Vector Fields in Bianchi Type I Space-Times....................... 150

4.3 Teleparallel Proper Homothetic Vector Fields in Non Static Plane Symmetric Space-Times . 180

4.4 Teleparallel Proper Homothetic Vector Fields in Static Cylindrically Symmetric Space-Times200


5. Proper Conformal Vector Fields in Non Conformally Flat Non

Static Cylindrically Symmetric, Kantowski-Sachs and Bianchi

Type III Space-Times 5.1 Introduction............................................................................................................................... 234

5.2 Proper Conformal Vector Fields in Non Conformally Flat Non Static Cylindrically Symmetric Space-Times ............................................................................................................................. 235

5.3 Proper Conformal Vector Fields in Non Conformally Flat Kantowski-Sachs and Bianchi type III Space-Times ........................................................................................................................ 263

vii


6. Self Similar Vector Fields in Kantowski-Sachs, Bianchi Type III,

Static Plane Symmetric and Static Spherically Symmetric Space-

Times 6.1 Introduction............................................................................................................................... 280

6.2 Self Similar Vector Fields in Kantowski-Sachs and Bianchi Type III Space-Times................ 281

6.3 Self Similar Vector Fields in Static Plane Symmetric Space-Times ........................................ 286

6.4 Self Similar Vector Fields in Static Spherically Symmetric Space-Times ............................... 292

6.5 Self Similar Vector Fields in Static cylindrically Symmetric Space-Times ............................. 301


References………………………………………………………………………………………...312

viii

Abstract

The aim of this thesis is to study some important Lie symmetries of well known space-times in

general relativity and teleparallel theory of gravitation. In teleparallel theory we investigated

teleparallel Killing vector fields for Bianchi types I, II, III, VIII, IX, static cylindrically

symmetric, Kantowski-Sachs and spatially homogeneous rotating space-times and proper

teleparallel homothetic vector fields for Bianchi type I, static cylindrically symmetric and non

static plane symmetric space-times. In general relativity conformal vector fields are studied for

non conformally flat Kantowski-Sachs, Bianchi type III and non static cylindrically symmetric

space-times. We also study self-similar vector fields in general relativity for Kantowski-Sachs,

Bianchi type III, static plane symmetric and static spherically symmetric space-times.

In teleparallel Killing vector fields for Bianchi types I, II, III, VIII, IX, static cylindrically

symmetric, Kantowski-Sachs and spatially homogeneous rotating space-times we have shown

that the number of teleparallel Killing vector fields for the above space-time are either same in

number or more to the Killing vector fields in general relativity. Interestingly, in the presence of

torsion it turns out that in some cases of Bianchi type I and static cylindrically symmetric space

times the dimension of teleparallel Killing vector fields is ten.

In proper teleparallel homothetic vector fields for Bianchi type I, static cylindrically symmetric

and non static plane symmetric space-times we have shown that the number of teleparallel proper

homothetic vector fields is always one which is the same in number to proper homothetic vector

field in general relativity. In general the dimension of the teleparallel homothetic vector fields is

always greater to the homothetic vector fields in general relativity.

In general relativity we have considered Kantowski-Sachs, Bianchi type III and non static

cylindrically symmetric space-times for proper conformal vector fields. We have shown that the

above space-times possess proper conformal vector fields for a special choice of metric

functions.

In general relativity we have investigated self similar vector fields for Kantowski-Sachs, Bianchi

type III, static plane symmetric, static spherically symmetric and static cylindrically symmetric

space-times. All the above space-times admit self-similar vector fields for a special choice of the

metric functions. Here, we have also discussed self similarity of first, second, zeroth and infinite

kind for both the tilted and non-tilted.

ix

ACKNOWLEDGMENTS

I have really no words to express the deepest sense of gratitude to Almighty Allah, who enabled

me to complete this thesis successfully. I wish to express my sincere gratitude to my respectable

and praiseworthy supervisor Dr. Ghulam Shabbir for giving me an opportunity to work under

his dynamic supervision. His motivation, continuous guidance, patience and valuable advices

have been fruitful in the completion of this uphill task.

I must acknowledge the valuable discussions with Dr. Muhammad Jamil Amir who’s guidance

enabled me to work in the field of teleparallel theory of gravitation. I am also indebted to my

father Haji Nawab for his encouragement, moral and financial support throughout my studies. I

should thank my wife for her patience and forbearance. Without her help it would have been

impossible for me to continue my studies. I am also thankful to my son Saad Ahmad and my

daughter Silah Suhail who spent most of their time without me. I am deeply indebted to my

mother, sisters, mother in law and brother Tahir Khan who always prayed for my brilliant

success. The name of my father in law Mr. Faramosh Khan needs to be specially mentioned here

for his moral and financial support he extended to me during my studies. Without his financial

support for my family it would not be possible for me to complete this task.

My thanks are due to some of my friends Abdul Mateen, Dr. Rana M. Ramzan, Dr. Amjad Ali,

Alamgeer Khan, Fazal Wahab, M. Tufail and Adam Khan who cooperated with me during my

studies.

Finally, I profoundly appreciate the financial support of Higher Education Commission of

Pakistan whose generous support made my dreams true. By awarding indigenous 5000

scholarship to me HEC has lightened my financial burden which allowed me to achieve a higher

goal. The transparency and generosity of HEC has inspired me to give back my best to the

community.

I would also thank Director Higher Education (Colleges) Khyber Pakhtunkhwa for granting me

study leave.

Suhail Khan

1

Chapter 1

Preliminaries

1.1 Introduction

Since long, man is trying to unfold mysteries of the universe and this quest for

exploring secretes of universe is increasing day by day. With all the modern

technology in hands man is still unable to resolve completely the mystery of gravity.

It is fact that no one can explain gravity at the most elementary level. In history,

gravity remained one of the most attractive topics to study. The common people

understand gravity just as a force to help them in staying on surface of earth. The

significance of gravity is much wider because it is responsible for the paths followed

by the planets around the Sun.

Among the ancient philosophers, Aristotle had a great insight of massive bodies. He

believed that universe was made up of five elements: earth, water, air, fire and ether.

He called ether the heavenly element and the rest of four as earthly elements. In his

law of terrestrial motion, he states that “all terrestrial bodies tend to go to their natural

state of rest”[1]. According to him anything taken from earth will alternately fall to

the earth, which is the natural state of rest for that body. Galileo Galilei has observed

the solar system through his telescope and concluded that Sun is the center of the

solar system and all the other planets revolve around it. He also discovered that

objects of different weights fall to the surface of earth with the same speed. Newton

was another notable figure who worked on gravity. He says that gravity travels

instantaneously throughout space. He thought that gravity is a force between the two

objects. According to Newton’s law of gravitation “The force of gravity is

proportional to the product of the two masses and inversely proportional to the square

of the distance between them”. The gravitational phenomena observed in the solar

2

system do not perfectly match with Newton’s law especially for Mercury’s orbit

where deviation of Mercury from precession is exceptional.

It was Albert Einstein who made the picture clear that in our universe everything is

relative. In 1905 he presented his theory of special relativity which he based on two

principles: (i) all inertial frames are physically equivalent and (ii) The speed of light

in vacuum is constant for all inertial observers. Since this theory is limited to deal

with linear motion, it has given the name special relativity. After spending 10 more

years Einstein proposed his second theory of relativity known as general theory of

relativity. The basic approach of this theory is to describe gravitational interaction

among massive objects by geometrizing the space-time.

It is the major aim of theoretical physics to procure a stable theory which unifies

general theory of gravity with the laws of quantum mechanics. One attempt to obtain

such a theory was made by Einstein in 1928. He used the mathematical structure of

distant or absolute parallelism also called teleparallelism. At each point of tangent

space of the four dimensional space-time he introduced a tetrad field. The tetrad field

was then used to compare the direction of the tangent vectors at different locations of

the space-time structure. This teleparallel theory involves sixteen components at each

point for the specification of four tetrad vectors while there are only ten components

for the specification of symmetric metric tensor. Einstein erroneously supposed that

the additional six degrees of freedom guaranteed by the tetrad is representing

electromagnetic field [2]. The attempt made by Einstein to unify general relativity

and electromagnetism failed but the concept introduced by him is still significant.

Today we study teleparallel theory as an independent theory of gravitation without

unifying if with the laws of quantum mechanics. Therefore, gravity can be studied

alternately in general relativity or teleparallel theory of gravitation.

Gravitation describes two equivalent descriptions because of the gravitational

property called universality, i.e. in nature every object feels gravity the same (at least

at classical level). This point can be described in both teleparallel and general theory

of relativity. In general relativity theory the role of gravitation is to give the same

3

acceleration to two objects of different masses. Because of this property, gravitation

is described through curvature of the space-time. On the other hand torsion in

teleparallel theory acts as a force on objects. In teleparallel theory of gravitation there

are no geodesics, but there are force equations [3].

To understand our universe we need to know the physical and geometrical features of

our space-times. To understand these features different theories have been developed

but general relativity seems more relativistic theory at least at classical level. General

relativity theory is governed by Einstein’s field equations. These field equations are

highly non linear. Therefore, to solve Einstein field equations certain symmetry

restrictions are required. Symmetries are therefore widely studied in general

relativity. In this chapter a brief introduction to the basic definitions involved in

general relativity and teleparallel theory of gravitation are given. All the stuff for

basic definitions involved in general relativity and teleparallel theory of gravitation

have been taken from [4-7]. The layout of this chapter is as follows: In section (1.2) a

brief description about manifolds is given. Notions of tangent spaces, tensors,

covariant derivative, parallel transport, Lie derivative and Riemann curvature tensor

are given. A brief introduction to symmetries in general relativity is given which

involves definition of a space-time, tetrad formalism and tangent space classification.

In section (1.3) some basic definitions in teleparallel theory, space-time structure and

Lie derivative are given. In section (1.4) literature review of conformal, homothetic,

Killing and self similar vector fields in general theory of relativity is also given. In the

last section (1.5) a brief literature review of teleparallel Killing and homothetic vector

fields are given in teleparallel theory of gravitation.

1.2 Basics of General Relativity 1.2.1 Manifolds

The aim of this section is to give a precise definition and some properties of

manifolds. A manifold is a set which is composed of the pieces that look like open

4

subsets of mℜ and these pieces can be combined mutually smoothly. One can define

manifold as [5]:

An m-dimensional, ,∞C real manifold M is a set together with a collection of

subsets λU satisfying the following properties:

(1) Each Mp∈ lies in at least one ,λU i.e., λU covers M .

(2) For each ,λ there is a bijective map ),(: λλλλ ϕϕ UU → where )( λλϕ U is an

open subset of mℜ .

(3) If any two sets λU and µU overlap, φµλ ≠∩UU (where φ denotes the empty

set), we can consider the map 1−λµ ϕϕ o (where o denotes composition) which takes

points in mUU ℜ⊆∩ ][ µλλϕ to points in .][ mUU ℜ⊆∩ µλµϕ

A schematic diagram showing the defining notions of a manifold is shown below in

the figure 1.2.1.

Figure 1.2.1

We require these subsets of mℜ to be open and this map to be ,∞C i.e., infinitely

continuously differentiable. Each map ),( λλ ϕU is known as a chart or a coordinate

5

system. If the domain of a chart is the whole M it is called global. The collection of

those charts whose union constitutes the whole manifold is known as an atlas of the

manifold. Basically an atlas gives differentiability structure on .M Since a manifold

has given a structure by coordinate system; we can now define smoothness and

differentiability of maps on manifolds. For such purposes suppose 1M and 2M are

manifolds and let ),( 11λλ ϕU and ),( 22

µµ ϕU denote the chart maps. A map

21: MMh → is said to be ∞C if for each λ and µ , the map 112 )( −λµ ϕϕ oo h taking

mV ℜ⊆1λ to mV ℜ⊆2

µ is ∞C in the sense used in advanced calculus. If 21: MMh →

is ,∞C bijective and has ∞C inverse, then h is called a diffeomorphism and ,1M 2M

are said to be diffeomorphic. Diffeomorphic manifolds have identical manifold

structure [5]. A one parameter group of diffeomorphism tψ over a manifold M is a

∞C map from MM →×ℜ such that for a fixed ℜ∈t MMt →:ψ is a

diffeomorphism and for all ,, ℜ∈st we have .stst +=ψψψ o Now we are interested

to define Hausdorff, compact and connected spaces. A manifold is said to be

Hausdorff if it satisfies the Hausdorff separation axiom: whenever sr, are two

distinct points in M , there exist open sets sr WW , in M such that φ=∩ sr WW and

., sr WsWr ∈∈ A topological space is said to be compact if each open cover of the

space has finite sub cover. M (as a topological space) is said to be connected if the

only subsets of M which are both open and closed are the empty set φ and the entire

set M itself [5].

1.2.2 Tangent Spaces

In this section it will be shown that at each point of an m-dimensional smooth

manifold M there is a well defined family of finite dimensional real vector spaces. In

,mℜ there is a one-to-one correspondence between directional derivatives and

vectors. A vector ),...,( 1 mwww = defines the directional derivative operator

6

∑ ∂∂ν

νν )/( xw and vice versa [5]. To characterize directional derivatives, its linearity

and ‘Leibnitz rule’ behavior is checked when acting on functions. On a manifold M

consider the collection Φ of ∞C maps from M into .ℜ A tangent vector at a point

Mp∈ is defined to be a map ℜ→Φ:η which obeys the Leibnitz rule and is linear

[5]. By linearity of tangent vector w we mean that ),()()( khkh ηβηαβαη +=+

for all .,and, ℜ∈Φ∈ βαkh The Leibnitz rule is defined as

).()()()()( hpkkphkh ηηη += A collection of tangent vectors at a point p denoted

by pW has a natural structure of a vector space called the tangent space. This tangent

space obeys the addition and scalar multiplication laws respectively as:

)()())(( 2121 hhh ηηηη +=+ and ).())(( hh ηαηα = An important property of the

tangent space is given in the following theorem:

Theorem:

Let M be an m-dimensional manifold. Let Mp∈ and let pW denote the tangent

space at .p Then mWp = dim [5].

1.2.3 Tensors

To define a tensor precisely we consider a finite dimensional vector space W and its

dual space .*W A tensor ,T of type ),( sl over W is a multilinear map [5]

.......: ** ℜ→××××× 434214434421sl

WWWWT This definition of tensor simply mean that for l

dual vectors and s ordinary vectors, T yields a number. A type )1,0( tensor is exactly

a dual vector and a type )0,1( tensor is an element of **W where **W is nothing but

an ordinary vector because we identify **W as .W The collection ),( slΦ which

represents tensors of type ),,( sl has a natural structure of vector space with the well

known rules of addition and scalar multiplication [5].

7

In general relativity we often apply the two operations of contraction and outer

product to the tensors. These two operations are explained in the following lines.

Contraction of tensors of ),( sl type is a map )1,1(),(: −−Φ→Φ slslO defined as

follows:

If T is a tensor of ),( lk type then ,,...),...,..,(...,1

*∑=

=m

wwTTOσ

σσ where *σw is the

dual basis of W which has basis σw and these vectors are inserted into the ith and

the jth slots of .T The tensor OT obtained here is independent of the choice of basis

µw so contraction operation is well defined [5].

The second operation on tensors is the outer product. Given any tensor 1T of ),( sl

type and another tensor 2T of ),( sl ′′ type, a new tensor can be constructed of type

),( ssll ′+′+ which is called the outer product of 1T and 2T denoted as 21 TT ⊗ by the

following simple rule. Given ll ′+ dual vectors **1 ,..., llww ′+ and ss ′+ vectors

,,...,1 ssww ′+ we define 21 TT ⊗ acting on these vectors to be the product of

),...,;,...,( 1**1

1 sl wwwwT and ).,...,;,...,( 1

**12 sss

lll wwwwT ′++′++ Construction of tensors

on one hand is to take outer products of vectors and dual vectors. A tensor which can

be expressed as such an outer product is called simple [5].

A metric is an important tensor constructed on a manifold. A metric tensor is defined

as a non-degenerate, second order symmetric type )2,0( tensor. By symmetric we

mean that ,αββα gg = where βαg denotes the components of the metric tensor

∑ ⊗=βα

βααβ

,.dxdxgg The metric at each point Mp∈ is a multilinear map from

.ℜ→× pp WW Basically, a metric is an inner product on the tangent space at each

point. Other notation which is used for the metric tensor is ,2ds thus in term of 2ds

we may write ∑=νµ

νµµν

,

2 .dxdxgds Plus and minus signs appearing in the metric

tensor is called the signature of the metric. Different types of metrics can be found

8

like positive definite metrics where the signature is ).,,,( ++++ Riemannian metrics

have positive definite signature. Space-time metric has signature ),,,( +++− and such

metrics are called Lorentzian [5]. From here onward we follow the Einstein’s

summation convention in which summation is understood. For example, µµ xa in a

four dimensional space-time simply means .3

0∑=µ

µµ xa

1.2.4 Covariant Derivative

A derivative operator ∇ on M is called covariant derivative if it map each smooth

tensor field of ),( sl type to a smooth tensor field of )1,( +sl type and satisfy the five

properties given as [5]:

1) Linearity: For all ),(, slKH Φ∈ and ,, ℜ∈µλ

.)( ......

......

......

......

11

11

11

11

lk

lk

lk

lk bb

aaebb

aaebb

aabb

aae KHKH ∇+∇=+∇ µλµλ

2) Leibnitz rule: For all ),,(),,( slKslH ′′Φ∈Φ∈

).()()( ......

......

......

......

......

......

11

11

11

11

11

11

lk

lk

lk

lk

lk

lk dd

ccebb

aadd

ccbb

aaedd

ccbb

aae KHKHKH ′

′′

′′

′ ∇+∇=∇

3) Commutativity with Contraction: For all ),,( slH Φ∈

.)( ............

............

11

11

lk

lk bcb

acadbcb

acad HH ∇=∇

This means that both the contraction and covariant differentiation operations

commute each other.

4) Consistency with the notion of tangent vectors as directional derivatives on

scalar fields: For all Φ∈H where Φ is the collection of ∞C maps and all

,pa W∈η .)( ,; a

aa

aa

aa

a HHHHH ηηηηη ===∇=

5) Torsion Free: For all ,Φ∈H .HH abba ∇∇=∇∇

9

1.2.5 Parallel Transport and Lie derivative

A connection µ∇ can be used for defining notion of parallel transport of a vector or

tensor along a curve Ω with tangent .aτ Mathematically, if a vector νw given at

each point on the curve Ω satisfies ,0=∇ νµ

µτ w then it is said to be transported

parallely along the curve [5].

We now turn our attention towards Lie derivative. Let M be a manifold and tψ be a

one parameter group of diffeomorphism. For every Mp∈ a vector field K over M

determines a unique curve )(tpΩ such that pp =Ω )0( and pK is the tangent vector

to curve. Along a curve )(tpΩ the local coordinates µs are the solutions of the

system of ordinary differential equations )( µνν

sxdt

ds= with initial value

).()0( pxs νν = To introduce a new type of differentiation we consider the map tψ

dragging each p with coordinates νx along the curve )(tpΩ through the point p

into the image )(tr pΩ= with coordinates ).(tsν When the parameter t is

sufficiently small, the map tψ is one-to-one and represents a map called pull-back

map Tt*ψ for any tensor .T The Lie derivative of T with respect to K is defined as

,lim *

0 ⎥⎦⎤

⎢⎣⎡ −

=→ t

TTTL t

tK

ψ where T

t*ψ and T are of the same type tensors and are

evaluated at the same point .p Generally, for a smooth tensor field T of type ),( sl

the components of TLK

becomes [5]

ss

sjli

jj

sjli

sjli

ll

sjli

ii

sjli

sjli

sl

sl

bbbaaa

bbbaaa

bbbaaaa

bbbaa

abbb

aaabbb

aabb

aabb

aa

K

KT

KTKTKT

KTKTKTTL

;............

;............

;............

;............

;............

;............

;......

......

11

11

11

11

11

11

11

11

11

11

...

......

...

µµ

µµ

µµµ

µ

µµ

µµµ

µ

++

+++−−

−−−=

10

The Lie derivative also satisfies the following useful properties [4]. In the following

h is some smooth function, H and K are smooth vector fields, ℜ∈ba, and ,S T

are smooth tensor fields.

(1) ,)( SLTTLSSTL HHH ⊗+⊗=⊗

(2) ,)( SLbTLaSbTaL HHH +=+

(3) ,TLTL HH ρρ = (4) ,TLbTLaTL KHbKaH +=+

(5) ],[ KHKLH = (6) ),(hHhLH =

where ρ denotes the contraction operation. The Lie derivative plays a vital role in

gravitational theories because we describe symmetries through it.

1.2.6 Riemann Curvature Tensor

We shall now define Riemann curvature tensor associated with metric tensor. Let α∇

be a derivative operator and αw be a dual vector field then the following relation

holds ,δγβαδ

γαβγβα wRww =∇∇−∇∇ where γβαδR is called the Riemann

curvature tensor and the relation itself is known as Ricci identity [78]. In terms of

christoffel symbols the Riemann curvature tensor is given as

.,,δαµ

µγβ

δβµ

µγα

δαγβ

δβγαγβα

δ ΓΓ−ΓΓ+Γ−Γ=R Riemann curvature tensor can be

decompose into two parts known as trace and trace free parts. The trace of Riemann

curvature tensor known as Ricci tensor can be obtained by contracting the first and

third indices of the Riemann tensor as .βγαγ

βα RR = The Ricci scalar R is given as

[5] .αδδα

αα RgRR == The Weyl tensor δγβαC known as the trace free part of the

Riemann curvature tensor for manifolds of dimension 3≥m can be obtained as [5]

( ) .)2)(1(

22

2][][][ βδγααδγββδγααβγδαβγδ ggR

mmRgRg

mCR

−−−−

−+= (1.2.1)

11

A space-time metric is said to be conformally flat if all the components of Weyl

tensor become zero and is said to be flat if all the components of Riemann curvature

tensor become zero.

1.2.7 Symmetries in General Relativity

Symmetry of the space-time is a local smooth diffeomorphism which preserves some

geometrical feature of M [4]. A diffeomorphism ψ will be symmetry of a tensor T

if the tensor remains unchanged when pulled back under ψ [4]. In general relativity

the symmetries of metric are important. A metric remains invariant when pulled back

through ψ i.e. .* ababt gg =ψ This type of diffeomorphism is known as isometry [5].

A vector field µK generates a one parameter family of isometries, called a Killing

vector field. Any vector field K on a manifold can be decomposed as [4]

ababba BhK +=21

; (1.2.2)

where abKbaab gLhh == )( and )( baab BB −= are symmetric and skew symmetric

tensors on ,M respectively. The vector field K is said to be conformal if the

associated tψ called the local diffeomorphism, with K maintain the metric structure

through conformal factor η i.e. ,ggt ηψ =∗ where ℜ→U:η is a smooth conformal

function on some open subset U of .M This is equivalent to [4]

,abab gh η= (1.2.3)

or

.,,, abcbac

cacb

ccababK

gKgKgKggL η=++≡ (1.2.4)

If η is a smooth conformal function on some open subset U of M then K is called

conformal vector field. If η becomes constant on ,M then K is called homothetic

vector field (proper homothetic vector field when 0≠η ) while if 0=η it becomes

12

Killing vector field. Also if the vector field K is not a homothetic vector field then it

is called proper conformal vector field. Another kind of symmetry which we will be

discussing in these thesis is self similar vector fields. A vector field K is said to be

self similar if it satisfies the two conditions [8] that

aaKuuL α= (1.2.5)

,2 ababKhhL δ= (1.2.6)

where au is the four velocity of the fluid satisfying 1±=aauu and baabab uugh ±=

is the projection tensor and ., ℜ∈δα If ,0≠δ the ratio ,/δα which is scale

independent, characterizes similarity transformation and known as similarity index. If

the above ratio gives unity, K turns out to be a homothetic vector field, which is

known as first kind self similarity. If 0=α and ,0≠δ similarity is called of the

zeroth kind. If the ratio is neither zero nor one, it is referred to as self-similarity of the

second kind. Self similarity is called of infinite kind when 0≠α and .0=δ If both

,0== αδ K turns out to be a Killing vector fields [9]. A self similar vector field K

can be tilted or non tilted to the four velocity vector .au When au is time like then

1−=aauu and .baabab uugh += In such case the self similar vector field K will be

parallel, orthogonal or tilted (that is neither parallel nor orthogonal) to the time like

vector field au when ,)(u

uFK∂∂

= x

xfK∂∂

= )( or x

xu

uK∂∂

+∂∂

+= )( βα

respectively [9]. The above theory also valid for space-like vector field .au

1.2.8 Space-Times and Tetrad

A space-time is a set ),( gM where M is a four dimensional smooth, connected,

compact and Hausdorff manifold and g is a lorentzian metric with signature

),,,,( +++− which is symmetric and non degenerate. A tetrad can be constructed at

each point of a manifold by taking a system of four linearly independent vectors.

13

Different types of tetrads can be formed over a point Mp∈ in the tangent space. For

instance, one is an orthonormal tetrad ),,,( zyxt and the other is called a real null

tetrad ).,,,( yxml The inner product for orthonormal tetrad members satisfy [4]

1====− aa

aa

aa

aa zzyyxxtt with all other products as zero. Also the inner

product of the members of real null tetrad ),,,( yxml satisfy 0== aa

aa mmll and

1=== aa

aa

aa yyxxml with all other products as zero. The null vectors l and m for

real null tetrad are given as )(2

1 aaa tzl += and ).(2

1 aaa tzm −= Given a tetrad

we can check it to be an orthonormal or real null tetrad if the completeness relation

for metric abg at p holds i.e. a tetrad will be an orthonormal if it satisfies

,zzyyxxttg abababaab +++−= (1.2.7)

and it will be a real null tetrad if it satisfies

.2 )( babaabbabababaab yyxxmlmlyyxxmlg +++=++= (1.2.8)

A complex null tetrad ),,,( ssml may also be introduce at a point ,Mp∈ where, l

and m are as defined in the real null tetrad and s and its conjugate s are defined

from the real null tetrad by )(2

1 aaa iyxs += and ).(2

1 aaa iyxs −= The inner

product among ,l ,m s and s are 1=aaml and 1=a

a ss with all other product as

zero. It is clear that as and as are complex null vectors. In this type of tetrad the

completeness relation at Mp∈ is given as [6]

.22 )()( babababababaab sssslmmlssmlg +++=+= (1.2.9)

14

1.2.9 Classification of tangent spaces

In Minkowski space-time a non zero vector w is called time like, space like or null

according as ),( wuη is negative, positive or zero [4], where η is defined as

.),( baab wuwu ηη = A one dimensional subspace of Minkowski space is called time

like, space like or null if it is spanned by a time like, space like or null vector

respectively. A set of two dimensional subspaces (2-spaces) of the tangent space

MTp is called space like, time like, null if it contains no null directions, exactly two

null directions, exactly one null direction respectively [6]. It is obvious to see that

these are the only possibilities for the 2-spaces because for a real null tetrad

),,,( yxml at p and for two independent vectors u and w at the point, if ),( wu

represents a 2-space then ),( nl is time-like, ),( yx is space-like and

),(),,(),,( xmylxl and ),( ym are null [6]. The set of those vectors which are

orthogonal to each member of the 2-space is called the orthogonal complement of that

set. Therefore, ),( xl and ),( yl are orthogonal complements as are ),( yx and ),( ml

also ),( xm and ),( ym . Similarly a three dimensional subspace (3-space) of

Minkowski space is called time like, space like or null if its orthogonal complement is

space like, time like or a null 1-space.

1.3 Basics of Teleparallel Theory of Gravitation 1.3.1 Tetrad in Teleparallel Theory

In teleparallel theory, a tetrad field represents the gravitational field. A tetrad field S

is basically a map ,: MTMS p→ where M is Minkowski space and MTp is the

tangent space at the point .p A tetrad and its dual are defined respectively as [7]

νν ∂= aa SS .ν

ν dxSS bb = (1.3.1)

15

Tetrad field νaS and its inverse field denoted by ν

aS satisfies the relations

,µν

µν δ=a

a SS ,abb

a SS δνν = (1.3.2)

where abδ is the kronecker delta. It is important to mention that the above equations

hold for non-trivial tetrad field. A trivial tetrad field and its dual can be written as:

,ννδ ∂= aae .ν

νδ dxe bb = (1.3.3)

If we choose a trivial tetrad field then all the torsion components will vanish. To work

in the teleparallel theory of gravity one needs to select a non trivial tetrad field.

1.3.2 Space-time structure

The Riemannian metric can be generated from the tetrad field as [7]

.νµνµ η baba SSg = (1.3.4)

where abη is the Minkowski metric given by ).1,1,1,1(diag −=abη A weitzenböck

connection can be defined through a non trivial tetrad field as [10]

.µνθθ

µνa

a SS ∂=Γ (1.3.5)

Unlike christoffel symbol weitzenböck connection is not symmetric in its lower two

indices and therefore generates torsion in the space-times. The teleparallel covariant

derivative ρ∇ of a covariant tensor of rank 2 in terms of weitzenböck connection is

defined as [7]

,, θνθµρθµ

θρνρνµµνρ AAAA Γ−Γ−=∇ (1.3.6)

where comma denotes the partial derivative and θρνΓ are weitzenböck connections

defined as above. The covariant derivative of the tetrad field is given by

., θθρνρννρ

aa SSS Γ−=∇ (1.3.7)

16

Now using equation (1.3.5) in equation (1.4.7) we get ,0=∇ νρaS which means that

in teleparallel theory tetrads are parallely transported. The weitzenböck and

christoffel symbols have the relation

,0νµ

θµν

θµν

θ

N+Γ=Γ (1.3.8)

where

],[21

νµθ

µθ

ννθ

µνµθ TTTN −+= (1.3.9)

is a tensor quantity called the contortion tensor and νµθ0Γ is the christoffel symbols

defined as

).(21

,,,0

σνµνµσµνσσθ

µν

θ

gggg −+=Γ (1.3.10)

As discussed above weitzenböck connections are not symmetric with respect to its

lower two indices, therefore their difference give us the torsion in the space-times.

Mathematically, torsion can be defined as [11]

,θµν

θνµµν

θ Γ−Γ=T (1.3.11)

which is anti symmetric with respect to its lower indices. The Riemann curvature

tensor in terms of weitzenböck connection in teleparallel theory is given as [12]

.,,λµσ

θνλ

λνσ

θµλ

θνµσ

θµσνσµν

θ ΓΓ−ΓΓ+Γ−Γ=R (1.3.12)

Now using equation (1.3.8) in equation (1.3.12) we have [3]

,00 =+= θµνσ

θµνσ

θµνσ GRR (1.3.13)

where θµνσ0R represents Riemann curvature tensor in general relativity and

,- λνσ

θµλ

λµσ

θνλ

θµσ

νθνσ

µθµνσ NNNNNNG +−∇∇= (1.3.14)

17

is the tensor quantity based on weitzenböck connection only. From (1.3.13) it is clear

that in teleparallel theory curvature of the space-time vanishes identically. In

teleparallel theory torsion is responsible for the gravitational interaction.

1.3.3 Teleparallel Lie Derivative

We have already discussed Lie derivative along a vector field in section (1.2.5). Lie

derivative plays an important role in both general and teleparallel theories because we

study symmetries of the space-times through it. The teleparallel version of the Lie

derivative was introduced in [13]. The teleparallel Lie derivative of a second rank

covariant tensor along a vector field K is given as

).(,,, νρσ

σλνλσ

ρσν

ρν

νλλν

ρνν

νρλρλ TETEKKEKEKEELK

T ++++= (1.3.15)

Similarly, for a second rank contravariant tensor teleparallel Lie derivative is given as

[13]

).(,,, νσρσλ

νσλρσν

νρλν

νλρνν

νρλρλ TETEKKEKEKEEL

K

T +−+−= (1.3.16)

A vector field K is said to be teleparallel homothetic vector field if it satisfies [72-

74]

,2)(,,, ρλνρσ

σλνλσ

ρσν

ρν

νλλν

ρνν

νρλρλ η gTgTgKKgKgKggLK

T =++++= (1.3.17)

where η is constant on .M The teleparallel vector field K is called proper

teleparallel homothetic vector field when 0≠η while if 0=η it becomes teleparallel

Killing vector fields.

18

1.4 Literature Review of Conformal, Homothetic, Killing and Self Similar Vector Fields in General Relativity

In general relativity the above mentioned symmetries can be used to study the laws of

conservation of energy, momentum and angular momentum [14]. These symmetry

restrictions not only give us the laws of conservation but provide some physical and

geometrical information about the space-time. Some geometrical and physical

features through symmetries are the study of cosmological voids, cosmological

perturbations, gravitational collapse, primordial black holes, star formation and

cosmic censorship [15]. In literature Killing, homothetic, conformal and self similar

symmetries of the space-times have been studied by different researchers. Bokhari et

al [16] classified static spherically symmetric space-times according to its Killing

vector fields. They explored that static spherically symmetric space-times admit

minimum 4 linearly independent Killing vector fields. Qadir and Ziad [17, 18]

worked to investigate the Killing isometry for static cylindrically symmetric space-

times and non static spherically symmetric space-times. They completely classified

these space-times according to their metrics and isometries. They showed that the

number of Killing vector fields for static cylindrically symmetric space-times can be

,3 ,4 ,5 ,6 7 or .10 In [19] the authors completely classified non static plane

symmetric space-times according to their Killing vector fields and metrics. Authors

obtained classes of metrics that admit Killing vector fields of dimension ,3 ,4 ,5 ,6

7 and .10 In [20] it is found that the dimension of the homothetic vector fields for

space-times admitting m Killing vector fields is .1+m In [21] authors explored that

the maximum possible dimension of homothetic vector fields is .11 In [22] authors

investigated spherically symmetric space-times according to its homothetic vector

fields. They have given complete classification and it is explored that this space-time

admit homothetic vector fields of dimension ,4 ,5 ,6 8 and .11 They showed that

19

the space-time becomes flat when admit 11 homothetic vector fields. In [23, 24]

authors completely classified static plane symmetric space-times and static

cylindrically symmetric space-times according to homothetic motions. In [23] they

showed that space-time admits ,5 ,6 8 and 11 homothetic vector fields. In [24]

authors completely classified static cylindrically symmetric space-times according to

its homothetic vector fields and metrics. They showed that there exist ,4 ,5 7 and 11

homotheties. They argued that some of the metrics admitting homothetic vector fields

were left in the classification of Hall and Steele [21] because some classes of metrics

admit homothetic vector fields which do not have global topological structure. They

also extended some local homotheties globally. On the same topic authors in [25-27]

have discovered homothetic vector fields and their metrics in Bianchi type I, static

cylindrically symmetric and non static plane symmetric space-times.

In [28, 29] the authors introduced the method of finding a conformal change in the

metric on a manifold. The conformal change is taken such that the Lie algebra of

conformal vector fields on the manifold with respect to one metric becomes Killing or

homothetic algebra of the other metric. According to the authors of these two papers,

the Lie algebra of conformal vector fields can be reduced to homothetic algebra under

some assumptions if the space-time is not conformally flat. The theorems stated in the

above two papers were not precise. Afterward in 1990, Hall [30] gave some counter

examples for the theorems given in [28, 29] by considering non conformally flat

space-times admitting proper homothetic vector fields with isolated zero. In [31] the

authors discussed conformal vector fields in a space-time and found out the maximum

dimension of conformal vector fields for non conformally flat space-times. In this

paper the authors discussed deficiencies in the theorems of [28, 29] and also corrected

them. They found that the maximum dimension of the conformal vector fields for non

conformally flat space-times is seven and for conformally flat space-times it is 15.

Kramer et al [32] considered rigidly rotating, stationary axisymmetric, perfect fluid

space-times admitting a proper conformal vector field under certain assumption of

Lie algebra and showed that no such solutions of Einstein field equations exist. By

20

considering static spherically symmetric space-times [33], authors obtained

conformal vector fields in non conformally flat and conformally flat cases. The

showed that non conformally flat static spherically symmetric space-times admit at

the most two proper conformal motions. They also listed all eleven proper

conformation vector fields for conformally flat static spherically symmetric space-

times. Hall [34] established some restrictions on existence of conformal and

homothetic vector fields in spaces admitting Killing vector fields. G. Shabbir et al

[35-39] explored conformal vector fields in Bianchi type I, static cylindrically

symmetric, non static spherically symmetric, Bianchi types VIII and IX and spatially

homogeneous rotating space-times by using direct integration method. Using the

same method we investigated proper conformal vector fields in non conformally flat

Kantowski-Sachs [40], Bianchi type III [41] and non static cylindrically symmetric

space-times [42]. We showed that all the above space-times possess proper conformal

vector fields for special choice of the metric functions.

Much work has been done to find self-similar solutions of the space-times. Coley [43]

in his paper discussed the self similarity differences between generalized similarity

and first kind self-similarity. He argued about the suitability of generalization of

homothety as self-similarity. He also discussed various mathematical and physical

properties of space-times admitting self-similarity. In his work Coley introduced the

governing equations for perfect fluid cosmological models and a set of integrability

conditions have derived for the existence of proper self-similar vector fields. In [8]

authors discussed some important symmetries of Bianchi type I space-times. They

determined the self similarities of Bianchi I metrics without any constraint on the type

of the fluid. They showed that Bianchi type I space-times admit self similarity of first

and zeroth kind. They explored that Bianchi type I space-times which admit self

similar vector field of second kind with co-moving fluid are only Kasner type space-

times. H. Maeda et al [44] classify all spherically symmetric space-times admitting

self-similar vector fields of the second, zeroth or infinite kind. They studied the cases

in which the self-similar vector field is not only tilted but also parallel or orthogonal

21

to the fluid flow. M. Sharif and S. Aziz [45, 46] published their work on self

similarity solutions for spherically symmetric and cylindrically symmetric space-

times, respectively. In [45] they explored some properties of the self similar solutions

of the first kind for spherically symmetric space-times and also checked the

singularities of these solutions. In [46] they studied the cylindrically symmetric

solutions which admit self-similar vector fields of second, zeroth and infinite kinds,

for the tilted fluid case, parallel and orthogonal cases. They showed that the parallel

case gives contradiction both in perfect fluid and dust cases and the orthogonal

perfect fluid case yields a vacuum solution while the orthogonal dust case gives

contradiction. Using an algebraic and direct integration method we investigated

proper self similar vector fields of first, second zeroth and infinite kind for Bianchi

type III [47], static plane symmetric [48], static spherically symmetric[49] and static

cylindrically symmetric space-times [80]. We showed that all the above space-times

possess proper self similar vector fields for special choice of the metric functions.

1.5 Literature Review of Teleparallel Killing and

Homothetic Vector Fields in Teleparallel

Theory of Gravitation

Although general relativity is a powerful theory of gravity, it has some controversial

and unresolved problems. One of the problems is the energy and momentum

localization. This theory has no consistent formalism to localize energy and

momentum. For this reason the problem of localization of energy was also considered

in teleparallel theory of gravitation [50-55]. As a second step in this theory

teleparallel versions of the exact solutions of general relativity have been studied by

many authors [56-63]. In teleparallel theory symmetries of the metric tensor were

ignored. As a pioneer work Sharif and Jamil [13] introduced the teleparallel version

of the Lie derivative for Killing vector fields and used those equations to find the

22

teleparallel Killing vector fields in Einstein universe. After that Sharif and Bushra

[64] explored teleparallel Killing vector fields for spherically symmetric static space-

times. The above authors did not classify the space-times according to their

teleparallel Killing vector fields. We started our work in this field by classifying

space-times according to their teleparallel Killing vector fields by using direct

integration method. We classify Bianchi types I, II, III, VIII, IX, Kantowski-Sachs,

static cylindrically symmetric, spatially homogeneous rotating and non static

cylindrically symmetric space-times according to their teleparallel Killing vector

fields [65-71]. Later we extended our work to the classification of Bianchi type I,

static cylindrically symmetric and non static plane symmetric space-times by

teleparallel proper homothetic vector fields [72-74] which are pioneer papers in this

field. In the above papers we have also established a brief comparison between

teleparallel Killing vector fields and Killing vector fields in general relativity.

23

Chapter 2

Teleparallel Killing Vector Fields in

Bianchi Types I, II, VIII and IX Space-

Times

2.1. Introduction

This chapter is devoted to investigate teleparallel Killing vector fields in Bianchi

types I, II, VIII and IX space-times by using direct integration technique. This chapter

is organized as follows: In section (2.2) teleparallel Killing vector fields of Bianchi

type I space-times are investigated. In section (2.3) teleparallel Killing vector fields in

Bianchi type II space-times in the context of teleparallel theory have been explored.

In the next section (2.4) teleparallel Killing vector fields of Bianchi type VIII and IX

space-times are explored in the context of teleparallel theory of gravitation. A brief

comparison of Killing vector fields in both the theories is also given. Last section

(2.5) of the chapter is dedicated to a detailed summary of the work.

24

2.2. Teleparallel Killing Vector Fields in Bianchi

Type I Space-Times

Bianchi type-I space-time is a spatially homogeneous space-time. In general relativity

it admits an abelian Lie algebra of isometries ,3G acting on space like hyper surfaces

generated by the space like Killing vector fields [78] .,,zyx ∂∂

∂∂

∂∂ The line element in

the usual coordinates ),,,( zyxt (labeled by ),,,,( 3210 xxxx respectively) is given by

[75, 78]

.2)(22)(22)(222 dzedyedxedtds tCtBtA +++−= (2.2.1)

where BA, and C are functions of t only. The tetrad components and its inverse

can be obtained by using the relation (1.3.4) as [65]

),,,,1diag( )()()( tCtBtAa eeeS =µ ).,,,1diag( )()()( tCtBtAa eeeS −−−=µ (2.2.2)

Using equation (1.3.5), the corresponding non-vanishing Weitzenböck connections

are obtained as

,101 •=Γ A ,20

2 •=Γ B ,303 •=Γ C (2.2.3)

where dot denotes the derivative with respect to .t The non vanishing torsion

components by using (1.3.11) are

.,, 303

033

202

022

101

011 ••• =−==−==−= CTTBTTATT (2.2.4)

Now using (2.2.1) and (2.2.4) in (1.3.17) we get the teleparallel Killing equations as:

03,3

2,2

1,1

0,0 ==== XXXX (2.2.5)

,01,2)(2

2,1)(2 =+ XeXe tBtA (2.2.6)

,01,3)(2

3,1)(2 =+ XeXe tCtA (2.2.7)

25

,02,3)(2

3,2)(2 =+ XeXe tCtB (2.2.8)

,01)(20,

1)(21,

0 =−− • XAeXeX tAtA (2.2.9)

,02)(20,

2)(22,

0 =−− • XBeXeX tBtB (2.2.10)

.03)(20,

3)(23,

0 =−− • XCeXeX tCtC (2.2.11)

Now integrating equation (2.2.5), we get

).,,(),,,(),,,(),,,(

4332

2110

yxtPXzxtPXzytPXzyxPX

==

== (2.2.12)

where ),,,(1 zyxP ),,,(2 zytP ),,(3 zxtP and ),,(4 yxtP are functions of integration

which are to be determined. In order to find solution for equations (2.2.5) to (2.2.11)

we will consider each possible form of the metric for Bianchi type I space-times and

then solve each possibility in turn. Following are the possible cases for the metric

where the above space-times admit teleparallel Killing vector fields:

(I) )(),(),( tCCtBBtAA === and .,, CBCABA ≠≠≠

(II)(a) ),(),( tBBtAA == and .tan tconsC =

(II)(b) ),(),( tCCtAA == and .tan tconsB =

(II)(c) ),(),( tCCtBB == and .tan tconsA =

(III)(a) )(),(),( tCCtBBtAA === and ).()( tCtB =

(III)(b) )(),(),( tCCtBBtAA === and ).()( tCtA =

(III)(c) )(),(),( tCCtBBtAA === and ).()( tBtA =

(IV) )(),(),( tCCtBBtAA === and ).()()( tCtBtA ==

(V)(a) )(),(,tan tCCtBBtconsA === and ).()( tCtB =

(V)(b) )(,tan),( tCCtconsBtAA === and ).()( tCtA =

(V)(c) tconsCtBBtAA tan),(),( === and ).()( tBtA =

26

(VI)(a) )(tAA = and .tan tconsCB ==

(VI)(b) )(tBB = and .tan tconsCA ==

(VI)(c) )(tCC = and .tan tconsBA ==

We will discuss each possibility in turn.

Case (I):

In this case we have ),(tAA = ),(tBB = ),(tCC = ,BA ≠ CA ≠ and .CB ≠ Now

substituting equation (2.2.12) in equation (2.2.6), we get

.0),,(),,( 2)(23)(2 =+ zytPezxtPe ytA

xtB (2.2.13)

Differentiating equation (2.2.13) with respect to ,x we get

),,(),(),,(0),,( 2133 ztEztxEzxtPzxtPxx +=⇒= where ),(1 ztE and ),(2 ztE are

functions of integration. Substituting back this value in equation (2.2.13) we get

⇒−= − ),(),,( 1)(2)(22 ztEezytP tAtBy ),,(),(),,( 31)(2)(22 ztEztEeyzytP tAtB +−= − where

),(3 ztE is a function of integration. Now refreshing the system of equations (2.2.12)

we get

).,,(),,(),(),,(),(),,,(

43212

31)(2)(2110

yxtPXztEztxEXztEztEeyXzyxPX tAtB

=+=

+−== −

(2.2.14)

Considering equation (2.2.7) and using equation (2.2.14) we get

.0)],(),([),,( 31)(2)(2)(24)(2 =+−+ − ztEztEeyeyxtPe zztAtBtA

xtC (2.2.15)


),,(),(),,(0),,( 5444 ytEytxEyxtPyxtPxx +=⇒= where ),(4 ytE and ),(5 ytE are

functions of integration. Substituting back the above value in (2.2.15) we get

.0),(),(),( 3)(21)(24)(2 =+− ztEeztEeyytEe ztA

ztBtC Differentiating this equation with

respect to y twice, we get ).()(),(0),( 2144 tKtyKytEytEyy +=⇒= Substituting

back the above value in equation (2.2.15) and solving, we get

27

)()(),( 31)(2)(21 tKtKezztE tBtC += − and ),()(),( 42)(2)(23 tKtKezztE tAtC +−= − where

),(1 tK ),(2 tK )(3 tK and )(4 tK are functions of integration. Substituting back the

above information in equation (2.2.14) we get

).,()()(),,()()(

),()()()(),,,(

5213

231)(2)(22

42)(2)(23)(2)(21)(2)(21

10

ytEtxKtKyxXztEtxKtKezxX

tKtKeztKeytKezyXzyxPX

tBtC

tAtCtAtBtAtC

++=

++=

+−−−=

=

−

−−−

(2.2.16)


.0),(),()(2 2)(25)(21)(2 =++ ztEeytEetKxe ztB

ytCtC (2.2.17)

Differentiating the above equation with respect to x we get .0)(1 =tK Substituting

the above value in (2.2.17) and differentiating the resulting equation with respect to

,y we get ),()(),(0),( 6555 tKtKyytEytEyy +=⇒= where )(5 tK and )(6 tK are

functions of integration. Substituting the above values back in (2.2.17) we get

),()(),( 75)(2)(22 tKtKezztE tBtC +−= − where )(7 tK is a function of integration.

Refreshing the system of equations (2.2.16) we get

).()()(),()()(),()()(

),,,(

652375)(2)(232

42)(2)(23)(2)(21

10

tKtKytxKXtKtKeztxKXtKtKeztKeyX

zyxPX

tBtC

tAtCtAtB

++=+−=

+−−=

=

−

−− (2.2.18)


.0)()(),,()()(

)()]()(2[)()()]()(2[4)(214)(22)(2

2)(23)(23)(2

=−+−+

−++−

tKetAzyxPtKetKez

tKetAtCztKeytKetAtBytA

txttA

ttC

tCttt

tBtBtt (2.2.19)

Differentiating (2.2.19) with respect to ,x we get

),,(),(),,(0),,( 7611 zyEzyxEzyxPzyxPxx +=⇒= where ),(6 zyE and ),(7 zyE

are functions of integration. Substituting the above value back in (2.2.19) and

differentiating twice with respect to ,y we get

28

),()(),(0),( 9866 zKzKyzyEzyEyy +=⇒= where )(8 zK and )(9 zK are functions

of integration. Substituting back the above value in (2.2.19) and differentiating with

respect to ,y we get .0)()()()]()(2[ 83)(23)(2 =++− zKtKetKetAtB ttBtB

tt

Differentiating this equation with respect to ,z we get .,)(0)( 1188 ℜ∈=⇒= cczKzK z

Substituting back this value in the above equation and solving we get

.,)( 2)(2)(

2)()(2)(

13 ℜ∈+−= −−− ∫ cecdteectK tBtAtAtBtA Now substituting all the above

information in (2.2.19) and solve after differentiating twice with respect to ,z we get

.,,)(0)( 434399 ℜ∈+=⇒= ccczczKzK zz Substituting back in equation (2.2.19) and

solve after differentiating with respect to ,z we get

.,)( 5)(2)(

5)()(2)(

32 ℜ∈+−= −−− ∫ cecdteectK tCtAtAtCtA Now substituting all the above

information in equation (2.2.19) and solving we get

.,)( 6)(

6)()(

44 ℜ∈+= −−− ∫ cecdteectK tAtAtA Refreshing the system of equation

(2.2.18) with the help of above information, we get

),()(

),()(

,

),,(

65)(2)(5

)()(2)(3

3

75)(2)(2)(2)(2

)()(2)(1

2

)(6

)()(4

)(5

)()(3

)(2

)()(1

1

7431

0

tKtKyexcdteexcX

tKtKzeexcdteexcX

ecdteecezcdteezceycdteeycX

zyExcxzcxycX

tCtAtAtCtA

tBtCtBtAtAtBtA

tAtAtAtAtAtAtAtAtA

+++−=

+−+−=

++−+−=

+++=

−−−

−−−−

−−−−−−−−−

∫∫

∫∫∫(2.2.20)

where .,,,,, 654321 ℜ∈cccccc Now considering equation (2.2.10) and using equation

(2.2.20) we get

.0)()()()()()]()(2[

)]()([)]()([),(27)(27)(25)(25)(2

)(2

)()(1

71

=−−+−+

−−−++ ∫ −

tKetBtKetKeztKetBtCz

etBtAcxdteetBtAxczyExctB

tttB

ttCtC

tt

tAtt

tAtAtty (2.2.21)


.0)]()([)]()([2 )(2

)()(11 =−−−+ ∫ − tA

tttAtA

tt etBtAcdteetBtAcc Solving this equation

and remember that in this case ,0)()( ≠− tBtA tt we get ,01 =c which on back

29

substitution give us .02 =c Substituting the above information in equation (2.2.21)

and differentiating with respect to ,y we get

),()(),(0),( 111077 zKzyKzyEzyEyy +=⇒= where )(10 zK and )(11 zK are

functions of integration. Substituting the above value in (2.2.21) and differentiating

with respect to z twice, we get .,,)(0)( 87871010 ℜ∈+=⇒= ccczczKzK zz

Substituting back this value in equation (2.2.21) and differentiating with respect to ,z

we get .0)()()]()(2[ 5)(25)(27 =+−+ tKetKetBtCc t

tCtCtt Solving this equation we get

.,)( 9)(2)(

9)()(2)(

75 ℜ∈+−= −−− ∫ cecdteectK tCtBtBtCtB Now substituting all the above

information in equation (2.2.21) we get ⇒=−− 0)()()( 7)(27)(28 tKetKetBc t

tBtBt

.,)( 10)(

10)()(

87 ℜ∈+= −−− ∫ cecdteectK tBtBtB Refreshing the system of equations

(2.2.20) we get

),(

,

,

),(

6)(2)(9

)()(2)(7

)(2)(5

)()(2)(3

3

)(10

)()(8

)(9

)()(7

2

)(6

)()(4

)(5

)()(3

1

118743

0

tKeyc

dteeycexcdteexcX

ecdteecezcdteezcX

ecdteecezcdteezcX

zKycyzcxcxzcX

tCtB

tBtCtBtCtAtAtCtA

tBtBtBtBtBtB

tAtAtAtAtAtA

++

−+−=

++−=

++−=

++++=

−

−−−−−

−−−−−−

−−−−−−

∫∫∫∫∫∫

(2.2.22)

where .,,,,,,, 109876543 ℜ∈cccccccc Now considering equation (2.2.11) and using

equation (2.2.22) we get

.0)()()(

))()(())()(())(

)(())()(()(22

6)(2)(26

)(9

)()(7

)(

5)()(

311

73

=−−

−−−+−

−−+++

∫∫

−

−

tKetCetK

etCtBycdteetCtBycetC

tAcxdteetCtAxczKycxc

tCt

tCt

tBtt

tBtBtt

tAt

ttAtA

ttz

(2.2.23)


.0)]()([)]()([2 )(5

)()(33 =−−−+ ∫ − tA

tttAtA

tt etCtAcdteetCtAcc Solving this equation

and remember that in this case ,0)()( ≠− tCtA tt we get ,03 =c which on back

30

substitution gives us .05 =c Substituting the above information in equation (2.2.23)


.0)]()([)]()([2 )(9

)()(77 =−−−+ ∫ − tB

tttBtB

tt etCtBcdteetCtBcc Solving this equation

and remember that in this case ,0)()( ≠− tCtB tt we get ,07 =c which on back

substitution gives us .09 =c Now substituting all the above values in equation

(2.2.23) and differentiating with respect to ,z we get

.,,)(0)( 121112111111 ℜ∈+=⇒= ccczczKzK zz Substituting the above value in

equation (2.2.23) we get ⇒=−− 0)()()( 6)(26)(211 tKetKetCc t

tCtCt

.,)( 13)(

13)()(

116 ℜ∈+= −−− ∫ cecdteectK tCtCtC Refreshing the system of equations

(2.2.22) we get

,,

,,)(

13)()(

113)(

10)()(

82

)(6

)()(4

1121184

0

tCtCtCtBtBtB

tAtAtA

ecdteecXecdteecX

ecdteecXczcycxcX−−−−−−

−−−

+=+=

+=+++=

∫∫∫

(2.2.24)

where .,,,,,, 13121110864 ℜ∈ccccccc The line element for Bianchi type I space-times is

given in equation (2.2.1). The above space-time admits seven linearly independent

teleparallel Killing vector fields which are ,t∂∂ ,)(

xe tA

∂∂− ,)(

ye tB

∂∂− ,)(

ze tC

∂∂−

,)(1

xtG

tx

∂∂

+∂∂

ytG

ty

∂∂

+∂∂ )(2 and ,)(3

ztG

tz

∂∂

+∂∂ where ,)( )()(1 ∫ −−= dteetG tAtA

∫ −−= dteetG tBtB )()(2 )( and .)( )()(3 ∫ −−= dteetG tCtC Killing vector fields in general

relativity are ,x∂∂

y∂∂ and .

z∂∂ It is evident that teleparallel Killing vector fields are

different and more in number to the Killing vector fields in general relativity.

Case (II)(a):

In this case we have tCtBBtAA tancos),(),( === and ).()( tBtA ≠ Now


31

.0),,(),,( 2)(23)(2 =+ zytPezxtPe ytA

xtB (2.2.25)






we get

).,,(),,(),(),,(),(),,,(

43212

31)(2)(2110


=+=

+−== −

(2.2.26)


.0)],(),([),,( 31)(2)(2)(24 =+−+ − ztEztEeyeyxtP zztAtBtA

x (2.2.27)




.0),(),(),( 3)(21)(24 =+− ztEeztEeyytE ztA

ztB Differentiating this equation with

respect to y twice we get ).()(),(0),( 2144 tKtyKytEytEyy +=⇒= Substituting


)()(),( 31)(21 tKtKezztE tB += − and ),()(),( 42)(23 tKtKezztE tA +−= − where ),(1 tK

),(2 tK )(3 tK and )(4 tK are functions of integration. Substituting back the above

information in equation (2.2.26) we get

).,()()(),,()()(

),()()()(),,,(

5213

231)(22

42)(23)(2)(21)(21

10

ytEtxKtKyxXztEtxKtKezxX


tB

tAtAtBtA

++=

++=

+−−−=

=

−

−−−

(2.2.28)


32

.0),(),()(2 2)(251 =++ ztEeytEtxK ztB

y (2.2.29)





),()(),( 75)(22 tKtKezztE tB +−= − where )(7 tK is a function of integration.


).()()(),()()(),()()(

),,,(

652375)(232

42)(23)(2)(21

10

tKtKytxKXtKtKeztxKXtKtKeztKeyX

zyxPX

tB

tAtAtB

++=+−=

+−−=

=

−

−− (2.2.30)


.0)()(),,()()(

)()()()()]()(2[4)(214)(22

23)(23)(2

=−+−+

−+−

tKetAzyxPtKetKz

tKtAztKeytKetAtBytA

txttA

t

tttBtB

tt (2.2.31)








tt



.,)( 2)(2)(

2)()(2)(




33


.,)( 5)(

5)()(

32 ℜ∈+−= ∫ − cecdteectK tAtAtA Now substituting all the above


.,)( 6)(

6)()(



),()(

),()(

,

),,(

65)(5

)()(3

3

75)(2)(2)(2

)()(2)(1

2

)(6

)()(4

)(5

)()(3

)(2

)()(1

1

7431

0

tKtKyexcdteexcX

tKtKzeexcdteexcX

ecdteecezcdteezceycdteeycX

zyExcxzcxycX

tAtAtA

tBtBtAtAtBtA

tAtAtAtAtAtAtAtAtA

+++−=

+−+−=

++−+−=

+++=

∫∫

∫∫∫

−

−−−−

−−−−−−−−−

(2.2.32)


(2.2.32) we get

.0)()()()()()(

)]()([)]()([),(27)(27)(255

)(2

)()(1

71

=−−+−

−−−++ ∫ −

tKetBtKetKztKtBz


tttB

tt

tAtt

tAtAtty (2.2.33)


.0)]()([)]()([2 )(2

)()(11 =−−−+ ∫ − tA

tttAtA









we get .0)()()( 557 =+− tKtKtBc tt Solving this equation we get

.,)( 9)(

9)()(

75 ℜ∈+−= ∫ − cecdteectK tBtBtB Now substituting all the above

34


tBtBt

.,)( 10)(

10)()(


(2.2.32) we get

),(

,

,

),(

6)(9

)()(7

)(5

)()(3

3

)(10

)()(8

)(9

)()(7

2

)(6

)()(4

)(5

)()(3

1

118743

0

tKeycdteeycexcdteexcX

ecdteecezcdteezcX

ecdteecezcdteezcX

zKycyzcxcxzcX

tBtBtBtAtAtA

tBtBtBtBtBtB

tAtAtAtAtAtA

++−+−=

++−=

++−=

++++=

∫∫∫∫∫∫

−−

−−−−−−

−−−−−−

(2.2.34)



.0)()()(

)()()(226)(

9)()(

7

)(5

)()(3

1173

=−−+

−+++

∫∫

−

−

tKetBycdteetByc

etAcxdteetAxczKycxc

ttB

ttBtB

t

tAt

tAtAtz

(2.2.35)


.0)()(2 )(5

)()(33 =−+ ∫ − tA

ttAtA

t etAcdteetAcc Solving this equation and remember that

in this case ,0)( ≠tAt we get ,03 =c which on back substitution gives us .05 =c

Substituting the above information in equation (2.2.35) and differentiating with

respect to ,y we get .0)()(2 )(9

)()(77 =−+ ∫ − tB

ttBtB

t etBcdteetBcc Solving this

equation and remember that in this case ,0)( ≠tBt we get ,07 =c which on back

substitution gives us .09 =c Now substituting all the above values in equation



equation (2.2.35) we get ⇒=− 0)(611 tKc t .,)( 131311

6 ℜ∈+= cctctK Refreshing the

system of equations (2.2.34) we get

35

,,

,,)(

13113)(

10)()(

82

)(6

)()(4

1121184

0

tCtBtBtB

tAtAtA

ectcXecdteecX

ecdteecXczcycxcX−−−−

−−−

+=+=

+=+++=

∫∫

(2.2.36)


given as

.22)(22)(222 dzdyedxedtds tBtA +++−= (2.2.37)

The above space-time admits seven linearly independent teleparallel Killing vector

fields which are ,t∂∂ ,)(

xe tA

∂∂− ,)(

ye tB

∂∂− ,

z∂∂ ,)(1

xtG

tx

∂∂

+∂∂

ytG

ty

∂∂

+∂∂ )(2

and ,z

tt

z∂∂

+∂∂ where ∫ −−= dteetG tAtA )()(1 )(

and .)( )()(2 ∫ −−= dteetG tBtB Killing

vector fields in general relativity are ,x∂∂

y∂∂ and .

z∂∂ It is evident that only one

teleparallel Killing vector field is the same and other are different from Killing vector

fields in general relativity. Also teleparallel Killing vector fields are more in number

to the Killing vector fields in general relativity. Cases (II)(b) and (II)(c) can be solved

exactly the same as in the above case.

Case (III)(a):

In this case we have )(),(),( tCCtBBtAA === and ).()( tCtB = Now substituting

equation (2.2.12) in equation (2.2.6), we get

.0),,(),,( 2)(23)(2 =+ zytPezxtPe ytA

xtB (2.2.38)





36


we get

).,,(),,(),(),,(),(),,,(

43212

31)(2)(2110


=+=

+−== −

(2.2.39)



xtB (2.2.40)





ztBtB Differentiating this equation with



)()(),( 311 tKtKzztE += and ),()(),( 42)(2)(23 tKtKezztE tAtB +−= − where ),(1 tK



).,()()(),,()()(

),()()()(),,,(

5213

2312

42)(2)(23)(2)(21)(2)(21

10

ytEtxKtKyxXztEtxKtKzxX


tAtBtAtBtAtB

++=

++=

+−−−=

=−−−

(2.2.41)


.0),(),()(2 251 =++ ztEytEtxK zy (2.2.42)





37

),()(),( 752 tKtKzztE +−= where )(7 tK is a function of integration. Refreshing

the system of equations (2.2.41) we get

).()()(),()()(),()()(

),,,(

65237532

42)(2)(23)(2)(21

10

tKtKytxKXtKtKztxKXtKtKeztKeyX

zyxPXtAtBtAtB

++=+−=

+−−=

=−− (2.2.43)


.0)()(),,()()(

)()]()(2[)()()]()(2[4)(214)(22)(2

2)(23)(23)(2

=−+−+

−++−

tKetAzyxPtKetKez

tKetAtBztKeytKetAtBytA

txttA

ttB

tBttt

tBtBtt (2.2.44)








tt



.,)( 2)(2)(

2)()(2)(





.,)( 5)(2)(

5)()(2)(



.,)( 6)(

6)()(



38

),()(

),()(

,

),,(

65)(2)(5

)()(2)(3

3

75)(2)(2

)()(2)(1

2

)(6

)()(4

)(5

)()(3

)(2

)()(1

1

7431

0

tKtKyexcdteexcX

tKtzKexcdteexcX

ecdteec

ezcdteezceycdteeycX

zyExcxzcxycX

tBtAtAtBtA

tBtAtAtBtA

tAtAtA

tAtAtAtAtAtA

+++−=

+−+−=

++

−+−=

+++=

−−−

−−−

−−−

−−−−−−

∫∫

∫∫∫

(2.2.45)


(2.2.45) we get

.0)()()()()()(

)]()([)]()([),(27)(25)(27)(25)(2

)(2

)()(1

71

=−+−+

−−−++ ∫ −

tKetBtKetzBtKetKez


ttB

tttB

ttB

tAtt

tAtAtty (2.2.46)


.0)]()([)]()([2 )(2

)()(11 =−−−+ ∫ − tA

tttAtA









we get .0)()()( 5)(25)(27 =+− tKetKtBec t

tBt

tB Solving this equation we get

.,)( 9)(

9)()(

75 ℜ∈+−= −−− ∫ cecdteectK tBtBtB Now substituting all the above


tBtBt

.,)( 10)(

10)()(


(2.2.45) we get

39

),(

,

,

),(

6)(9

)()(7

)(2)(5

)()(2)(3

3

)(10

)()(8

)(9

)()(7

2

)(6

)()(4

)(5

)()(3

1

118743

0

tKeyc

dteeycexcdteexcX

ecdteecezcdteezcX

ecdteecezcdteezcX

zKycyzcxcxzcX

tB

tBtBtBtAtAtBtA

tBtBtBtBtBtB

tAtAtAtAtAtA

++

−+−=

++−=

++−=

++++=

−

−−−−−

−−−−−−

−−−−−−

∫∫∫∫∫∫

(2.2.47)



.0)()()()]()([

)]()([)(226)(26)(2)(

5

)()(3

1173

=−−−−

−+++ ∫ −

tKtBetKeetBtAcx

dteetBtAxczKycxc

ttB

ttBtA

tt

tAtAttz (2.2.48)


.0)()([)()([2 )(5

)()(33 =−−−+ ∫ − tA

tttAtA




and differentiating with respect to ,y we get .07 =c Now substituting all the above

values in equation (2.2.48) and differentiating with respect to ,z we get


equation (2.2.48) we get ⇒=−− 0)()()( 6)(26)(211 tKtBetKec t

tBt

tB

.,)( 13)(

13)()(


(2.2.47) we get

,

,

,,

)(13

)(9

)()(11

3

)(10

)(9

)()(8

2

)(6

)()(4

1121184

0

tBtBtBtB

tBtBtBtB

tAtAtA

eceycdteecX

ecezcdteecX

ecdteecXczcycxcX

−−−−

−−−−

−−−

++=

+−=

+=+++=

∫∫

∫ (2.2.49)


given as

40

).( 22)(22)(222 dzdyedxedtds tBtA +++−= (2.2.50)

The above space-time admits eight linearly independent teleparallel Killing vector


xe tA

∂∂− ,)(

ye tB

∂∂− ,)(

ze tB

∂∂− ,)(1

xtG

tx

∂∂

+∂∂

,)(2

ytG

ty

∂∂

+∂∂

ztG

tz

∂∂

+∂∂ )(2 and ),()(

yz

zye tB

∂∂

−∂∂− where

∫ −−= dteetG tAtA )()(1 )(

and .)( )()(2 ∫ −−= dteetG tBtB Killing vector fields in general

relativity are ,x∂∂ ,

y∂∂

z∂∂ and .

yz

zy

∂∂

−∂∂ Teleparallel Killing vector fields are

different and more in number to the Killing vector fields in general relativity. Cases

(III)(b) and (III)(c) can be solved exactly the same as in the above case.

Case (IV):

In this case we have )(),(),( tCCtBBtAA === and ).()()( tCtBtA == Now


.0),,(),,( 23 =+ zytPzxtP yx (2.2.51)

Differentiating equation (2.2.51) with respect to ,y we get

),(),(),,(0),,( 2122 ztEztyEzytPzytPyy +=⇒= ),(1 ztE and ),(2 ztE are functions

of integration. Substituting back this value in equation (2.2.51) we get

),,(),(),,(),(),,( 31313 ztEztxEzxtPztEzxtPx +−=⇒−= where ),(3 ztE is a

function of integration. Now refreshing the system of equations (2.2.12) we get

).,,(),,(),(),,(),(),,,(

43312

21110

yxtPXztEztxEXztEztEyXzyxPX

=+−=

+== (2.2.52)


.0)],(),(),,( 214 =++ ztEztEyyxtP zzx (2.2.53)

41




.0),(),(),( 214 =++ ztEztEyytE zz Differentiating this equation with respect to y

twice we get ).()(),(0),( 2144 tKtyKytEytEyy +=⇒= Substituting back the above

value in equation (2.2.53) and solving, we get )()(),( 311 tKtKzztE +−= and

),()(),( 422 tKtKzztE +−= where ),(1 tK ),(2 tK )(3 tK and )(4 tK are functions of

integration. Substituting back the above information in equation (2.2.53) we get

).,()()(),,()()(),()()()(),,,(

52133312

4231110

ytEtxKtKyxXztEtxKtKzxXtKtKztKytKzyXzyxPX

++=+−=

+−+−== (2.2.54)


.0),(),()(2 351 =++ ztEytEtxK zy (2.2.55)





),()(),( 753 tKtKzztE +−= where )(7 tK is a function of integration. Refreshing the


).()()(),()()(),()()(),,,(

65237532

423110

tKtKytxKXtKtKztxKXtKtKztKyXzyxPX

++=+−−=

+−== (2.2.56)


.0)()(),,()()(

)()()()()(4)(214)(22)(2

2)(23)(23)(2

=−+−+

+−−

tKetAzyxPtKetKez

tKetAztKeytKetAytA

txttA

ttA

tAtt

tAtAt (2.2.57)



42





respect to ,y we get .0)()()()( 83)(23)(2 =+−− zKtKetKetA ttAtA

t Differentiating this

equation with respect to ,z we get .,)(0)( 1188 ℜ∈=⇒= cczKzK z Substituting back

this value in the above equation and solving we get

.,)( 2)(

2)()(

13 ℜ∈+= −−− ∫ cecdteectK tAtAtA Now substituting all the above




.,)( 5)(

5)()(

32 ℜ∈+−= −−− ∫ cecdteectK tAtAtA Now substituting all the above


.,)( 6)(

6)()(



),()(

),()(

,

),,(

65)(5

)()(3

3

75)(2

)()(1

2

)(6

)()(4

)(5

)()(3

)(2

)()(1

1

7431

0

tKtKyexcdteexcX

tKtzKexcdteexcX

ecdteec

ezcdteezceycdteeycX

zyExcxzcxycX

tAtAtA

tAtAtA

tAtAtA

tAtAtAtAtAtA

+++−=

+−−−=

++

−++−=

+++=

−−−

−−−

−−−

−−−−−−

∫∫

∫∫∫

(2.2.58)


(2.2.58) we get

.0)()()()()()(),(2 7)(25)(27)(25)(271 =−+−++ tKetAtKetzAtKetKezzyExc tA

ttA

tttA

ttA

y (2.2.59)

Differentiating equation (2.2.59) with respect to ,x we get .01 =c Substituting the

above information in equation (2.2.59) and differentiating with respect to ,y we get

43





we get .0)()()( 5)(25)(27 =++ tKetKtAec t

tAt


.,)( 9)(

9)()(

75 ℜ∈+−= −−− ∫ cecdteectK tAtAtA Now substituting all the above

information in equation (2.2.59) we get ⇒=−− 0)()()( 7)(27)(28 tKetKetAc t

tAtAt

.,)( 10)(

10)()(

87 ℜ∈+= −−− ∫ cecdteectK tAtAtA Refreshing the system of equations

(2.2.58) we get

),(

,

,

),(

6)(9

)()(7

)(5

)()(3

3

)(10

)()(8

)(9

)()(7

)(2

2

)(6

)()(4

)(5

)()(3

)(2

1

118743

0

tKeycdteeycexcdteexcX

ecdteecezcdteezcexcX

ecdteecezcdteezceycX

zKycyzcxcxzcX

tAtAtAtAtAtA

tAtAtAtAtAtAtA

tAtAtAtAtAtAtA

++−+−=

++−+−=

++−+=

++++=

−−−−−−

−−−−−−−

−−−−−−−

∫∫∫∫∫∫

(2.2.60)



.0)()()()(22 6)(26)(21173 =−−++ tKtAetKezKycxc t

tAt

tAz (2.2.61)

Differentiating equation (2.2.61) with respect to x and y respectively, we get

.073 == cc Substituting the above information in equation (2.2.61) and

differentiating with respect to ,z we get


equation (2.2.61) we get ⇒=−− 0)()()( 6)(26)(211 tKtAetKec t

tAt

tA

.,)( 13)(

13)()(

116 ℜ∈+= −−− ∫ cecdteectK tAtAtA Refreshing the system of equations

(2.2.60) we get

44

,

,

,

,

)(13

)(9

)(5

)()(11

3

)(10

)(9

)(2

)()(8

2

)(6

)(5

)(2

)()(4

1

1211840

tAtAtAtAtA

tAtAtAtAtA

tAtAtAtAtA

eceycexcdteecX

ecezcexcdteecX

ecezceycdteecX

czcycxcX

−−−−−

−−−−−

−−−−−

+++=

++−−=

++−+=

+++=

∫∫∫

(2.2.62)

where .,,,,,,,,, 13121110986542 ℜ∈cccccccccc The line element for Bianchi type I

space-times is given as

).( 222)(222 dzdydxedtds tA +++−= (2.2.63)

The above space-time admits ten linearly independent teleparallel Killing vector


xe tA

∂∂− ,)(

ye tA

∂∂− ,)(

ze tA

∂∂− ,)(1

xtG

tx

∂∂

+∂∂

,)(1

ytG

ty

∂∂

+∂∂ ,)(1

ztG

tz

∂∂

+∂∂ ),()(

xz

zxe tA

∂∂

−∂∂− ),()(

yx

xye tA

∂∂

−∂∂− and

),()(

yz

zye tA

∂∂

−∂∂− where .)( )()(1 ∫ −−= dteetG tAtA

Killing vector fields in general


y∂∂ ,

z∂∂ ),(

yx

xy

∂∂

−∂∂ )(

zy

yz

∂∂

−∂∂ and ).(

zx

xz

∂∂

−∂∂ It is

clear that teleparallel Killing vector fields are different and more in number to the

Killing vector fields in general relativity.

Case (V)(a):

In this case we have )(),(,tan tCCtBBtconsA === and ).()( tCtB = Now


.0),,(),,( 23)(2 =+ zytPzxtPe yxtB (2.2.64)




⇒−= ),(),,( 1)(22 ztEezytP tBy ),,(),(),,( 31)(22 ztEztEeyzytP tB +−= where

45


we get

).,,(),,(),(),,(),(),,,(

43212

31)(2110

yxtPXztEztxEXztEztEeyXzyxPX tB

=+=

+−== (2.2.65)


.0),(),(),,( 31)(24)(2 =+− ztEztEeyyxtPe zztB

xtB (2.2.66)




.0),(),(),( 31)(24)(2 =+− ztEztEeyytEe zztBtB Differentiating this equation with



)()(),( 311 tKtKzztE += and ),()(),( 42)(23 tKtKezztE tB +−= where ),(1 tK



).,()()(),,()()(

),()()()(),,,(

5213

2312

42)(23)(21)(21

10

ytEtxKtKyxXztEtxKtKzxX


tBtBtB

++=

++=

+−−−=

=

(2.2.67)


.0),(),()(2 251 =++ ztEytEtxK zy (2.2.68)





46



).()()(),()()(),()()(),,,(

65237532

42)(23)(2110

tKtKytxKXtKtKztxKXtKtKeztKeyXzyxPX tBtB

++=+−=

+−−== (2.2.69)


.0),,()()(

)()(2)()()(2142)(2

2)(23)(23)(2

=+−+

++

zyxPtKtKez

tKetBztKeytKetBy

xtttB

tBtt

tBtBt (2.2.70)







respect to ,y we get .0)()()()(2 83)(23)(2 =++ zKtKetKetB ttBtB

t Differentiating this

equation with respect to ,z we get .,)(0)( 1188 ℜ∈=⇒= cczKzK z Substituting back

this value in the above equation and solving we get

.,)( 2)(2

2)(2

13 ℜ∈+−= −− cecetctK tBtB Now substituting all the above information

in (2.2.70) and solve after differentiating twice with respect to ,z we get



.,)( 5)(2

5)(2

32 ℜ∈+−= −− cecetctK tBtB Now substituting all the above information

in equation (2.2.70) and solving we get .,)( 6644 ℜ∈+= cctctK Refreshing the

system of equation (2.2.69) with the help of above information, we get

47

),()(

),()(

,),,(

65)(25

)(23

3

75)(22

)(21

2

64532117

4310

tKtKyexcetxcX

tKtzKexctexcX

ctczctzcyctycXzyExcxzcxycX

tBtB

tBtB

+++−=

+−+−=

++−+−=+++=

−−

−− (2.2.71)


(2.2.71) we get

.0)()()()()()(

)()(),(27)(25)(27)(25)(2

217

1

=−+−+

+−+

tKetBtKetzBtKetKez

tBcxtBtxczyExctB

ttB

tttB

ttB

tty (2.2.72)


.0)()(2 211 =+− tBctBtcc tt Solving this equation and remember that in this case

,0)( ≠tBt we get ,01 =c which on back substitution gives us .02 =c Substituting the






we get .0)()()( 5)(25)(27 =+− tKetKtBec t

tBt


.,)( 9)(

9)()(

75 ℜ∈+−= −−− ∫ cecdteectK tBtBtB Now substituting all the above


tBtBt

.,)( 10)(

10)()(


(2.2.71) we get

),(

,

,

),(

6)(9

)()(7

)(25

)(23

3

)(10

)()(8

)(9

)()(7

2

64531

118743

0

tKeycdteeycexcectxX

ecdteecezcdteezcX

ctczcctzX

zKycyzcxcxzcX

tBtBtBtBtB

tBtBtBtBtBtB

++−+−=

++−=

++−=

++++=

−−−−−

−−−−−−

∫∫∫

(2.2.73)

48



.0)()()()()()(22 6)(26)(253

1173 =−−+−++ tKtBetKetBtcxtBtxczKycxc t

tBt

tBttz (2.2.74)


.0)()(2 533 =+− tBctBtcc tt Solving this equation and remember that in this case

,0)( ≠tBt we get ,03 =c which on back substitution gives us .05 =c Substituting the


.07 =c Now substituting all the above values in equation (2.2.74) and differentiating

with respect to ,z we get .,,)(0)( 121112111111 ℜ∈+=⇒= ccczczKzK zz Substituting

the above values in equation (2.2.74) we get ⇒=−− 0)()()( 6)(26)(211 tKtBetKec t

tBt

tB

.,)( 13)(

13)()(


(2.2.73) we get

,

,

,,

)(13

)(9

)()(11

3

)(10

)(9

)()(8

2

641

1211840

tBtBtBtB

tBtBtBtB

eceycdteecX

ecezcdteecX

cctXczcycxcX

−−−−

−−−−

++=

+−=

+=+++=

∫∫ (2.2.75)

where .,,,,,,, 131211109864 ℜ∈cccccccc The line element for Bianchi type I space-

times is given as

).( 22)(2222 dzdyedxdtds tB +++−= (2.2.76)


fields which are ,t∂∂ ,

x∂∂ ,)(

ye tB

∂∂− ,)(

ze tB

∂∂− ,

xt

tx

∂∂

+∂∂ ,)(1

ytG

ty

∂∂

+∂∂

ztG

tz

∂∂

+∂∂ )(1 and ),()(

yz

zye tB

∂∂

−∂∂− where

.)( )()(1 ∫ −−= dteetG tBtB Killing

vector fields in general relativity are ,x∂∂ ,

y∂∂

z∂∂ and .

yz

zy

∂∂

−∂∂ Only one

49

teleparallel Killing vector field is the same as in general relativity and other

teleparallel Killing vector fields are different and more in number to the Killing

vector fields in general relativity. Cases (V)(b) and (V)(c) can be solved exactly the

same as in the above case.

Case (VI)(a):

In this case we have )(tAA = and .tan tconsCB == Substituting equation (2.2.12)

in equation (2.2.6), we get

.0),,(),,( 2)(23 =+ zytPezxtP ytA

x (2.2.77)




⇒−= − ),(),,( 1)(22 ztEezytP tAy ),,(),(),,( 31)(22 ztEztEeyzytP tA +−= − where


we get

).,,(),,(),(),,(),(),,,(

43212

31)(2110

yxtPXztEztxEXztEztEeyXzyxPX tA

=+=

+−== −

(2.2.78)


.0)],(),(),,( 3)(214 =+− ztEeztEyyxtP ztA

zx (2.2.79)




.0),(),(),( 3)(214 =+− ztEeztEyytE ztA

z Differentiating this equation with respect to

y twice we get ).()(),(0),( 2144 tKtyKytEytEyy +=⇒= Substituting back the

above value in equation (2.2.79) and solving, we get )()(),( 311 tKtKzztE += and

50

),()(),( 42)(23 tKtKezztE tA +−= − where ),(1 tK ),(2 tK )(3 tK and )(4 tK are

functions of integration. Substituting back the above information in equation (2.2.78)

we get

).,()()(),,()()(),()()()(

),,,(

52132312

42)(2)3)(21)(2)1

10

ytEtxKtKyxXztEtxKtKzxXtKtKeztKeytKezyX

zyxPXtAtAtA

++=++=

+−−−=

=−−− (2.2.80)


.0),(),()(2 251 =++ ztEytEtxK zy (2.2.81)







).()()(),()()(),()()(

),,,(

65237532

42)(23)(21

10

tKtKytxKXtKtKztxKXtKtKeztKeyX

zyxPXtAtA

++=+−=

+−−=

=−− (2.2.82)


.0)()(),,()()(

)()()()()(4)(214)(22

233

=−+−+

−+−

tKetAzyxPtKetKz

tKtAztKytKtAytA

txttA

t

ttt (2.2.83)






51


respect to ,y we get .0)()()()( 833 =++− zKtKtKtA tt Differentiating this equation

with respect to ,z we get .,)(0)( 1188 ℜ∈=⇒= cczKzK z Substituting back this value

in the above equation and solving we get .,)( 2)(

2)()(

13 ℜ∈+−= ∫ − cecdteectK tAtAtA

Now substituting all the above information in (2.2.83) and solve after differentiating

twice with respect to ,z we get .,,)(0)( 434399 ℜ∈+=⇒= ccczczKzK zz

Substituting back in equation (2.2.83) and solve after differentiating with respect to

,z we get .,)( 5)(

5)()(

32 ℜ∈+−= ∫ − cecdteectK tAtAtA Now substituting all the above


.,)( 6)(

6)()(



),()(

),()(

,

),,(

65)(5

)()(3

3

75)(2

)()(1

2

)(6

)()(4

)(5

)()(3

)(2

)()(1

1

7431

0

tKtKyexcdteexcX

tKtzKexcdteexcX

ecdteec

ezcdteezceycdteeycX

zyExcxzcxycX

tAtAtA

tAtAtA

tAtAtA

tAtAtAtAtAtA

+++−=

+−+−=

++

−+−=

+++=

∫∫∫

∫∫

−

−

−−−

−−−−−−

(2.2.84)


(2.2.84) we get

.0)()()()(),(2 75)(2

)()(1

71 =−+−++ ∫ − tKtKzetAcxdteetAxczyExc tt

tAt

tAtAty (2.2.85)


.0)()(2 )(2

)()(11 =−+ ∫ − tA

ttAtA




respect to ,y we get ),()(),(0),( 111077 zKzyKzyEzyEyy +=⇒= where )(10 zK

52

and )(11 zK are functions of integration. Substituting the above value in (2.2.85) and

differentiating with respect to z twice, we get

.,,)(0)( 87871010 ℜ∈+=⇒= ccczczKzK zz Substituting back this value in equation

(2.2.85) and differentiating with respect to ,z we get ⇒=+ 0)(57 tKc t

.,)( 9975 ℜ∈+−= cctctK Now substituting all the above information in equation

(2.2.85) we get ⇒=− 0)(78 tKc t .,)( 10108

7 ℜ∈+= cctctK Refreshing the system of

equations (2.2.84) we get

),(

,

,

),(

697

)(5

)()(3

3

108972

)(6

)()(4

)(5

)()(3

1

118743

0

tKyctycexcdteexcX

ctczctzcX

ecdteecezcdteezcX

zKycyzcxcxzcX

tAtAtA

tAtAtAtAtAtA

++−+−=

++−=

++−=

++++=

∫

∫∫

−

−−−−−−

(2.2.86)



.0)()()()(22 6)(5

)()(3

1173 =−−+++ ∫ − tKetAcxdteetAxczKycxc t

tAt

tAtAtz (2.2.87)


.0)()(2 )(5

)()(33 =−+ ∫ − tA

ttAtA




respect to ,y we get .07 =c Now substituting all the above values in equation


.,,)(0)( 121112111111 ℜ∈+=⇒= ccczczKzK zz Substituting the above values in

equation (2.2.87) we get ⇒=− 0)(611 tKc t .,)( 131311

6 ℜ∈+= cctctK Refreshing the


53

,,

,,

139113

10982

)(6

)()(4

1121184

0

cyctcXczctcX

ecdteecXczcycxcX tAtAtA

++=+−=

+=+++= −−− ∫ (2.2.88)

where .,,,,,,, 131211109864 ℜ∈cccccccc The line element for Bianchi type I space-

times is given as

).( 222)(222 dzdydxedtds tA +++−= (2.2.89)



xe tA

∂∂− ,

y∂∂ ,

z∂∂ ,)(1

xtG

tx

∂∂

+∂∂ ,

yt

ty

∂∂

+∂∂

zt

tz

∂∂

+∂∂

and ),(y

zz

y∂∂

−∂∂ where .)( )()(1 ∫ −−= dteetG tAtA



y∂∂

z∂∂ and .

yz

zy

∂∂

−∂∂ It is evident that three teleparallel

Killing vector fields are the same as Killing vector fields in general relativity and all

other teleparallel Killing vector fields are different. Teleparallel Killing vector fields

are more in number to the Killing vector fields in general relativity. Cases (VI)(b) and

(VI)(c) can be solved exactly the same as in the above case.


Type II Space-Times

Consider Bianchi type II space-times in usual coordinates ),,,( zyxt (labeled by

),,,,( 3210 xxxx respectively) with the line element [76, 78]

,)(2)]()([)()( 22222222222 dydztxBdztCtBxdytBdxtAdtds −++++−= (2.3.1)

where ,A B and C are no where zero functions of t only. The tetrad components

and its inverse components by using the relation (1.3.4) is obtained as [66]

54

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−=

)(000)()(00

00)(00001

tCtxBtB

tAS a

µ .

)()(000)(0000)(00001

11

1

1

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

−−

−

−

tCtxCtB

tASa

µ (2.3.2)

The non vanishing torsion components are obtained by (1.3.11) as

,011

AAT

•

= ,022

BBT

•

= ,033

CCT

•

= ,032

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

••

BB

CCxT ,113

2 −=T (2.3.3)

where ‘dot’ denotes differentiation with respect to .t A vector field X is said to be

teleparallel Killing vector field if it satisfies equation (1.3.17). One can write (1.3.17)

explicitly using (2.3.1) and (2.3.3) as

,0,0 1,1

0,0 == XX (2.3.4)

,02,3

2,2 =− XxX (2.3.5)

,011,

00,

12 =−− • XAAXXA (2.3.6)

,0322,

00,

320,

22 =−+−− •• XBBxXBBXXxBXB (2.3.7)

,0322,

121,

321,

22 =−+− XBXAXBxXB (2.3.8)

( ),0332

20,

32223,

00,

22

=−−

++−+••

•

XCCXBBxXBxBXCBxXXBx

(2.3.9)

( ) ,0321,

32221,

223,

12 =+++− XBxXCBxXxBXA (2.3.10)

( ) ,0,332

2,3222

2,22

3,22 =−++− XBxXCBxXxBXB (2.3.11)

( ) ,0,322

3,3222 =−+ XBxXCBx (2.3.12)

Solving equation (2.3.4), (2.3.5) and (2.3.11) we get

55

),,,(),,(

),,,(),,(),,(

),,,(),,,(

432

3

3432

2

2110

zxtPzxtPCByX

zxtPzxtxPzxtPCByxX

zytPXzyxPX

z

z

+⎟⎠⎞

⎜⎝⎛−=

++⎟⎠⎞

⎜⎝⎛−=

==

(2.3.13)

where ),,,(1 zyxP ),,,(2 zytP ),,,(3 zxtP and ),,,(4 zxtP are functions of integration

which are to be determined. Considering equation (2.3.12) and using equation

(2.3.13) we get

.0),,()(),,()(),,()( 324232 =−+− zxtPtBxzxtPtCzxtPtyB zzzz (2.3.14)

Differentiating (2.3.14) with respect to ,y we get

),,(),(),,(0),,( 2133 xtKxtKzzxtPzxtPzz +=⇒= where ),(1 xtK and ),(2 xtK are

functions of integration. Substituting back this value in equation (2.3.14) and

differentiating the resulting equation with respect to ,z we get


functions of integration. Substituting back the above values we get

).,()()(),( 1

23 xtK

tCtBxxtK ⎟⎟⎠

⎞⎜⎜⎝

⎛= Therefore ).,(),(

)()(),,( 41

24 xtKxtK

tCtBzxzxtP +⎟⎟⎠

⎞⎜⎜⎝

⎛=

Refreshing the system of equations (2.3.13) we have

).,(),(][

),,(),(),(][

),,,(),,,(

4122

3

24122

22

2110

xtKxtKCBy

CBzxX

xtKxtxKxtKzCByx

CBzxX

zytPXzyxPX

+⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

+++⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

==

(2.3.15)


.0),,()(),()(),()( 222212 =++ zytPtAxtKtBxtKtzB yxx (2.3.16)

56

Differentiating (2.3.16) first with respect to z and then with respect to ,x we get

),()(),(0),( 2111 tDtDxxtKxtK xx +=⇒= where )(1 tD and )(2 tD are functions of

integration. Substituting back the above value in (2.3.16) then solve after

differentiating with respect to ,z we get

),,(),()()()(),,( 651

22 ytKztKtD

tAtByzzytP ++⎟⎟⎠

⎞⎜⎜⎝

⎛−= where ),(5 ztK and ),(6 ztK are

functions of integration. Substituting back these values in (2.3.16) and solving we get

)()(),( 432 tDtDxxtK += and ),()()()(),( 53

26 tDtD

tAtByytK +⎟⎟⎠

⎞⎜⎜⎝

⎛−= where ),(3 tD

)(4 tD and )(5 tD are functions of integration. Substituting all these information in

(2.3.15) we get

).,())()(]([

),()(),())()((][

),()()()(),()(

)()(

),,,(

42122

3

4342122

22

532

512

1

10

xtKtDtxDCBy

CBzxX

tDtxDxtxKtDtxDzCByx

CBzxX

tDtDtAtByztKtD

tAtBzyX

zyxPX

++⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

+++++⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

+⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

=

(2.3.17)


.0),()(),()()()()()()()]()(2[

4252

3222122

=++

−++−

xtKtCztKtAtDtxBtDtBztDtzBxtyB

xz

(2.3.18)

Differentiating (2.3.18) with respect to ,z we get

.0),()()()()()( 522212 =++ ztKtAtDtBtDtBx zz (2.3.19)

Differentiating the above equation with respect to ,x we get .0)(1 =tD Substituting

back in (2.3.19) we get ⇒=+ 0),()()()( 5222 ztKtAtDtB zz

57

),()()()()(

21),( 762

225 tDtzDtD

tAtBzztK ++⎟⎟⎠

⎞⎜⎜⎝

⎛−= where )(6 tD and )(7 tD are

functions of integration. Now substituting back the above value in (2.3.18) then

differentiating the resulting equation with respect to x and solving we get

),()()()()(

21),( 983

224 tDtxDtD

tCtBxxtK ++⎟⎟⎠

⎞⎜⎜⎝

⎛= )(

)()()( 8

26 tD

tAtCtD ⎟⎟⎠

⎞⎜⎜⎝

⎛−= and

),()()()()(

)()(

21),( 78

22

225 tDtD

tAtCztD

tAtBzztK +⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−= where )(8 tD and )(9 tD

are functions of integration. Refreshing the system of equations (2.3.17) we get

).()()()()(

21)(][

),()()]()(

)()()(

21[)(][

),()()()()(

)()()(

)()(

21

),,,(

9832

2222

3

4398

32

2222

22

1032

82

22

21

10

tDtxDtDtCtBxtD

CBy

CBzxX

tDtxDtDtxD

tDtCtBxxtDz

CByx

CBzxX

tDtDtAtBytD

tAtCztD

tAtBzX

zyxPX

++⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

++++

⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟

⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

+⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

=

(2.3.20)

Considering equaiton (2.3.7) and using equation (2.3.20) we get

.0)()()()()()(

)()()(),,()()()()()()(43

21423222

=++

+−++

tDtBtBtDtBtxB

tDtBtzBzyxPtDtBtDtBxtDtzB

tt

tyttt (2.3.21)

Differentiating (2.3.21) with respect to ,z we get

.0)()()(),,()()( 2122 =+− tDtBtBzyxPtDtB tzyt In order to solve this equation we

consider ),,(),(),,(0),,( 8711 yxKzxKzyxPzyxP zy +=⇒= where ),(7 zxK and

),(8 yxK are functions of integration. Substituting back this value in the above

equation we get .,)(

1)(0)()()()( 11222 ℜ∈=⇒=+ cc

tBtDtDtBtDtB tt Substituting

back the above values in (2.3.21) and differentiating with respect to ,x we get

58

.0)()()(),,()()( 3832 =+− tDtBtBzyxKtDtB tyxt Solving this equation by considering

0),,(8 =zyxK yx we get ),()(),(0),,( 121188 yDxDyxKzyxK yx +=⇒= where

)(11 xD and )(12 yD are functions of integration and .,)(

1)( 223 ℜ∈= cc

tBtD Now

substituting back all the above values in (2.3.21) and solving we get ,)(

1)( 34 c

tBtD =

and .,,)(0)( 153151212 ℜ∈=⇒= cccyDyD Refreshing the system of equations

(2.3.20) by using the above information, we get

).()()(

)(21]

)()(

)()([

,)(

1)()(

])(

)(21[]

)()(

)()([

),()(

)()()()(

)()(

21

,)(),(

9822

2122

3

3982

222

12222

1022

82

1221

151170

tDtxDctC

tBxctC

tBytC

tBzxX

ctB

txDtDx

cxtC

tBxxcztC

tByxtC

tBzxX

tDctAtBytD

tAtCzc

tAtBzX

cxDzxKX

++⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

+++

+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

+⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

++=

(2.3.22)

Now differentiating (2.3.6) with respect to y and using equation (2.3.22) we get

( ) .0)()()()(2 =− tBtAtAtBc tt We will discuss here three different possibilities to

solve this equation:

(I) ( ) ,0)()()()( ≠− tBtAtAtB tt ,02 =c (II) ( ) ,0)()()()( =− tBtAtAtB tt ,02 ≠c

(III) ( ) ,0)()()()( =− tBtAtAtB tt .02 =c

We will discuss each case in turn. In case (III) we will obtain the same teleparallel

Killing vector fields as in case (II) with the reduction of one teleparallel Killing

vector fileds generated from .2c Hence we will not discuss the case (III).

59

Case (I):

In this case we have ( ) 0)()()()( ≠− tBtAtAtB tt and .02 =c Substituting 02 =c in


).()(])(

)()(

)([

,)(

1)()(])(

)()(

)([

),()()()(

)()(

21

,)(),(

98122

3

3982

12222

1082

1221

151170

tDtxDctC

tBytC

tBzxX

ctB

txDtDxcztC

tByxtC

tBzxX

tDtDtAtCzc

tAtBzX

cxDzxKX

++⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

++++⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

+⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

++=

(2.3.23)

Differentiating equation (2.3.6) with respect to z and using (2.3.23), we get

.0),()()(

)()()()()()(2)()()()(

782

82

1

=−−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−⎟

⎠⎞

⎜⎝⎛ −

−

zxKtDtC

tDA

tAtCtCtCtAA

tAtBtBtAcz

zxt

tttt

(2.3.24)

We solve this equation by letting ),()(),(0),( 141377 zDxDzxKzxK zx +=⇒= where

)(13 xD and )(14 zD are functions of integration. Substituting this value back in

(2.3.24) and differentiating with respect to ,z we get .01 =c We are also left with

equation which we will solve later on.

.0)()()()()()()()(2 828

2

=+⎟⎟⎠

⎞⎜⎜⎝

⎛ −tDtCtD

AtAtCtCtCtA

ttt (2.3.25)

Refreshing the system of equations with the help of above information, we get

),()(,)(

1)()(

),()()()(,)()(

9833

9822

1082

115

14150

tDtxDXctB

txDtDxX

tDtDtAtCzXczDxDX

+=++=

+⎟⎟⎠

⎞⎜⎜⎝

⎛−=++=

(2.3.26)

60

where ).()()( 111315 xDxDxD += Considering equation (2.3.9) and using equation

(2.3.26) we get

.0)()()()()()()()()()()( 98928214 =−−−− tDtCtCtDtCtxCtDtCtDtxCzD ttttz (2.3.27)

Differentiating this equation with respect to ,x we get ⇒=+ 0)()()()( 88 tDtCtDtC tt

.,)(

1)( 448 ℜ∈= cc

tCtD Now using this value in (2.3.27) and taking

ℜ∈=⇒= 551414 ,)(0)( cczDzDz we get .,

)(1)( 66

9 ℜ∈= cctC

tD Now substituting

back these values in (2.3.25) we get ( ) .0)()()()(4 =− tAtCtCtAc tt In order to solve

this equation we consider the following three cases:

(a) ( ) ,0)()()()( ≠− tAtCtCtA tt ,04 =c (b) ( ) ,0)()()()( =− tAtCtCtA tt ,04 ≠c

(c) ( ) ,0)()()()( =− tAtCtCtA tt .04 =c

We will discuss each case in turn. In case (I)(c) we will obtain the same teleparallel

Killing vector fields as in case (I)(b) with the reduction of one teleparallel Killing

vector fileds generated from .4c Hence we will not discuss the case (I)(c).

Case (I)(a):

In this case we have ( ) ,0)()()()( ≠− tBtAtAtB tt ( ) 0)()()()( ≠− tAtCtCtA tt and

.04 =c Substituting 04 =c with all the above information in system of equations

(2.3.26) we get

,)(

1,)(

1)(

1),(,)( 63

362101

16150 c

tCXc

tBcx

tCXtDXcxDX =+==+= (2.3.28)

where .,, 1551663 ℜ∈+= ccccc Considering equation (2.3.9) and using equation

(2.3.28) we get 715 )( cxD = and .,,

)(1)( 878

10 ℜ∈= ccctA

tD The line element for

61

Bianchi type II space-times is given in equation (2.3.1) and the solution of equations

from (2.2.4) to (2.3.12) is [66]

,)(

1,)(

1)(

1,)(

1, 63

362

81

90 c

tCXc

tBcx

tCXc

tAXcX =+=== (2.3.29)

where .,,, 1679863 ℜ∈+= cccccc The above space-times (2.3.1) admits four linearly

independent teleparallel Killing vector fields which are ,t∂∂ ,

)(1

xtA ∂∂

ytB ∂∂

)(1 and

).()(

1zy

xtC ∂

∂+

∂∂ Killing vector fields in general relativity are ,

y∂∂ ,

z∂∂

).(y

zx ∂

∂+

∂∂ One can easily see that the teleparallel Killing vector fields are different

from Killing vector fields in general relativity.

Case (I)(b):

In this case we have ( ) ,0)()()()( ≠− tBtAtAtB tt ( ) 0)()()()( =− tAtCtCtA tt and

.04 ≠c Equation ( ) ).()(0)()()()( tCtAtAtCtCtA tt =⇒=− System of equations

(2.3.26) become

,)(

1)(

1,)(

1)(

1)(

1

),()(

1,)(

643

36422

104

116

150

ctC

ctC

xXctB

cxtC

cxtC

X

tDctC

zXcxDX

+=++=

+−=+= (2.3.30)

where .,,, 16643 ℜ∈cccc Considering equation (2.3.9) and using (2.3.30) we get

715 )( cxD = and .,,

)(1)( 878

10 ℜ∈= ccctA

tD The line element for Bianchi type II

space-times takes the form

,)(2)]()([)()( 22222222222 dydztxBdztAtBxdytBdxtAdtds −++++−= (2.3.31)

Solution of equations from (2.2.4) to (2.3.12) is given as [66]

62

,)(

1)(

1,)(

1)(

1)(

1

,)(

1)(

1,

643

36422

841

170

ctA

ctA

xXctB

cxtA

cxtA

X

ctA

ctA

zXcX

+=++=

+−== (2.3.32)

where .,,,, 167178643 ℜ∈+= ccccccc The above space-time (2.3.31) admits five

linearly independent teleparallel Killing vector fields which can be written as ,t∂∂

,)(

1xtA ∂∂ ,

)(1

ytB ∂∂ )(

)(1

zyx

tA ∂∂

+∂∂ and ).(

)(1 2

zx

yx

xz

tA ∂∂

+∂∂

+∂∂

− Killing

vector fields in general relativity are ,y∂∂ ,

z∂∂ )(

yz

x ∂∂

+∂∂ and

).)22

(22

zx

yxz

xz

∂∂

+∂∂

++∂∂ One can easily compare that teleparallel Killing vector

fields are different from Killing vector fields in general relativity.

Case (II):

In this case we have ( ) 0)()()()( =− tBtAtAtB tt and .02 ≠c Equation

( ) ).()(0)()()()( tBtAtBtAtAtB tt =⇒=− The system of equations (2.3.22) can be

written as

).()()(

)(21]

)()(

)()([

,)(

1)()(

)(1]

)()(

21[]

)(1

)()(

)()([

),()(

1)()()(

)(1

21

,)(),(

9822

2122

3

3982

22

22

12222

102

82

121

151170

tDtxDctCtBxc

tCtBy

tCtBzxX

ctB

txDtDx

ctB

xtCtBxxc

tBz

tCtByx

tCtBzxX

tDctB

ytDtAtCzc

tBzX

cxDzxKX

++⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

+++

+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

+−⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

++=

(2.3.33)

63

Differentiating (2.3.6) with respect to z then using (2.3.33) and remember that in this

case ),()( tBtA = we get ),()(),( 13127 zDxDzxK += where )(12 xD and )(13 zD are

functions of integration and

.0)()()()()()()()(2 828

2

=−⎟⎟⎠

⎞⎜⎜⎝

⎛ −− tDtCtD

AtAtCtCtCtA

ttt (2.3.34)

We will solve this equation later on. Refreshing the system of equations with the

above information, we get

).()()(

)(21]

)()(

)()([

,)(

1)()(

)(1]

)()(

21[]

)(1

)()(

)()([

),()(

1)()()(

)(1

21

,)()(

9822

2122

3

3982

22

22

12222

102

82

121

1513140

tDtxDctCtBxc

tCtBy

tCtBzxX

ctB

txDtDx

ctB

xtCtBxxc

tBz

tCtByx

tCtBzxX

tDctB

ytDtAtCzc

tBzX

czDxDX

++⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

+++

+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

+−⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

++=

(2.3.35)

where ).()()( 121114 xDxDxD += Considering equation (2.3.9) and using equation

(2.3.35) then after differentiating with respect to y we get

( ) .0)()()()(1 =− tCtBtBtCc tt Here we have to discuss three possible cases:

(d) ( ) ,0)()()()( ≠− tBtCtCtB tt ,01 =c (e) ( ) ,0)()()()( =− tBtCtCtB tt ,01 ≠c

(f) ( ) ,0)()()()( =− tBtCtCtB tt .01 =c

We will discuss each case in turn. In case (II)(f) we will obtain the same teleparallel

Killing vector fields as in case (II)(e) with the reduction of one teleparallel Killing

vector fileds generated from .1c Hence we will not discuss the case (II)(f).

64

Case (II)(d):

In this case we have ),()( tBtA = ( ) 0)()()()( ≠− tCtBtBtC tt and .01 =c Substituting

all the above information in (2.3.9) and differentiating twice with respect to ,x we

reach to the equation ( ) .0)()()()(2 =− tCtBtBtCc tt In this case neither 02 =c nor

( ) .0)()()()( =− tCtBtBtC tt Hence this case leads to a contradiction and is not

possible.

Case (II)(e):

In this case we have ),()( tBtA = ( ) 0)()()()( =− tCtBtBtC tt and .01 ≠c Equation

( ) ).()(0)()()()( tCtBtCtBtBtC tt =⇒=− Therefore we have ).()()( tCtBtA ==

Now substituting all the above information in (2.3.9) and differentiating the resulting

equation first with respect to z then with respect to x we get ,)( 1613 czD =

ℜ∈= 1716178 ,,

)(1)( ccctB

tD and .,)(

1)( 18189 ℜ∈= cc

tBtD Refreshing the system of


.)(

1)(

1)(

121

)(1)(

,)(

1)(

1)(

1)(

1]21[

)(1][

),()(

1)(

1)(

121

,)(

181722

13

318172

23

122

102171

21

19140

ctB

ctB

xctB

xctB

yzxX

ctB

ctB

xctB

x

ctB

xxctB

zyxzxX

tDctB

yctB

zctB

zX

cxDX

+++−=

+++

+++−=

+−−⎟⎟⎠

⎞⎜⎜⎝

⎛−=

+=

(2.3.36)

Solve after considering equation (2.3.6) and using equation (2.3.36) we get

2014 )( cxD = and .,,

)(1)( 212021

10 ℜ∈= ccctB

tD The line element is this case

becomes

65

].2)1([)( 2222222 dzxdydzxdydxtAdtds −++++−= (2.3.37)

Refreshing the system of equations we get [66]

.)(

1)(

1)(

121

)(1)(

,)(

1)(

1)(

1)(

1)21(

)(1)(

,)(

1)(

1)(

1)(

121,

181722

13

318172

23

122

21217121

220

ctB

ctB

xctB

xctB

yzxX

ctB

ctB

xctB

x

ctB

xxctB

zyxzxX

ctB

ctB

yctB

zctB

zXcX

+++−=

+++

+++−=

+−−⎟⎟⎠

⎞⎜⎜⎝

⎛−==

(2.3.38)

where .,,,,,, 201922211817321 ℜ∈+= ccccccccc Here the above space-time (2.3.37)

admits seven linearly independent teleparallel Killing vector fields which can be

written as ,t∂∂ ,

)(1

xtA ∂∂ ),(

)(1

zyx

tA ∂∂

+∂∂ ),(

)(1 2

zx

yx

xz

tA ∂∂

+∂∂

+∂∂

−

),2

)2

(()(

1 23

zx

yxx

xy

tA ∂∂

+∂∂

++∂∂

− ytA ∂∂

)(1 and

].)()(2

[)(

1 22

zyxz

yzxyzx

xz

tA ∂∂

−+∂∂

+−+∂∂

− Killing vector fields in general

relativity are ,y∂∂ ,

z∂∂ )(

yz

x ∂∂

+∂∂ and ).)

22(

22

zx

yxz

xz

∂∂

+∂∂

++∂∂ One can

easily see that teleparallel Killing vector fields are different from general relativity in

this case and there are three more teleparallel Killing vector fields in teleparallel

theory of gravitation.

In the following we will consider now the case when .0\, ℜ∈=== λλCBA

The line element is this case after a suitable rescaling, becomes

.2)1( 222222 dzxdydzxdydxdtds −++++−= (2.3.39)

In this case the only non vanishing torsion components are .1 313

132 TT −=−= Our

system of equations (2.3.13) for the space-time (2.3.39) becomes

66

).,,(),,(

),,,(),,(),,(

),,,(),,,(

433

3432

2110

zxtPzxtPyX

zxtPzxtxPzxtPyxX

zytPXzyxPX

z

z

+−=

++−=

==

(2.3.40)

Substituting (2.3.40) in (2.3.12) and differentiating with respect to ,y we get


functions of integration. Substituting the above value back in (2.3.12) we get

),,(),(),,(0),(),,( 41414 xtKxtKzxzxtPxtKxzxtPz +=⇒=− where ),(4 xtK is a

function of integration. Refreshing the system of equations (2.3.40) we get

),,(),(][),,(),(),(][

),,,(),,,(

413

24122

2110

xtKxtKyzxXxtKxtxKxtKzyxzxX

zytPXzyxPX

+−=

+++−=

==

(2.3.41)


.0),,(),(),( 221 =++ zytPxtKxtzK yxx Differentiating this equation first with respect to

z and then with respect to ,x we get ),()(),(0),( 2111 tDtDxxtKxtK xx +=⇒=

where )(1 tD and )(2 tD are functions of integration. Substituting back we get

),,(),()(),,(0),,()( 651221 ytKztKtDyzzytPzytPtD zy ++−=⇒=+ where

),(5 ztK and ),(6 ytK are functions of integration. Once again substituting back

these values in the above equation we get .0),(),( 62 =+ ytKxtK yx Differentiating this

equation with respect to ,x we get ),()(),(0),( 4322 tDtDxxtKxtK xx +=⇒= where

)(3 tD and )(4 tD are functions of integration. Substituting back and solving we get

),()(),( 536 tDtDyytK +−= where )(5 tD is a functions of integration. Substituting

all these information in (2.3.41) we get

67

).,())()(]([),()(),())()((][

),()(),()(),,,(

4213

4342122

53511

10

xtKtDtxDyzxXtDtxDxtxKtDtxDzyxzxX

tDtyDztKtDzyXzyxPX

++−=

+++++−=

+−+−=

=

(2.3.42)


0),(),()()()(]2[ 45321 =++−++− xtKztKtxDtDztDzxy xz (2.3.43)

Differentiating (2.3.43) with respect to ,y we get .0)(1 =tD Now differentiating

(2.3.43) with respect to ,z we get ⇒=+ 0),()( 52 ztKtD zz

),()()(21),( 76225 tDtzDtDzztK ++−= where )(6 tD and )(7 tD are functions of

integration. Now substituting back this value in (2.3.43) then differentiating with

respect to x and solving, we get ),()()(21),( 98324 tDtxDtDxxtK ++= where

)(8 tD and )(9 tD are functions of integration and

).()()(21),()()( 7822586 tDtzDtDzztKtDtD +−−=⇒−= Refreshing the system of


).()()(21)(][

),()()]()()(21[)(][

),()()()(21),,,(

983223

439832222

103822110

tDtxDtDxtDyzxX

tDtxDtDtxDtDxxtDzyxzxX

tDtyDtzDtDzXzyxPX

+++−=

++++++−=

+−−+−==

(2.3.44)


.0),,()()()( 1432 =−++ zyxPtDtDxtzD yttt Differentiating the above equation with

respect to z and ,t we get .,,)(0)( 212122 ℜ∈+=⇒= ccctctDtDtt Substituting

back this value in the above equation and differentiating with respect to ,z we get

),,(),(),,(),,( 871

11

1 zxKyxKzcyzyxPczyxP zy ++=⇒= where ),(7 yxK and

68

),(8 zxK are functions of integration. Substituting back these values in the above

equation and solving we get ,,,)( 43433 ℜ∈+= ccctctD

),()(),( 12113

7 yDxDcxyyxK ++= where )(11 xD and )(12 yD are functions of

integration, ℜ∈+= 65654 ,,)( ccctctD and ,,)( 775

12 ℜ∈+−= ccycyD Refreshing


).()()(21)]([

,)()()](21[)(][

),()()()(21

,),()(

9843

221

3

65982

432

2122

1043

821

21

78

511

310

tDtxDctcxctcyzxX

ctctxDtDxctcxxxctczyxzxX

tDctcytzDctczX

czxKycxDcyxczyX

+++++−=

+++++++++−=

++−−++−=

++−++=

(2.3.45)

Considering equation (2.3.6) and using equation (2.3.45) then differentiating the

resulting equation with respect to y we get .03 =c Substituting back the above value

and differentiating the resulting equation first with respect to x then with respect to

z twice we get ),()()(),( 1312118 zDzxDxDzxK ++−= where )(13 zD is a function

of integration and .,,21)( 98981

212 ℜ∈++−= cccczczzD Substituting back the

above values we get ,)( 1088 ctczD += ℜ∈+= 1110119

10 ,,)( ccctctD and

.2)( 813 czzD = Refreshing the system of equations (2.3.45) we get

),()(21)]([

,)()(]21[)(][

,)()(21

,221

91084

221

3

659

1082

42

2122

119108212

41

789812

510

tDctcxcxctcyzxX

ctctxDctcxcxxxctczyxzxX

ctcctczctczycX

czcxcxzccxzcyczyX

+++++−=

+++++++++−=

+++−+−−=

++++−−=

(2.3.46)

where .,,,,,,,,, 111098765421 ℜ∈cccccccccc Considering equation (2.3.9) and using

equaiton (2.3.46) then differentiating the resulting equation with respect to ,y z and

69

x we get ,021 == cc 85 cc = and .,21)( 15158

29 ℜ∈+= cccttD Now considering

equations (2.3.6) and (2.3.7) and using (2.3.46) we get .085 == cc Refreshing the

system (2.3.46) teleparallel Killing vector fields are obtained as [66]

.21

,]21[

,,

1510423

615102

422

1191041

790

cxccxX

cxccxcxxxX

ctczcycXcxcX

++=

++++=

++−−=+=

(2.3.47)

where .,,,,,, 1511109764 ℜ∈ccccccc Here the above space-time (2.3.39) admits seven


,x∂∂ ),(

zyx

∂∂

+∂∂ ),( 2

zx

yx

xz

∂∂

+∂∂

+∂∂

− ),2

)2

((23

zx

yxx

xy

∂∂

+∂∂

++∂∂

− y∂∂

and ].)()(2

[ 22

zyxz

yzxyzx

xz

∂∂

−+∂∂

+−+∂∂

− Killing vector fields in general

relativity are ,y∂∂ ,

z∂∂ )(

yz

x ∂∂

+∂∂ and ).)

22(

22

zx

yxz

xz

∂∂

+∂∂

++∂∂ One can

easily see that teleparallel Killing vector fields are different and more in number from



Types VIII and IX Space-Times

Consider Bianchi types VIII and IX space-times in usual coordinates ),,,( zyxt

(labeled by ),,,,( 3210 xxxx respectively) with the line element [38, 78]

70

,)()(2)()()()()()( 2222222

dzdxygtAdzyGtBygtAdytBdxtAdtds

+++++−=

(2.4.1)

where A and B are no where zero functions of t only. The above space-time (2.4.1)

is called Bianchi type VIII if yyg cosh)( = and yyG sinh)( = while (2.4.1) becomes

Bianchi type IX when yyg cos)( = and .sin)( yyG = The above space-times admit

four linearly independent Killing vector fields in general relativity which are ,x∂∂

,z∂∂

zz

yGyg

yz

xz

yG ∂∂

−∂∂

+∂∂ sin

)()(cossin

)(1

and

.cos)()(sincos

)(1

zz

yGyg

yz

xz

yG ∂∂

+∂∂

+∂∂

− The tetrad components and its inverse can be

obtained by using the relation (1.3.4) as [67]

,

sin)(0000)(00

cos)(0)(00001

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

ytBtB

ytAtAS a

µ

⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛−

=

ytB

tB

ytBy

tASa

sin)(1000

0)(

100

sin)(cos0

)(10

0001

µ (2.4.2)


are obtained as [67]

,2

101

AA•

=Γ ,2

202

BB•

=Γ ,2

303

BB•

=Γ ,)()(

323

yGyg

=Γ ,)(21

301

BB

AAyg

••

−=Γ

,)(

132

1

yG−=Γ where dot denotes the derivative with respect to .t The non vanishing

torsion components by using equation (1.3.11) are ,2

011

AAT

•

=

,)(21

031

BB

AAygT

••

−= ,2

022

BBT

•

= ,2

033

BBT

•

= ,)(

123

1

yGT −= .

)()(

233

yGygT =

71

A vector field X is said to be teleparallel Killing vector field if it satisfies equation

(1.3.17). One can write (1.3.17) explicitly using (2.4.1) and the above torsion

components as:

,0,,0, 22

00 == XX (2.4.3)

,0,)(, 13

11 =+ XygX (2.4.4)

,0,2,2 22

00

2 =+− • XBXXB (2.4.5)

,0)()(,, 32,

32

11

2 =−++ XyGAXygAXAXB (2.4.6)

,0)(21)(

21, 3

0,31

1,0

01 =+++− •• XygAXygAXAXXA (2.4.7)

,0)()(21

)(21)()(,)(

32

13,

00,

3220

1

=++

+−++

••

•

XyGBygA

XygAXXyBGyAgXygA (2.4.8)

,0)()()(,)( 3,3

3,1

1,322

11 =++++ XygAAXXyBGyAgXygA (2.4.9)

,0)()()()()(,)( 33,

22,

3222

1 =−−+++ XyGygBABXXyBGyAgXygA (2.4.10)

.0)()(,)( 3,322

31 =++ XyBGyAgXygA (2.4.11)

Integrating equation (2.4.3) we get

,),,(),,,( 2210 zxtEXzyxEX == (2.4.12)

where ),,(1 zyxE and ),,(2 zxtE are functions of integration which are to be

determined. Considering equation (2.4.9) and using equation (2.4.4) we get

.0)()( 3,3

3,1

1,32 =++ XygXAXyGB (2.4.13)

Now differentiating equation (2.4.4) with respect to ,z we get

0,)(, 133

131 =+ XygX (2.4.14)

72

Differentiating equation (2.4.13) with respect to x and using equation (2.4.14) we get

.011,3 =X Now integrating this equation with respect to ,x we get

),,,(),,( 433 zytEzytExX += (2.4.15)

where ),,(3 zytE and ),,(4 zytE are functions of integration. Substituting back

equation (2.4.15) into equation (2.4.4), we get

),,,(),,()( 531 zytEzytEyxgX +−= (2.4.16)

where ),,(5 zytE is a functions of integration which is to be determined. We have the

following system of equations:

).,,(),,(,),,(),,,(),,()(),,,(

43322

53110

zytEzytExXzxtEXzytEzytEyxgXzyxEX

+==

+−== (2.4.17)


resulting equation with respect to y we get

),,(),(),,(0),,( 2111 zxFzxFyzyxEzyxEyy +=⇒= where ),(1 zxF and ),(2 zxF

are functions of integration. Refreshing the system of equations (2.4.17) we get

).,,(),,(,),,(),,,(),,()().,,(),(

43322

531210

zytEzytExXzxtEXzytEzytEyxgXzxFzxFyX

+==

+−=+= (2.4.18)

Considering equation (2.4.6) and using equation (2.4.18) then differentiating with

respect to ,x we get ),,(),(),,(0),,( 4322 ztFztxFzxtEzxtExx +=⇒= where

),(3 ztF and ),(4 ztF are functions of integration. Refreshing (2.4.18) we get

).,,(),,(),,(),(),,,(),,()().,,(),(

433432

531210

zytEzytExXztFztxFXzytEzytEyxgXzxFzxFyX

+=+=

+−=+= (2.4.19)

Considering equation (2.4.7)and using equation (2.4.19) then differentiating the

resulting equation with respect to ,x we get .0),(),( 21 =+ zxFzxyF xxxx Differentiating

the above equation with respect to ,y we get )()(),(0),( 2111 zKzxKzxFzxFxx +=⇒=

73

and ),()(),(0),( 4322 zKzxKzxFzxFxx +=⇒= where ),(1 zK ),(2 zK )(3 zK and

)(4 zK are functions of integration. Substituting these values in (2.4.19) we get

( )

).,,(),,(),,(),(

),,,(),,()(),()()()(

433

432

531

43210

zytEzytExXztFztxFX

zytEzytEyxgXzKzxKzKzxKyX

+=

+=

+−=

+++=

(2.4.20)


resulting equation with respect to y we get

( ) ),,(),(),,()(),,(0),,()(),,( 654522,

45 ztFztyFzytEygzytEzytEygzytE ++−=⇒=+

where ),(5 ztF and ),(6 ztF are functions of integration. Equation (2.4.20) becomes

( )

).,,(),,(),,(),(

),,(),(),,()(),,()(),()()()(

433

432

65431

43210

zytEzytExXztFztxFX

ztFztyFzytEygzytEyxgXzKzxKzKzxKyX

+=

+=

++−−=

+++=

(2.4.21)

Considering equation (2.4.11) and using (2.4.21) then differentiating with respect to

,x we get ),,(),,(0),,( 733 ytFzytEzytEz =⇒= where ),(7 ytF is a function of

integration. Now using these information in (2.4.8) and differentiate first with respect

to x then with respect to ,z we get .0)()( 31 =+ zKzyK zzzz Solving this equation by

differentiating with respect to ,y we get .,,)(0)( 212111 ℜ∈+=⇒= ccczczKzK zz

Substituting back we get .,,)(0)( 434333 ℜ∈+=⇒= ccczczKzK zz The system of

equations (2.4.21) become

( )

),,,(),,(),,(),(),,(),()],,(),([)(

),()()()(

473432

65471

443

221

0

zytEzytFxXztFztxFXztFztyFzytEytxFygX

zKczcxzKczcxyX

+=+=

++−−=

+++++=

(2.4.22)

where .,,, 4321 ℜ∈cccc Considering equation (2.4.10) and using (2.4.22) then

differentiating the resulting equation first with respect to x then with respect to ,z we

74

get ),()(),(0),( 2133 tRtzRztFztFzz +=⇒= where )(1 tR and )(2 tR are functions of

integration. Also considering equation (2.4.10) and using equation (2.4.22) then

differentiating the resulting equation first with respect to x then with respect to ,y

we get

.0),())()((),()()(3),()( 722772 =−++ ytFyGygytFyGygytFyG yyy (2.4.23)

We will make use of this equation later on. Refreshing the system (2.4.22) we get

( )

).,,(),,(),,()]()([),,(),()],,(),([)(

),()()()(

4734212

65471

443

221

0

zytEzytFxXztFtRtzRxXztFztyFzytEytxFygX

zKczcxzKczcxyX

+=++=

++−−=

+++++=

(2.4.24)

Considering equation (2.4.9) and using equation (2.4.24), we get

.0),(),(),()( 6572 =++ ztAFztFyAytFyGB zz (2.4.25)

Differentiating (2.4.25) with respect to ,z we get .0),(),( 65 =+ ztFztyF zzzz Solving

this equation by differentiating with respect to ,y we get

),()(),(0),( 4355 tRtzRztFztFzz +=⇒= where )(3 tR and )(4 tR are functions of

integration. Substituting back the above value we get ⇒= 0),(6 ztFzz

),()(),( 656 tRtzRztF += where )(5 tR and )(6 tR are functions of integration. Using

the above values in (2.4.25) and differentiating twice with respect to ,y we get

.0))()()(,(2),()()(4),()( 227772 =−++ yGygytFytFyGygytFyG yyy (2.4.26)

Now subtracting equation (2.4.23) from (2.4.26) we get

.0))()()(,(),()()( 2277 =−+ yGygytFytFyGyg y (2.4.27)

Suppose ,0),(7 ≠ytF then ),()()(

1),()()(

)()(),(),( 77

22

7

7

tRyGyg

ytFyGyg

ygyGytFytFy =⇒

−=

where )(7 tR is the function of integration. Now substituting back all the above

information in (2.4.9) and differentiating the resulting equation with respect to ,y we

75

get 0)(7 =tR which implies ,0),(7 =ytF a contradiction to our assumption. Hence

we must take .0),(7 =ytF From equation (2.4.25) we see that ⇒= 0)(3 tR

),(),( 45 tRztF = ).(),(0)( 665 tRztFtR =⇒= Substituting back the above values in

(2.4.10) and differentiating with respect to ,x we get .0)(1 =tR Also substituting

back the above values in (2.4.5) and differentiating first with respectg to x then with

respcect to ,z we get .01 =c Refreshing (2.4.24) with all the abvoe information, we

get

( )

).,,(),,()(),()(),,()(

),()()(

43422

6441

443

22

0

zytEXztFtxRXtRtyRzytEygX

zKczcxzKxcyX

=+=

++−=

++++=

(2.4.28)

Considering equation (2.4.11) and using (2.4.28) then solving we get

⇒= 0),,(4 zytEz ),,(),,( 84 ytFzytE = where ),(8 ytF is a function of integration.

Refreshing the system of equations (2.4.28) with the help of above information, we

get

( )

).,(),,()(),()(),()(

),()()(

83422

6481

443

22

0

ytFXztFtxRXtRtyRytFygX

zKczcxzKxcyX

=+=

++−=

++++=

(2.4.29)

Differentiating equation (2.4.10) first with respect to x then with respect to z and

using (2.4.29) we get ),()(),(0),( 9844 tRtzRztFztFzz +=⇒= where )(8 tR and

)(9 tR are functions of integration. The system of equations (2.4.29) become

( )

),,(),()()(),()(),()(

),()()(

8310822

6481

443

22

0

ytFXtRtzRtxRXtRtyRytFygX

zKczcxzKxcyX

=++=

++−=

++++=

(2.4.30)

where ).()()( 9610 tRtRtR += Now considering equation (2.4.7) and using equation

(2.4.30) then differentiating the resulting equation with respect to z we get .03 =c

Also considering equation (2.4.8) and using equation (2.4.30) then differentiating the

76

resulting equation with respect to z we get .0)()( 42 =+ zKzyK zzzz Solving this

equation by differentiating with respect to ,y we get ⇒= 0)(2 zK zz

.,,)( 65652 ℜ∈+= ccczczK Substituting back we get ,)(0)( 87

44 czczKzKzz +=⇒=

., 87 ℜ∈cc Refreshing equation (2.4.30) we get

( )

),,(),()()(),()(),()(

,

8310822

6481874652

0

ytFXtRtzRtxRXtRtyRytFygX

czcxcczcxcyX

=++=

++−=

+++++=

(2.4.31)

where .,,,,, 876542 ℜ∈cccccc Considering equation (2.4.6) and using equation

(2.4.31) we get

.0)()( 42 =+ tARtBR (2.4.32)

In order to sovle this equation we will consider the following possibilities:

(I) ,BA ≠ constant≠A and constant.≠B

(II) ,BA ≠ constant≠A and .0\, ℜ∈= ηηB

(III) ,BA ≠ 0\, ℜ∈= λλA and constant.≠B

(IV) ,BA = constant≠A and constant.≠B

(V) ,BA ≠ 0\, ℜ∈= λλA and .0\, ℜ∈= ηηB

(VI) .0\, ℜ∈== λλBA

We will discuss each case in turn.

Case I:

In this case we have ,BA ≠ constant≠A and constant.≠B From equation (2.4.32)

we get .0)()( 42 == tRtR Substituting the above values in (2.4.5) and differentiating

with respect to ,x we get .02 =c Now substituting 02 =c in (2.4.5) and

differentiating with respect to ,z we get ⇒=−+ 02)()(2 588 ctRBtBR tt 05 =c and

77

.,)(

1)( 998 ℜ∈= cc

tBtR Substituting all the above information back in (2.4.5) we

get ⇒=−+ 02)()(2 61010 ctRBtBR tt 06 =c and .,

)(1)( 1010

10 ℜ∈= cctB

tR Also

using the above information in (2.4.7) we get ⇒=−+ 0)(21)( 4

66 ctRAtAR tt 04 =c

and .,)(

1)( 11116 ℜ∈= cc

tAtR Refreshing the system of equations (2.4.31) we get

),,(,)(

1)(

1

,)(

1),()(,

83109

2

1181

870

ytFXctB

ctB

zX

ctA

ytFygXczcX

=+=

+−=+=

(2.4.33)


⇒=−⎥⎦⎤

⎢⎣⎡ + 0)(),(

21),( 7

288 cyGytFBytBF tt 07 =c and 0),(21),( 88 =+ ytFBytBF tt

.)(),( 21

18 −=⇒ ByDytF Considering equation (2.4.10) and using equation (2.4.33)

with all the above information we get

.)(

1)()(

)(1ln)(

)()()()()( 9

1911

yGyGyg

yGcyD

yGc

yDygyDyG y ⎥⎦

⎤⎢⎣

⎡+=⇒−=+ The

solution of equations (2.4.3) to (2.4.11) is given as [67]

,)(

1)()(

)(1ln

)(,

)(1

)(1

,)(

1)()(

)()(

)(1ln

)(

,

93109

2

1191

80

yGyGyg

yGtBc

XctB

ctB

zX

ctAyG

ygyGyg

yGtBc

X

cX

⎥⎦

⎤⎢⎣

⎡+=+=

+⎥⎦

⎤⎢⎣

⎡+−=

=

(2.4.34)

where .,,, 111098 ℜ∈cccc In this case the space-time (2.4.1) admits four linearly

independent teleparallel Killing vector fields, which can be written as ,t∂∂ ,1

xA ∂∂

78

yB ∂∂1 and ].

)(1

)()(

)(1ln

)()(

)()(

)(1ln[1

zyGyGyg

yGyz

xyGyg

yGyg

yGB ∂∂

++∂∂

+∂∂

+−

Killing vector fields in general relativity are ,x∂∂ ,

z∂∂

zz

yGyg

yz

xz

yG ∂∂

−∂∂

+∂∂ sin

)()(cossin

)(1 and .cos

)()(sincos

)(1

zz

yGyg

yz

xz

yG ∂∂

+∂∂

+∂∂

− On

comparison to the Killing vector fields in general relativity we see that all the

teleparallel Killing vector fields are different from Killing vector fields in general

relativity.

Case II:

In this case we have ,BA ≠ constant≠A and .0\, ℜ∈= ηηB From the above

equation (2.4.32) we get .0)()( 42 == tRtR Substituting these values in (2.4.5) and

differentiating with respect to ,x we get .02 =c Now substituting 02 =c in (2.4.5)

and differentiating with respect to ,z we get 05 =c and .,)( 998 ℜ∈= cctR

Substituting all the above information back in (2.4.5) we get 06 =c and

.,)( 101010 ℜ∈= cctR Also using the above information in (2.4.7) we get 04 =c and

.,)(

1)( 11116 ℜ∈= cc

tAtR Refreshing the system of equations (2.4.31) we get

),,(,

,)(

1),()(,

83109

2

1181

870

ytFXczcX

ctA

ytFygXczcX

=+=

+−=+= (2.4.35)

Considering equation (2.4.8) and using equation (2.4.35) we get 07 =c and

).((),(0),( 2188 yDytDytFytFtt +=⇒= Substituting the above information and the

system of equations (2.4.35) in (2.4.8) we get 0)(1 =yD and .07 =c Now using the

above values in (2.4.10) we get

79

.)(

1)()(

)(1ln)(

)()()()()( 9

2922

yGyGyg

yGcyD

yGc

yDygyDyG y ⎥⎦

⎤⎢⎣

⎡+=⇒−=+ The

solution of equations (2.4.3) to (2.4.11) is given as

,)(

1)()(

)(1ln,

,)(

1)()(

)()(

)(1ln,

93

1092

1191

80

yGyGyg

yGcXczcX

ctAyG

ygyGyg

yGcXcX

⎥⎦

⎤⎢⎣

⎡+=+=

+⎥⎦

⎤⎢⎣

⎡+−==

(2.4.36)

where .,,, 111098 ℜ∈cccc In this case the line element for Bianchi type VIII and IX

space-time becomes

,)()(2)()()()( 2222222

dzdxygtAdzyGygtAdydxtAdtds

+++++−= ηη

(2.4.37)

The above space-time (2.4.37) admits four linearly independent teleparallel Killing

vector fields, which can be written as ,t∂∂ ,1

xA ∂∂

y∂∂ and

.)(

1)()(

)(1ln

)()(

)()(

)(1ln

zyGyGyg

yGyz

xyGyg

yGyg

yG ∂∂

++∂∂

+∂∂

+− Killing vector

fields in general relativity are ,x∂∂ ,

z∂∂

zz

yGyg

yz

xz

yG ∂∂

−∂∂

+∂∂ sin

)()(cossin

)(1 and

.cos)()(sincos

)(1

zz

yGyg

yz

xz

yG ∂∂

+∂∂

+∂∂

− On comparison to the Killing vector fields in

general relativity we see that all the teleparallel Killing vector fields are different

from Killing vector fields in general relativity.

Case III:

In this case we have ,BA ≠ 0\, ℜ∈= λλA and constant.≠B In this case the

line element for Bianchi type VIII and IX space-time becomes

,)(2)()()()( 2222222 dzdxygdzyGtBygdytBdxdtds λλλ +++++−= (2.4.38)

80

This case can be solved similarly as the two cases above and the result is given as

[67]

,)(

1)()(

)(1ln

)(,

)(1

)(1

,)()(

)()(

)(1ln

)(

,

93109

2

1191

80

yGyGyg

yGtBc

XctB

ctB

zX

cyGyg

yGyg

yGtBc

X

cX

⎥⎦

⎤⎢⎣

⎡+=+=

+⎥⎦

⎤⎢⎣

⎡+−=

=

(2.4.39)

where .,,, 111098 ℜ∈cccc In this case the above space-time (2.4.38) admits four

linearly independent teleparallel Killing vector fields, which can be written as ,t∂∂

,x∂∂

yB ∂∂1 and ].

)(1

)()(

)(1ln

)()(

)()(

)(1ln[1

zyGyGyg

yGyz

xyGyg

yGyg

yGB ∂∂

++∂∂

+∂∂

+−

Killing vector fields in general relativity are ,x∂∂ ,

z∂∂

zz

yGyg

yz

xz

yG ∂∂

−∂∂

+∂∂ sin

)()(cossin

)(1 and .cos

)()(sincos

)(1

zz

yGyg

yz

xz

yG ∂∂

+∂∂

+∂∂

− On

comparison to the Killing vector fields in general relativity we see that in this case

only one teleparallel Killing vector field is the same as in general relativity and all

others are different from Killing vector fields in general relativity.

Case IV:

In this case we have ,BA = constant≠A and constant.≠B From the above

equation (2.4.32) we get ).()( 42 tRtR −= Using these information in (2.4.7) and solve

after differentiating with respect to ,y we get 02 =c and ℜ∈= 994 ,

)(1)( cc

tAtR

which in turn implies .)(

1)( 92 c

tAtR −= Now substituting these information in

81

(2.4.7) and solving we get 04 =c and .,)(

1)( 10106 ℜ∈= cc

tAtR Our system of

equations (2.4.31) becomes

( )

),,(),()()(

1

,)(

1)(

1),()(

,

831089

2

10981

87650

ytFXtRtzRctA

xX

ctA

ctA

yytFygX

czcczcyX

=++−=

++−=

+++=

(2.4.40)

Considering equation (2.4.5) and using equation (2.4.40) then solve after

differentiating with respect to ,z we get 05 =c and .,)(

1)( 11118 ℜ∈= cc

tAtR Now

using this value in (2.4.5) we get 06 =c and .,)(

1)( 121210 ℜ∈= cc

tAtR Now using

all the above information in (2.4.8) and solving we get 07 =c and ,)()(),(

18

tAyDytF =

where )(1 yD is a function of integration. Now using the above information in

(2.4.10) we get [ ] ⇒−−=)(

1)()()(sin 1192,

1

yGc

yGygcyyD

.)(

1)()(

)(1ln)(ln)( 119

1

yGyGyg

yGcyGcyD ⎥

⎦

⎤⎢⎣

⎡++−= In this case the line element for

Bianchi type VIII and IX space-time becomes

,)()(2)]()([)(])[( 2222222 dzdxygtAdzyGygtAdydxtAdtds +++++−= (2.4.41)

Teleparallel Killing vector fields for the above space-time is given as [67]

82

,)(

1)()(

)(1ln)(ln1

,

,)()(

)()(

)(1ln)(ln1

,

1193

121192

109119

1

80

yGyGyg

yGcyGc

AX

Ac

Azc

Axc

X

Ac

Acy

yGyg

yGyg

yGcyGc

AX

cX

++=

++−=

+−++−−=

=

(2.4.42)

where .,,,, 12111098 ℜ∈ccccc The above space-time (2.4.41) admits five linearly

independent teleparallel Killing vector fields, which can be written as ,t∂∂ ,1

xA ∂∂

,1yA ∂∂ ]

)(1

)()(

)(1ln

)()(

)()(

)(1ln[1

zyGyGyg

yGyz

xyGyg

yGyg

yGA ∂∂

++∂∂

+∂∂

+−

and ].)(ln1)(ln1[z

yGAyA

xxA

yyGA ∂

∂+

∂∂

−∂∂

− Killing vector fields in

general relativity are ,x∂∂ ,

z∂∂

zz

yGyg

yz

xz

yG ∂∂

−∂∂

+∂∂ sin

)()(cossin

)(1 and

.cos)()(sincos

)(1

zz

yGyg

yz

xz

yG ∂∂

+∂∂

+∂∂


general relativity we see that all the teleparallel Killing vector fields are different

from Killing vector fields in general relativity. It is important to note that in this case

the space-time admits one more teleparallel Killing vector field than Killing vector

fields in general relativity.

Case V:

In this case we have ,BA ≠ 0\, ℜ∈= λλA and .0\, ℜ∈= ηηB From the

above equation (2.4.32) we get ).()( 42 tRtRηλ

−= Using these information in (2.4.7)

and solve after differentiating with respect to ,y we get 02 =c and

ℜ∈= 994 ,)( cctR which in turn implies .)( 9

2 ctRηλ

−= Now substituting these

83

information in (2.4.7) and solving we get 04 =c and .,)( 10106 ℜ∈= cctR Our system

of equations (2.4.31) becomes

( )

),,(),()(

,),()(,

831089

2

10981

87650

ytFXtRtzRxcX

cycytFygXczcczcyX

=++−=

++−=+++=

ηλ (2.4.43)

Considering equation (2.4.5) and using eqution (2.4.43) then solve after

differentiating with respect to ,z we get 05 =c and .,)( 11118 ℜ∈= cctR Now using

this value in (2.4.5) and solving we get 06 =c and .,)( 121210 ℜ∈= cctR Now using

all the above information in (2.4.8) and solving we get 07 =c and ),(),( 18 yDytF =

where )(1 yD is a function of integration. Now using the above information in

equation (2.4.10) we get [ ] ⇒−−=)(

1)()()(sin 1192,

1

yGc

yGygcyyD

ηλ

.)(

1)()(

)(1ln)(ln)( 119

1

yGyGyg

yGcyGcyD ⎥

⎦

⎤⎢⎣

⎡++−=

ηλ In this case the line element

for Bianchi type VIII and IX space-time becomes

,)(2)()( 2222222 dzdxygdzyGygdydxdtds ληληλ +++++−= (2.4.44)

Teleparallel Killing vector fields for the above space-time are given as [67]

.)(

1)()(

)(1ln)(ln

,

,)()(

)()(

)(1ln)(ln,

1193

121192

1091191

80

yGyGyg

yGcyGcX

czccxX

ccyyGyg

yGyg

yGcyGcXcX

++−=

++−=

++++−−==

ηλ

ηλ

ηλ

(2.4.45)


independent teleparallel Killing vector fields, which can be written as ,t∂∂ ,

x∂∂

84

,y∂∂ ]

)(1

)()(

)(1ln

)()(

)()(

)(1ln[

zyGyGyg

yGyz

xyGyg

yGyg

yG ∂∂

++∂∂

+∂∂

+− and

].))(

1()(ln)(ln[zyG

yGy

xx

yyG∂∂

−+∂∂

−∂∂

+ηλ

ηλ

ηλ Killing vector fields in

general relativity are ,x∂∂ ,

z∂∂

zz

yGyg

yz

xz

yG ∂∂

−∂∂

+∂∂ sin

)()(cossin

)(1 and

.cos)()(sincos

)(1

zz

yGyg

yz

xz

yG ∂∂

+∂∂

+∂∂


general relativity we see that only one teleparallel Killing vector field is the same and

all others are different from Killing vector fields in general relativity. It is important

to note that in this case the space-time admits one more teleparallel Killing vector

field.

Case VI:

In this case we have .0\, ℜ∈== λλBA The space-time in this case becomes

,)(2)()()( 2222222 dzdxygdzyGygdydxdtds λλλ +++++−= (2.4.46)

This case can be solved exactly the same as case (IV). Teleparallel Killing vector

fields in this case are given as

.)(

1)()(

)(1ln)(ln

,

,)()(

)()(

)(1ln)(ln,

1193

121192

1091191

80

yGyGyg

yGcyGcX

czccxX

ccyyGyg

yGyg

yGcyGcXcX

++−=

++−=

++++−−==

(2.4.47)


independent teleparallel Killing vector fields, which can be written as ,t∂∂ ,

x∂∂

,y∂∂ ]

)(1

)()(

)(1ln

)()(

)()(

)(1ln[

zyGyGyg

yGyz

xyGyg

yGyg

yG ∂∂

++∂∂

+∂∂

+− and

85

].))(

1()(ln)([lnzyG

yGy

xx

yyG∂∂

−+∂∂

−∂∂

+ Killing vector fields in general


z∂∂

zz

yGyg

yz

xz

yG ∂∂

−∂∂

+∂∂ sin

)()(cossin

)(1 and

.cos)()(sincos

)(1

zz

yGyg

yz

xz

yG ∂∂

+∂∂

+∂∂


general relativity we see that only one teleparallel Killing vector field is the same and

all others are different from Killing vector fields in general relativity. It is important

to note that in this case the space-time admits one more teleparallel Killing vector

field.

2.5. Summary of the Chapter

In this chapter we explored Killing vector fields for Bianchi types I, II, VIII and IX

space-times in the context of teleparallel theory of gravitation using direct integration

technique. Following results are obtained from the above study:

(1) In Bianchi type I: Different possibilities for the existence of teleparallel Killing

vector fields have been found by using direct integration technique. It turns out that

the above space-times admit 7, 8 or 10 teleparallel Killing vector fields. When the

space-time becomes Minkowski then all the torsion components become zero and the

teleparallel Lie derivative for the metric gives the same equations as in general

relativity, hence the Killing vector fields are same in both the theories. The results for

above space time when it admits seven independent teleparallel Killing vector fields

are given in equations (2.2.24) and (2.2.36). When the above space time admits eight

independent teleparallel Killing vector fields, results are given in equations (2.2.49),

(2.2.75) and (2.2.88). When the above space time admits ten independent teleparallel

Killing vector fields, result is given in equation (2.2.62). It is important to note that

even the space time possesses ten linearly independent teleparallel Killing vector

fields, it does not become Minkowski and has torsion.

86

(2) In Bianchi type II: Teleparallel Killing vector fields have been explored by direct

integration technique. The above space-time admits teleparallel Killing vector fields

in different cases. It turns out that the above space-times admit 4, 5 or 7 teleparallel

Killing vector fields. It is important to note that when all the metric functions are

constant the torsion does not become zero and hence the teleparallel Killing vector

fields are different than Minkowski space-times in general relativity. When the above

space time admits four independent teleparallel Killing vector fields, result is given in

equation (2.3.29). The result for five independent teleparallel Killing vector fields is

given in equation (2.3.32) and the results for seven independent teleparallel Killing

vector fields are given in equations (2.3.38) and (2.3.47).

(3) In Bianchi type VIII and IX: Teleparallel Killing vector fields have been shown

by using direct integration technique. It turns out that the above space-times admit 4

or 5 teleparallel Killing vector fields. It is important to note that when all the metric

functions are constant the torsion does not become zero and hence the teleparallel

Killing vector fields are different than Killing vector fields in general relativity due to

non vanishing of torsion components. A brief comparison of Killing vector fields in

both the theories for Bianchi types VIII and IX space-times are also given and it

comes out that Killing vector fields in both the theories are different. The result for

four independent teleparallel Killing vector fields are given in equations (2.4.34),

(2.4.36) and (2.4.39). Also the results for five independent teleparallel Killing vector

fields are given in equations (2.4.42), (2.4.45) and (2.4.47).

87

Chapter 3

Teleparallel Killing Vector Fields in

Kantowski-Sachs, Bianchi Type III, Static

Cylindrically Symmetric and Spatially

Homogeneous Rotating Space-Times

3.1. Introduction

This chapter is devoted to investigate teleparallel Killing vector fields in Kantowski-

Sachs, Bianchi type III, static cylindrically symmetric and spatially homogenous

rotating space-times by using direct integration technique. This chapter is organized

as follows: In section (3.2) Killing vector fields of Kantowski-Sachs and Bianchi type

III space-times are investigated. In the next section (3.3) Killing vector fields in

cylindrically symmetric static space-times in the context of teleparallel theory have

been explored. In section (3.4) Killing vector fields of spatially homogeneous rotating

space-times are explored in the context of teleparallel theory. In the same section we

have also discussed some special cases of spatially homogeneous rotating space-

times. Last section (3.5) of the chapter is devoted to a detailed summary of the work.

88

3.2. Teleparallel Killing Vector Fields in

Kantowski-Sachs and Bianchi Type III

Space-Times

Consider Kantowski-Sachs and Bianchi type III space-times in usual coordinates

),,,( φθrt (labeled by ),,,,( 3210 xxxx respectively) with the line element [77]

),)(()()( 22222222 φθθ dfdtBdrtAdtds +++−= (3.2.1)

where A and B are no where zero functions of t only. For θθ sinh)( =f the space-

time (3.2.1) represents Bianchi type III space-time and for θθ sin)( =f it becomes

Kantowski-Sachs space-times. The above space-times (3.2.1) admit at least four

independent Killing vector fields [79], which are

,r∂∂ ,

φ∂∂ ,sin

)()(cos

φφ

θθ

θφ

∂∂′

−∂∂

ff ,cos

)()(sin

φφ

θθ

θφ

∂∂′

+∂∂

ff (3.2.2)

where prime denotes the derivative with respect to .θ The tetrad components µaS

and its inverse µaS can be obtained by using the relation (1.3.4) as [68]

)),()(),(),(,1(diag θµ ftBtBtAS a = ).)()(

1,)(

1,)(

1,1(diagθ

µ

ftBtBtASa = (3.2.3)

The corresponding non-vanishing Weitzenböck connections for (3.2.3) are obtained

by using the relation (1.3.5) as

,101

AA•

=Γ ,202

BB•

=Γ ,303

BB •

=Γ (3.2.4)

where “dot” denotes the derivative with respect to .t The non vanishing torsion

components can be obtained by using (1.3.11) as [68]

89

,011

AAT

•

= ,022

BBT

•

= .033

BBT

•

= (3.2.5)


(1.3.17). One can write (1.3.17) explicitly using (3.2.1) and (3.2.5) as

,0,0,0 2,2

1,1

0,0 === XXX (3.2.6)

,02,12

1,22 =+ XAXB (3.2.7)

,0)( 1,322

3,12 =+ XfBXA θ (3.2.8)

,0)( 2,32

3,2 =+ XfX θ (3.2.9)

,0)()(

3,32 =+

′XX

ffθθ (3.2.10)

,011,

00,

12 =+− • XAAXXA (3.2.11)

,022,

00,

22 =+− • XBBXXB (3.2.12)

.0)()( 323,

00,

322 =+− • XfBBXXfB θθ (3.2.13)

Equation (3.2.6) and (3.2.10) gives

),,,(),,()()(,),,(

),,,(),,,(

43332

2110

θφφθθφ

φθφθ

rtPdrtPffXrtPX

tPXrPX

+′

−==

==

∫ (3.2.14)

where ),,,(1 φθrP ),,,(2 φθtP ),,(3 φrtP and ),,(4 θrtP are functions of integration

which are to be determined. Considering equation (3.2.7) and using equation (3.2.14)

then differentiating the resulting equation with respect to ,θ we get

),,(),(),,(0),,( 2122 φφθφθφθθθ tKtKtPtP +=⇒= where ),(1 φtK and ),(2 φtK are

functions of integration. Also differentiating (3.2.7) with respect to ,r we get

),,(),(),,(0),,( 4333 φφφφ tKtKrrtPrtPrr +=⇒= where ),(3 φtK and ),(4 φtK are

90

functions of integration. Now substituting back the above information in (3.2.7) we

get

.0),(),( 3212 =+ φφ tKBtKA (3.2.15)

In order to solve this equation we impose the following conditions.

(I) ),(tAA = )(tBB = and ,BA ≠ (II) )(tAA = and 0),\R( ∈= ηηB

(III) )(tBB = and 0),\R( ∈= ηηA (IV) ),(tAA = )(tBB = and ,BA =

(V) β=A and η=B where 0).\R,( ∈ηβ

Case (I):

In this case we have ),(tAA = )(tBB = and .BA ≠ The space-time is given in

equation (3.2.1). Substituting the above information in equation (3.2.15) and solving

we get .0),(),( 31 == φφ tKtK The system of equations (3.2.14) becomes

).,,(),()()(,),(

),,(),,,(

44342

2110

θφφθθφ

φφθ

rtPdtKffXtKX

tKXrPX

+′

−==

==

∫ (3.2.16)


resulting equation with respect to ,φ we get ⇒=+ 0),(),( 44 φφφφ tKtK

,sin)(cos)(),( 214 φφφ tEtEtK += where )(1 tE and )(2 tE are functions of

integration. Now substituting back the above value in equation (3.2.9) we get

),,(),,(0),,( 544 rtKrtPrtP =⇒= θθθ where ),(5 rtK is a function of integration.

Substituting back all the above information in (3.2.16), we get

( ) ).,(cos)(sin)()()(

,sin)(cos)(),,(),,,(

5213

2122110

rtKtEtEffX

tEtEXtKXrPX

+−′

−=

+===

φφθθ

φφφφθ (3.2.17)


respect to ,θ we get ),(),(0),( 355 tErtKrtKr =⇒= where )(3 tE is a function of

91

integration. Substituting back the above value in (3.2.8) we get

),(),(0),( 422 tEtKtK =⇒= φφφ where )(4 tE is a function of integration.

Refreshing the system of equations (3.2.17), we get

( ) ).(cos)(sin)()()(

,sin)(cos)(),(),,,(

3213

2124110

tEtEtEffX

tEtEXtEXrPX

+−′

−=

+===

φφθθ

φφφθ (3.2.18)

Considering equation (3.2.11) and using equation (3.2.18) then solving, we get

),,(),,(0),,( 611 φθφθφθ KrPrPr =⇒= where ),(6 φθK is a function of integration

and .,1)( 114 ℜ∈= cc

AtE Our system of equations (3.2.18) become

( ) ).(cos)(sin)()()(

,sin)(cos)(,1),,(

3213

2121

160

tEtEtEffX

tEtEXcA

XKX

+−′

−=

+===

φφθθ

φφφθ (3.2.19)


),(),(0),( 566 φφθφθθ EKK =⇒= where )(5 φE is a function of integration.

Substituting back the above value and solving we get 21 1)( c

BtE = and ,1)( 3

2 cB

tE =

., 32 ℜ∈cc Refreshing the system of equations (3.2.19) we get

( )

( ) ).(sincos)()(1

,sincos1,1),(

323

3

322

1150

tEccff

BX

ccB

XcA

XEX

+−′

=

+===

φφθθ

φφφ (3.2.20)

Considering equation (3.2.13) and using equation (3.2.20) then solving we get

0)(5 =φE and .,1)( 443 ℜ∈= cc

BtE Refreshing equation (3.2.20) we get

92

( )

( ) .1sincos)()(1

,sincos1,1,0

5233

322

110

cB

ccff

BX

ccB

XcA

XX

+−′

=

+===

φφθθ

φφ (3.2.21)

In this case the above space-time (3.2.1) admits four linearly independent teleparallel

Killing vector fields which are ,)(

1rtA ∂∂ ,

)(1

φ∂∂

tB )sin

)()((cos

)(1

φφ

θθ

θφ

∂∂′

−∂∂

ff

tB

and ).cos)()((sin

)(1

φφ

θθ

θφ

∂∂′

+∂∂

ff

tB Killing vector fields in general relativity are

,r∂∂ ,

φ∂∂

φφ

θθ

θφ

∂∂′

−∂∂ sin

)()(cos

ff and .cos

)()(sin

φφ

θθ

θφ

∂∂′

+∂∂

ff Comparison

shows that Killing vector fields in both theories are same in number and teleparallel

Killing vector fields are multiple of the corresponding elements of the inverse tetrad

field.

Case (II):

In this case we have )(tAA = and 0).\( ℜ∈= ηηB The line element for Bianchi

type III and Kantowski-Sachs space-time in this case is given as

),)(()( 22222222 φθθη dfddrtAdtds +++−= (3.2.22)

Substituting the above information in equation (3.2.15) we get

).,(),( 32

21 φηφ tK

AtK −= Now substituting all the above information in equation

(3.2.14), we get

( ) ).,,(),(),()()(

,),(),(),,(),(),,,(

4433

432232

2110

θφφφθθ

φφφφηθφθ

rtPdtKtrKffX

tKtrKXtKtKA

XrPX

++′

−=

+=+−==

∫ (3.2.23)


resulting equation first with respect to r then with respect to ,φ we get

93

⇒=+ 0),(),( 33 φφφφ tKtK ,sin)(cos)(),( 213 φφφ tEtEtK += where )(1 tE and

)(2 tE are functions of integration. Substituting back the above value in (3.2.9) we get

.0),,()(),(),( 4244 =++ ∫ θθφφφ θφ rtPfdtKtK Differentiating this equation with

respect to ,r we get ),,(),(),,(0),,( 6544 rtKtKrtPrtPr +=⇒= θθθθ where

),(5 θtK and ),(6 rtK are functions of integration. Substituting back the above values

in (3.2.9) and differentiating with respect to ,φ we get


)(4 tE are functions of integration. Substituting back the above values in (3.2.9) we

get ),(),(0),( 555 tEtKtK =⇒= θθθ where )(5 tE is a function of integration. Now

refreshing the system of equations (3.2.23) we get

( )( )

( ) ( )).,()(

]sin)(cos)(sin)(cos)([)()(

,sin)(cos)(sin)(cos)(

),,(sin)(cos)(),,,(

65

43213

43212

2212

2110

rtKtE

dtEtEtEtErffX

tEtEtEtErX

tKtEtEA

XrPX

++

+++′

−=

+++=

++−==

∫ φφφφφθθ

φφφφ

φφφηθφθ

(3.2.24)


resulting equation twice with respect to θ then with respect to φ and solving, we get

.0)()( 21 == tEtE Substituting back the above values in (3.2.8) and solving we get

)(),( 66 tErtK = and ),(),( 72 tEtK =φ where )(6 tE and )(7 tE are functions of

integration. Refreshing the system of equations (3.2.24) we get

( ) ),(sin)(cos)()()(

,sin)(cos)(),(),,,(

8433

4327110

tEdtEtEffX

tEtEXtEXrPX

++′

−=

+===

∫ φφφθθ

φφφθ (3.2.25)

where ).()()( 658 tEtEtE += Considering equation (3.2.12) and using equation

(3.2.25) then solving the resulting equation we get ,)(0)( 133 ctEtEt =⇒=

94

ℜ∈=⇒= 21244 ,,)(0)( ccctEtEt and ),,(),,(0),,( 811 φφθφθθ rKrPrP =⇒=

where ),(8 φrK is a function of integration. Our system of equations (3.2.25) become

( ) ).(cossin)()(

,sincos),(),,(

821

3

2127180

tEccffX

ccXtEXrKX

+−′

−=

+===

φφθθ

φφφ (3.2.26)


ℜ∈=⇒= 3388 ,)(0)( cctEtEt and ),(),(0),( 988 rErKrK =⇒= φφφ where )(9 rE

is a function of integration. Now considering equation (3.2.11) and using equation

(3.2.26) then solving, we get 0)(9 =rE and .,1)( 447 ℜ∈= cc

AtE The solution of

equations (3.2.6) to (3.2.13) becomes [68]

( ) ,cossin)()(

,sincos,1,0

3213

212

510

cccffX

ccXcA

XX

+−′

−=

+===

φφθθ

φφ (3.2.27)

where .,,, 4321 ℜ∈cccc The above space-time (3.2.22) admits four linearly

independent teleparallel Killing vector fields which are ,)(

1rtA ∂∂ ,

φ∂∂

φφ

θθ

θφ

∂∂′

−∂∂ sin

)()(cos

ff and .cos

)()(sin

φφ

θθ

θφ

∂∂′

+∂∂

ff Killing vector fields in

general relativity are ,r∂∂ ,

φ∂∂

φφ

θθ

θφ

∂∂′

−∂∂ sin

)()(cos

ff and

.cos)()(sin

φφ

θθ

θφ

∂∂′

+∂∂

ff Comparison shows that only one teleparallel Killing

vector field is different from Killing vector fields in general relativity which is a

multiple of the corresponding element of the inverse tetrad field.

95

Case (III):

In this case we have )(tBB = and 0.\, ℜ∈= ηηA The line element for

Kantowski-Sachs and Bianchi type III space-time after a suitable rescaling of ,r is

given as

),)(()( 2222222 φθθ dfdtBdrdtds +++−= (3.2.28)

Substituting the above values in equation (3.2.15) we get ).,(),( 12

23 φηφ tK

BtK −=

Now substituting all the above information in (3.2.14) we get

).,,(),(),()()(

,),(),(),,(),(),,,(

4412

23

412

2221110

θφφφηθθ

φφηφφθφθ

rtPdtKtKBr

ffX

tKtKBrXtKtKXrPX

+⎟⎟⎠

⎞⎜⎜⎝

⎛+−

′−=

+−=+==

∫ (3.2.29)


resulting equation first with respect to φ then with respect to r and solving we get


)(2 tE are functions of integration. Substituting back the above value in equation

(3.2.9) and differentiating with respect to ,φ we get ⇒=+ 0),(),( 44 φφφφ tKtK

φφφ sin)(cos)(),( 434 tEtEtK += where )(3 tE and )(4 tE are functions of

integration. Once again substituting back the above information in equation (3.2.9)

we get ),,(),,(0),,( 544 rtKrtPrtP =⇒= θθθ where ),(5 rtK is the function of

integration. Now refreshing the system of equations (3.2.29) we get

( )( )

( ) ( )).,(


,sin)(cos)(sin)(cos)(

),,(sin)(cos)(),,,(

5

43212

23

43212

22

221110

rtK

dtEtEtEtEBr

ffX

tEtEtEtEBrX

tKtEtEXrPX

+

+++−′

−=

+++−=

++==

∫ φφφφφηθθ

φφφφη

φφφθφθ

(3.2.30)

96

Considering equation (3.2.8) and using equation (3.230) then differentiating the


.0)()( 21 == tEtE Substituting back the above values in (3.2.8) we get

⇒= 0),(2 φφφ tK ),()(),( 652 tEtEtK += φφ where )(5 tE and )(6 tE are functions of

integration. Now substituting back all the above information in (3.2.8) and solving we

get 0)(5 =tE and ),(),(0),( 1055 tErtKrtKr =⇒= where )(10 tE is a function of


( ) ).(sin)(cos)()()(

,sin)(cos)(),(),,,(

10433

4326110

tEdtEtEffX

tEtEXtEXrPX

++′

−=

+===

∫ φφφθθ

φφφθ (3.2.31)


),,(),(),,(0),,( 7611 φθφθφθφθ KKrrPrPrr +=⇒= where ),(6 φθK and ),(7 φθK

are functions of integration. Substituting back the above value in (3.2.11) and solving

we get ),(),(0),( 766 φφθφθθ EKK =⇒= where )(7 φE is the function of integration.

Substituting all the above information in (3.2.11) we get 0)(7 =φE and

.,)( 336 ℜ∈= cctE Our system of equations (3.2.31) become

( ) ).(sin)(cos)()()(

,sin)(cos)(,),,(

10433

4323

170

tEdtEtEffX

tEtEXcXKX

++′

−=

+===

∫ φφφθθ

φφφθ (3.2.32)


),(),(0),( 877 φφθφθθ EKK =⇒= where )(8 φE is the function of integration and

,1)( 43 c

BtE = .,,1)( 545

4 ℜ∈= cccB

tE Refreshing the system of equations we get

( )

( ) ),(cossin)()(1

,sincos1,),(

1054

3

542

3180

tEccff

BX

ccB

XcXEX

+−′

−=

+===

φφθθ

φφφ (3.2.33)

97

where .,, 543 ℜ∈ccc Now considering equation (3.2.13) and using equation (3.2.33)

then solving we get ,0)(8 =φE .,1)( 6610 ℜ∈= cc

BtE Finally, the solution of

equations (3.2.6) to (3.2.13) becomes

( )

( ) ,1cossin)()(1

,sincos1,,0

6543

542

310

cB

ccff

BX

ccB

XcXX

+−′

−=

+===

φφθθ

φφ (3.2.34)


independent teleparallel Killing vector fields which are ,r∂∂ ,

)(1

φ∂∂

tB

)sin)()((cos

)(1

φφ

θθ

θφ

∂∂′

−∂∂

ff

tB and ).cos

)()((sin

)(1

φφ

θθ

θφ

∂∂′

+∂∂

ff

tB In this case

Killing vector fields in general relativity are ,r∂∂ ,

φ∂∂

φφ

θθ

θφ

∂∂′

−∂∂ sin

)()(cos

ff and

.cos)()(sin

φφ

θθ

θφ

∂∂′

+∂∂

ff Teleparallel Killing vector field are same in number and

one is exactly the same as in general relativity. Other three teleparallel Killing vector

fields are different from Killing vector fields in general relativity.

Case (IV):

In this case we have ),(tAA = )(tBB = and .BA = The line element for Kantowski-

Sachs and Bianchi type III space-time in this case is given as

).)(()( 2222222 φθθ dfddrtAdtds +++−= (3.2.35)

Substituting back the above information in equation (3.2.15) we get

).,(),( 13 φφ tKtK −= Now substituting all the above information in (3.2.14) we get

( ) ).,,(),(),()()(

,),(),(),,(),(),,,(

4413

41221110

θφφφθθ

φφφφθφθ

rtPdtKtrKffX

tKtrKXtKtKXrPX

++−′

−=

+−=+==

∫ (3.2.36)

98










( )( )

( ) ( )).,(


,sin)(cos)(sin)(cos)(),,(sin)(cos)(),,,(

5

43213

43212

221110

rtK

dtEtEtEtErffX

tEtEtEtErXtKtEtEXrPX

+

+++−′

−=

+++−=

++==

∫ φφφφφθθ

φφφφ

φφφθφθ

(3.2.37)

Now considering equation (3.2.8) and using equation (3.2.37) then differentiating the


.0)()( 21 == tEtE Substituting back the above values in (3.2.8) and differentiating

with respect to ,φ we get ⇒= 0),(2 φφφ tK ),()(),( 652 tEtEtK += φφ where )(5 tE

and )(6 tE are functions of integration. Now substituting back all the above

information in (3.2.8) and solving we get ,0)(5 =tE

),(),(0),( 1055 tErtKrtKr =⇒= where )(10 tE is a function of integration.


( ) ).(sin)(cos)()()(

,sin)(cos)(),(),,,(

10433

4326110

tEdtEtEffX

tEtEXtEXrPX

++′

−=

+===

∫ φφφθθ

φφφθ (3.2.38)

99



are functions of integration. Substituting back the above value in (3.2.11) and solving

we get ),(),(0),( 766 φφθφθθ EKK =⇒= where )(7 φE is the function of integration.

Substituting all the above information in (3.2.11) we get 0)(7 =φE and

.,)(

1)( 336 ℜ∈= cc

tAtE Our system of equations (3.2.38) becomes

( ) ).(sin)(cos)()()(

,sin)(cos)(,)(

1),,(

10433

4323

170

tEdtEtEffX

tEtEXctA

XKX

++′

−=

+===

∫ φφφθθ

φφφθ (3.2.39)


),(),(0),( 877 φφθφθθ EKK =⇒= where )(8 φE is the function of integration,

43 1)( c

AtE = and .,,1)( 545

4 ℜ∈= cccA

tE Refreshing the above system of equations

we get

( )

( ) ),(cossin)()(1

,sincos1,)(

1),(

1054

3

542

3180

tEccff

AX

ccA

XctA

XEX

+−′

−=

+===

φφθθ

φφφ (3.2.40)

where .,, 543 ℜ∈ccc Considering equation (3.2.13) and using equation (3.2.40) then

solving we get 0)(8 =φE and .,1)( 6610 ℜ∈= cc

AtE Finally, the solution of

equations (3.2.6) to (3.2.13) becomes [68]

( )

( ) ,)(

1cossin)()(

)(1

,sincos)(

1,)(

1,0

6543

542

310

ctA

ccff

tAX

cctA

XctA

XX

+−′

−=

+===

φφθθ

φφ (3.2.41)

100


independent teleparallel Killing vector fields which are ,)(

1rtA ∂∂ ,

)(1

φ∂∂

tA

)sin)()((cos

)(1

φφ

θθ

θφ

∂∂′

−∂∂

ff

tA and ).cos

)()((sin

)(1

φφ

θθ

θφ

∂∂′

+∂∂

ff

tA Killing vector

fields in general relativity are ,r∂∂ ,

φ∂∂

φφ

θθ

θφ

∂∂′

−∂∂ sin

)()(cos

ff and

.cos)()(sin

φφ

θθ

θφ

∂∂′

+∂∂

ff Comparison shows that Killing vector fields in both

theories are same in number and the teleparallel Killing vector fields are multiple of

the corresponding elements of the inverse tetrad field.

Case (V):

In this case we have β=A 0\ℜ∈β and η=B 0.\ℜ∈η The space-time

(3.2.1) in this case after a suitable rescaling of ,r becomes

),)(( 2222222 φθθη dfddrdtds +++−= (3.2.42)


).,(),( 123 φηφ tKtK −= Substituting back all the above information in (3.2.14) we get

( ) ).,,(),(),()()(

,),(),(),,(),(),,,(

44123

412221110

θφφφηθθ

φφηφφθφθ

rtPdtKtKrffX

tKtKrXtKtKXrPX

++−′

−=

+−=+==

∫ (3.2.43)







101




( )( )

( ) ( )).,(


,sin)(cos)(sin)(cos)(),,(sin)(cos)(),,,(

5

432123

432122

221110

rtK

dtEtEtEtErffX

tEtEtEtErXtKtEtEXrPX

+

+++−′

−=

+++−=

++==

∫ φφφφφηθθ

φφφφη

φφφθφθ

(3.2.44)

Now considering equation (3.2.8) and using equation (3.2.44) then differentiating the


.0)()( 21 == tEtE Substituting back the above values in (3.2.8) and differentiating

with respect to ,φ we get ⇒= 0),(2 φφφ tK ),()(),( 652 tEtEtK += φφ where )(5 tE

and )(6 tE are functions of integration. Now substituting back all the above values in

(3.2.8) and solving, we get 0)(5 =tE and ),(),(0),( 1055 tErtKrtKr =⇒= where

)(10 tE is a function of integration. Refreshing the system of equations (3.2.44) we get

( ) ).(sin)(cos)()()(

,sin)(cos)(),(),,,(

10433

4326110

tEdtEtEffX

tEtEXtEXrPX

++′

−=

+===

∫ φφφθθ

φφφθ (3.2.45)



are functions of integration. Substituting back the above value in (3.2.11) we get

⇒=⇒=− 0)(0),()( 666 tEKtE ttt φθ .,,)( 43436 ℜ∈+= ccctctE Substituting back

this value we get .),( 36 cK =φθ Our system of equations (3.2.45) becomes

( ) ).(sin)(cos)()()(

,sin)(cos)(,),,(

10433

43243

173

0

tEdtEtEffX

tEtEXctcXKrcX

++′

−=

+=+=+=

∫ φφφθθ

φφφθ (3.2.46)

102


),(),(0),( 877 φφθφθθ EKK =⇒= where )(8 φE is the function of integration,

53 )( ctE = and .,,)( 656

4 ℜ∈= ccctE Refreshing the above system of equations

(3.2.46) we get

( )

( ) ),(cossin)()(

,sincos,),(

1065

3

652

4318

30

tEccffX

ccXctcXErcX

+−′

−=

+=+=+=

φφθθ

φφφ (3.2.47)

where .,,, 6543 ℜ∈cccc Now considering equation (3.2.13) and using equation

(3.2.47) then solving we get 78 )( cE =φ and .,,)( 878

10 ℜ∈= ccctE Finally the

solution of equations (3.2.6) to (3.2.13) becomes [68]

( )

( ) ,cossin)()(

,sincos,,

8653

652

431

730

cccffX

ccXctcXcrcX

+−′

−=

+=+=+=

φφθθ

φφ (3.2.48)

where .,,,,, 876543 ℜ∈cccccc In this case the above space-time (3.2.42) admits six

linearly independent teleparallel Killing vector fields which are ,t∂∂ ,

r∂∂ ,

φ∂∂

,t

rr

t∂∂

+∂∂

φφ

θθ

θφ

∂∂′

−∂∂ sin

)()(cos

ff and

φφ

θθ

θφ

∂∂′

+∂∂ cos

)()(sin

ff which are

exactly the same as Killing vector fields in general relativity.

3.3. Teleparallel Killing Vector Fields in Static

Cylindrically Symmetric Space-Times

Consider cylindrically symmetric static space-times in usual coordinates ),,,( zrt θ


,2)(2)(22)(2 dzededrdteds rCrBrA +++−= θ (3.3.1)

103

where ,A B and C are functions of r only. In general relativity the space-time

(3.3.1) admits minimum three Killing vector fields [36] which are .,,φθ ∂∂

∂∂

∂∂t

The

tetrad components and its inverse can be obtained by using the relation (1.3.3) as [69]

),,,1,(diag 2)(

2)(

2)( rCrBrA

a eeeS =µ ).,,1,(diag 2)(

2)(

2)( rCrBrA

a eeeS−−−

=µ (3.3.2)


are obtained as

,2

010 A′

=Γ ,2

212 B′

=Γ ,2

313 C ′

=Γ ,2221

B

e−=Γ (3.3.3)

where dash denotes the derivative with respect to .r The non vanishing torsion

components by using (1.3.11) are [69]

,2

100 AT

′= ,

213

3 CT′

= .2

122 BT

′= (3.3.4)

Now substituting (3.3.1) and (3.3.4) in (1.3.17) we get the teleparallel Killing

equations as follows:

,0,0,0,0 3,3

2,2

1,1

0,0 ==== XXXX (3.3.5)

,02,0)(

0,2)( =− XeXe rArB (3.3.6)

,03,0)(

0,3)( =− XeXe rArC (3.3.7)

,03,2)(

2,3)( =+ XeXe rBrC (3.3.8)

,02

0)(1,

0)(0,

1 =′

−− XeAXeX rArA (3.3.9)

,02

2)(2,

11,

2)( =′

++ XeBXXe rBrB (3.3.10)

,02

3)(3,

11,

3)( =′

++ XeCXXe rCrC (3.3.11)

104

Now integrating equations (3.3.5), we get

),,,(),,,(),,,(),,,(

4332

2110

θ

θθ

rtPXzrtPXztPXzrPX

==

== (3.3.12)

where ),,,(1 zrP θ ),,,(2 ztP θ ),,(3 zrtP and ),,(4 θrtP are functions of integration

which are to be determined. In order to find solution for equations (3.3.5) to (3.3.11)

we will consider each possible form of the metric for static cylindrically symmetric

space-times and then solve each possibility in turn. Following are the possible cases

for the metric where the above space-times admit teleparallel Killing vector fields:

(I) )(),(),( rCCrBBrAA === and .,, CBCABA ≠≠≠

(II)(a) ),(),( rCCrBB == and .tan tconsA =

(II)(b) ),(),( rCCrAA == and .tan tconsB =

(II)(c) ),(),( rBBrAA == and .tan tconsC =

(III)(a) )(),(),( rCCrBBrAA === and ).()( rCrB =

(III)(b) )(),(),( rCCrBBrAA === and ).()( rCrA =

(III)(c) )(),(),( rCCrBBrAA === and ).()( rBrA =

(IV) )(),(),( rCCrBBrAA === and ).()()( rCrBrA ==

(V)(a) )(),(,tan rCCrBBtconsA === and ).()( rCrB =

(V)(b) )(,tan),( rCCtconsBrAA === and ).()( rCrA =

(V)(c) tconsCrBBrAA tan),(),( === and ).()( rBrA =

(VI)(a) )(rAA = and .tan tconsCB ==

(VI)(b) )(rBB = and .tan tconsCA ==

(VI)(c) )(rCC = and .tan tconsBA ==


105

Case (I):

In this case we have ),(rAA = ),(rBB = ),(rCC = ,BA ≠ CA ≠ and .CB ≠ Now


.0),,(),,( 1)(3)( =− zrPezrtPe rAt

rB θθ (3.3.13)

Differentiating equation (3.3.13) with respect to ,t we get

),,(),(),,(0),,( 2133 zrEzrEtzrtPzrtPtt +=⇒= where ),(1 zrE and ),(2 zrE are

functions of integration. Substituting back the above value in equation (3.3.13) we get

⇒= − ),(),,( 11 zrEezrP ABθθ ),,(),(),,( 311 zrEzrEezrP AB += −θθ where ),(3 zrE is

a function of integration. Refreshing the system of equations (3.3.12) we get

).,,(),,(),(),,,(),,(),(

43212

21310

θ

θθ

rtPXzrEzrEtXztPXzrEzrEeX AB

=+=

=+= −

(3.3.14)


.0)],(),([),,( 31)()()(4)( =+− − zrEzrEeertPe zzrArBrA

trC θθ (3.3.15)


),,(),(),,(0),,( 5444 θθθθ rErEtrtPrtPtt +=⇒= where ),(4 θrE and ),(5 θrE are


.0)],(),([),( 31)()()(4)( =+− − zrEzrEeerEe zzrArBrArC θθ Differentiating the above

equation with respect to θ twice, we get ),()(),(0),( 2144 rKrKrErE +=⇒= θθθθθ

where )(1 rK and )(2 rK are functions of integration. Substituting back the above

value in equation (3.3.15) and solving we get )()(),( 31)()(1 rKrKezzrE rBrC += − and

),()(),( 42)()(3 rKrKezzrE rArC += − where )(3 rK and )(4 rK are functions of

integration. Substituting all the above information in (3.3.14) we get

106

( )( )

( ) ).,()()(),,()()(),,,(

),()()()(

5213

231)()(221

42)()(3)()(1)()(0

θθ

θ

θ

rErKrKtXzrErKrKeztXztPX

rKrKezrKerKezXrBrC

rArCrArBrArC

++=

++==

+++=−

−−−

(3.3.16)


.0),(),()(2 2)(5)(1)( =++ zrEerEerKte zrBrCrC θθ (3.3.17)

Differentiating the above equation first with respect to t and then with respect to ,θ

we get 0)(1 =rK and ),()(),(0),( 6555 rKrKrErE +=⇒= θθθθθ where )(5 rK and

)(6 rK are functions of integration. Substituting back the above values in (3.3.17) we

get ),()(),()(),( 75)()(25)()(2 rKrKezzrErKezrE rBrCrBrCz +−=⇒−= −− where )(7 rK

is a function of integration. Substituting all the above information in (3.3.16) we get

( )).()()(),()()(

),,,(),()()(652375)()(32

2142)()(3)()(0

rKrKrtKXrKrKezrKtXztPXrKrKezrKeX

rBrC

rArCrArB

++=+−=

=++=−

−−

θ

θθ (3.3.18)


.0)()(21)()(

)(21)()(

21),,(

4)(4)(2)(

2)(3)(3)(2

=−−−

⎟⎠⎞

⎜⎝⎛ −−−⎟

⎠⎞

⎜⎝⎛ −−

rKerArKerKez

rKeACzrKerKeABztP

rArr

rAr

rC

rCrrr

rBrBrrt θθθ

(3.3.19)

Differentiating equation (3.3.19) with respect to t we get ⇒= 0),,(2 ztPtt θ

),,(),(),,( 762 zEzEtztP θθθ += where ),(6 zE θ and ),(7 zE θ are functions of

integration. Substituting back the above value in (3.3.19) and differentiating with

respect to ,θ we get

.0)()(21),( 3)(3)(6 =−⎟

⎠⎞

⎜⎝⎛ −− rKerKeABzE r

rBrBrrθθ (3.3.20)

Now differentiating the above equation with respect to ,θ we get

),()(),(0),( 9866 zKzKzEzE +=⇒= θθθθθ where )(8 zK and )(9 zK are functions

107


respect to ,z we get .,)(0)( 1188 ℜ∈=⇒= cczKzKz Now substituting this value in

equation (3.3.20) and solving we get,

.,)( 2

)(2

)(

22

)()(2

)(

13 ℜ∈+=

−−

−

∫ cecdreecrKrBrArArBrA

Substituting all the above

information in equation (3.3.19) and differentiating twice with respect to ,z we get

.,,)(0)( 434399 ℜ∈+=⇒= ccczczKzKzz Substituting back this value in (3.3.19) and


.,)( 5

)(2

)(

52

)()(2

)(

32 ℜ∈+=

−−

−

∫ cecdreecrKrCrArArCrA

Now substituting all the above

information in equation (3.3.19) and solving we get,

.,)( 62

)(

62

)(2

)(

44 ℜ∈+=

−−−

∫ cecdreecrKrArArA

Refreshing the system of equations

(3.3.18) we get

).()(

),()(

),,(

,

65)(2

)(

52

)()(2

)(

33

75)()()(2

)(

22

)()(2

)(

12

7431

1

2)(

62

)(2

)(

42

)(

52

)(2

)(

32

)(

22

)(2

)(

10

rKrKetcdreectX

rKrKezectdreectX

zEtcctzctX

ecdreececzdreeczecdreecX

rCrArArCrA

rBrCrBrArArBrA

rArArArArArArArArA

+++=

+−+=

+++=

+++++=

−−

−

−−−

−

−−−−−−−−−

∫

∫

∫∫∫

θ

θθ

θθ

(3.3.21)


.0)()(21)()()()](

21)([

)]()([21)]()([

21),(2

7)(7)(5)(5)(

2)(

22

)(2

)(

17

1

=++−−−

−+−++ ∫−

rKerBrKerKezrKerBrCz

erBrActdreerBrActzEtc

rBrr

rBr

rCrCrr

rA

rr

rArA

rrθθ (3.3.22)


.0)]()([21)]()([

212 2

)(

22

)(2

)(

11 =−+−+ ∫− rA

rr

rArA

rr erBrAcdreerBrAcc Solving this

equation and remember that in this case ,0)()( ≠− rBrA rr we get ,01 =c which on

108

back substitution gives us .02 =c Substituting the above information in equation

(3.3.22) and differentiating with respect to ,θ we get


of integration. Now substituting the above information in equation (3.3.22) and

differentiating twice with respect to ,z we get

.,,)(0)( 87871010 ℜ∈+=⇒= ccczczKzKzz Substituting back these information in

equation (3.3.22) and differentiating the resulting equation with respect to ,z we get

,)(0)()(21 )(

2)(

92

)()(2

)(

755)(5)(

7

rCrBrBrCrB

rrCrC

rr ecdreecrKrKerKeBCc−

−−

+=⇒=−⎟⎠⎞

⎜⎝⎛ −− ∫

.9 ℜ∈c Now substituting back all the above information in equation (3.3.22) we get

.,)(0)()()(21

102

)(

102

)(2

)(

877)(7)(

8 ℜ∈+−=⇒=++−−−

∫ cecdreecrKrKerKerBcrBrBrB

rrBrB

r


).(

,

),(

,

6)(2

)(

92

)()(2

)(

7

)(2

)(

52

)()(2

)(

33

2)(

102

)(2

)(

82

)(

92

)(2

)(

72

118743

1

2)(

62

)(2

)(

42

)(

52

)(2

)(

30

rKecdreecetcdreectX

ecdreececzdreeczX

zKczctcctzX

ecdreececzdreeczX

rCrBrBrCrBrCrArArCrA

rBrBrBrBrBrB

rArArArArArA

++++=

+−−−=

++++=

+++=

−−

−−−

−

−−−−−−

−−−−−−

∫∫

∫∫

∫∫

θθ

θθ

(3.3.23)


.0)()(21)()]()([

21

)]()([21)]()([

21

)]()([21)(22

6)(6)(2)(

9

2)(

2)(

72

)(

5

2)(

2)(

311

73

=++−+

−+−+

−+++

∫

∫−

−

rKerCrKeerCrBc

dreerCrBcerCrAct

dreerCrActzKctc

rCrr

rCrB

rr

rBrB

rr

rA

rr

rArA

rrz

θ

θ

θ

(3.3.24)


.0)]()([21)]()([

212 2

)(

52

)(2

)(

33 =−+−+ ∫− rA

rr

rArA

rr erCrAcdreerCrAcc Solving this

109

equation and remember that in this case ,0)()( ≠− rCrA rr we get ,03 =c which on



0)]()([21)]()([

212 2

)(

92

)(2

)(

77 =−+−+ ∫− rB

rr

rBrB

rr erCrBcdreerCrBcc Solving this

equation and remember that in this case ,0)()( ≠− rCrB rr we get ,07 =c which on

back substitution gives us .09 =c Now substituting the above information in equation

(3.3.24) and differentiating twice with respect to ,z we get



,)(0)()()(21 2

)(

132

)(2

)(

1166)(6)(

11

rCrCrC

rrCrC

r ecdreecrKrKerKerCc−−−

+−=⇒=++ ∫

.13 ℜ∈c Refreshing the system of equations (3.3.23) we get

,,

,,

2)(

132

)(2

)(

1132

)(

102

)(2

)(

82

12118412

)(

62

)(2

)(

40

rCrCrCrBrBrB

rArArA

ecdreecXecdreecX

czcctcXecdreecX−−−−−−

−−−

+−=+−=

+++=+=

∫∫

∫ θ

(3.3.25)

where .,,,,,, 13121110864 ℜ∈ccccccc The line element for static cylindrically

symmetric space-times is given in equation (3.3.1). The above space-time admits

seven linearly independent teleparallel Killing vector fields which are ,r∂∂ ,2

)(

te

rA

∂∂−

,2)(

θ∂∂− rB

e ,2)(

ze

rC

∂∂−

,)(1

trM

rt

∂∂

+∂∂

θθ

∂∂

+∂∂ )(2 rMr

and ,)(3

zrM

rz

∂∂

+∂∂

where ,)( 2)(

2)(

1 ∫−−

= dreerMrArA

dreerMrBrB

∫−−

−= 2)(

2)(

2 )( and

.)( 2)(

2)(

3 ∫−−

−= dreerMrCrC

Killing vector fields in general relativity are ,t∂∂

θ∂∂ and

110

.z∂∂ It is evident that teleparallel Killing vector fields are different and more in

number to the Killing vector fields in general relativity.

Case (II)(a):

In this case we have ,constant=A ),(rBB = )(rCC = and .CB ≠ Now


.0),,(),,( 13)( =− zrPzrtPe trB θθ (3.3.26)




⇒= ),(),,( 11 zrEezrP Bθθ ),,(),(),,( 311 zrEzrEezrP B +=θθ where ),(3 zrE is a


).,,(),,(),(),,,(),,(),(

43212

21310

θ

θθ

rtPXzrEzrEtXztPXzrEzrEeX B

=+=

=+= (3.3.27)


.0),(),(),,( 31)(4)( =−− zrEzrEertPe zzrB

trC θθ (3.3.28)




.0),(),(),( 31)(4)( =−− zrEzrEerEe zzrBrC θθ Differentiating the above equation with

respect to θ twice, we get ),()(),(0),( 2144 rKrKrErE +=⇒= θθθθθ where )(1 rK

and )(2 rK are functions of integration. Substituting back the above value in equation

(3.3.28) and solving we get )()(),( 31)()(1 rKrKezzrE rBrC += − and

),()(),( 42)(3 rKrKezzrE rC += where )(3 rK and )(4 rK are functions of integration.

Substituting all the above information in (3.3.27) we get

111

( )( )

( ) ).,()()(),,()()(),,,(

),()()()(

5213

231)()(221

42)(3)(1)(0

θθ

θ

θ

rErKrKtXzrErKrKeztXztPX

rKrKezrKerKezXrBrC

rCrBrC

++=

++==

+++=− (3.3.29)



Differentiating the above equation first with respect to t and then with respect to ,θ

we get 0)(1 =rK and ),()(),(0),( 6555 rKrKrErE +=⇒= θθθθθ where )(5 rK and

)(6 rK are functions of integration. Substituting back the above values in (3.3.30) we

get ),()(),()(),( 75)()(25)()(2 rKrKezzrErKezrE rBrCrBrCz +−=⇒−= −− where )(7 rK

is a function of integration. Substituting all the above information in (3.3.29) we get

).()()(),()()(),,,(),()()(

652375)()(32

2142)(3)(0

rKrKrtKXrKrKezrKtXztPXrKrKezrKeX

rBrC

rCrB

++=+−=

=++=− θ

θθ (3.3.31)


.0)()()()()()()(),,( 42)(2)(3)(3)(2 =−−−−− rKrKezrKerCzrKerKerBztP rrrCrC

rrrBrB

rt θθθ (3.3.32)





.0)()()(),( 3)(3)(6 =−− rKerKerBzE rrBrB

rθθ (3.3.33)





equation (3.3.33) and solving we get, .,)( 2)(

2)(

13 ℜ∈+= −− cecrecrK rBrB

112

Substituting all the above information in equation (3.3.32) and differentiating twice

with respect to ,z we get .,,)(0)( 434399 ℜ∈+=⇒= ccczczKzKzz Substituting back

this value in (3.3.32) and solve after differentiating with respect to ,z we get

.,)( 5)(

5)(

32 ℜ∈+= −− cecrecrK rCrC Now substituting all the above information in

equation (3.3.32) and solving we get, .,)( 6644 ℜ∈+= ccrcrK Refreshing the


).()(),()(

),,(,65)(

5)(

3375)()()(

2)(

12

7431

1645321

0

rKrKetcectrXrKrKezecterctX

zEtcctzctXcrcczcrzcrcXrCrCrBrCrBrB +++=+−+=

+++=+++++=−−−−− θ

θθθθ(3.3.34)


.0)()(21)()(

)()](21)([)(

21)(

21),(2

7)(7)(5)(

5)(21

71

=++−

−−−−+

rKerBrKerKez

rKerBrCzrBctrBctrzEtc

rBrr

rBr

rC

rCrrrrθθ

(3.3.35)


.0)(21)(

212 211 =−− rBcrBcrc rr Solving this equation and remember that in this case

,0)( ≠rBr we get ,01 =c which on back substitution gives us .02 =c Substituting

the above information in equation (3.3.35) and differentiating with respect to ,θ we

get ),()(),(0),( 111077 zKzKzEzE +=⇒= θθθθθ where )(10 zK and )(11 zK are

functions of integration. Now substituting the above information in equation (3.3.35)

and differentiating twice with respect to ,z we get



,)(0)()(21 )(

2)(

92

)()(2

)(

755)(5)(

7

rCrBrBrCrB

rrCrC

rr ecdreecrKrKerKeBCc−

−−

+=⇒=−⎟⎠⎞

⎜⎝⎛ −− ∫

.9 ℜ∈c Now substituting back all the above information in equation (3.3.35) we get

113

.,)(0)()()(21

102

)(

102

)(2

)(

877)(7)(

8 ℜ∈+−=⇒=++−−−


rrBrB

r


).(

,

),(,

6)(2

)(

92

)()(2

)(

7)(

5)(

33

2)(

102

)(2

)(

82

)(

92

)(2

)(

72

118743

16453

0

rKecdreecetcerctX

ecdreececzdreeczX

zKczctcctzXcrcczcrzX

rCrBrBrCrBrCrC

rBrBrBrBrBrB

++++=

+−−−=

++++=+++=

−−

−−−

−−−−−−

∫

∫∫θθ

θθ

(3.3.36)


.0)()(21)()]()([

21

)]()([21)(

21)(

21)(22

6)(6)(2)(

9

2)(

2)(

75311

73

=++−+

−+−−++ ∫−

rKerCrKeerCrBc

dreerCrBcrCctrCctrzKctc

rCrr

rCrB

rr

rBrB

rrrrz

θ

θθ

(3.3.37)


.0)(21)(

212 533 =−− rCcrCrcc rr Solving this equation and remember that in this case

,0)( ≠rCr we get ,03 =c which on back substitution gives us .05 =c Substituting the

above information in equation (3.3.37) and differentiating with respect to ,θ we get

0)]()([21)]()([

212 2

)(

92

)(2

)(

77 =−+−+ ∫− rB

rr

rBrB

rr erCrBcdreerCrBcc Solving this

equation and remember that in this case ,0)()( ≠− rCrB rr we get ,07 =c which on

back substitution gives us .09 =c Now substituting the above information in equation

(3.3.37) and differentiating twice with respect to ,z we get



,)(0)()()(21 2

)(

132

)(2

)(

1166)(6)(

11

rCrCrC

rrCrC

r ecdreecrKrKerKerCc−−−

+−=⇒=++ ∫.13 ℜ∈c Refreshing the system of equations (3.3.36) we get

114

,,

,,

2)(

132

)(2

)(

1132

)(

102

)(2

)(

82

1211841

640

rCrCrCrBrBrB

ecdreecXecdreecX

czcctcXcrcX−−−−−−

+−=+−=

+++=+=

∫∫

θ

(3.3.38)

where .,,,,,, 13121110864 ℜ∈ccccccc The line element for static cylindrically

symmetric space-times is given as

,2)(2)(222 dzededrdtds rCrB +++−= θ (3.3.39)

The above space-time admits seven linearly independent teleparallel Killing vector

fields which are ,r∂∂ ,

t∂∂ ,2

)(

θ∂∂− rB

e ,2)(

ze

rC

∂∂−

,t

rr

t∂∂

+∂∂

θθ

∂∂

+∂∂ )(1 rMr

and

,)(2

zrM

rz

∂∂

+∂∂ where dreerM

rBrB

∫−−

−= 2)(

2)(

1 )( and .)( 2)(

2)(

2 ∫−−

−= dreerMrCrC

Killing vector fields in general relativity are ,t∂∂

θ∂∂ and .

z∂∂ It is evident that

teleparallel Killing vector fields are different and more in number to the Killing

vector fields in general relativity. Cases (II)(b) and (II)(c) can be solved exactly the


Case (III)(a):

In this case we have ),(rAA = ),(rBB = ),(rCC = ,BA ≠ CA ≠ and .CB = Now


.0),,(),,( 1)(3)( =− zrPezrtPe rAt

rB θθ (3.3.40)




⇒= − ),(),,( 11 zrEezrP ABθθ ),,(),(),,( 311 zrEzrEezrP AB += −θθ where ),(3 zrE is

a function of integration. Refreshing the system of equations (3.3.12) we get

115

).,,(),,(),(),,,(),,(),(

43212

21310

θ

θθ

rtPXzrEzrEtXztPXzrEzrEeX AB

=+=

=+= −

(3.3.41)



trB θθ (3.3.42)




.0)],(),([),( 31)()()(4)( =+− − zrEzrEeerEe zzrArBrArB θθ Differentiating the above

equation with respect to θ twice, we get ),()(),(0),( 2144 rKrKrErE +=⇒= θθθθθ

where )(1 rK and )(2 rK are functions of integration. Substituting back the above

value in equation (3.3.42) and solving we get )()(),( 311 rKrKzzrE += and

),()(),( 42)()(3 rKrKezzrE rArB += − where )(3 rK and )(4 rK are functions of


( )( )

( ) ).,()()(),,()()(),,,(

),()()()(

5213

231221

42)()(3)()(1)()(0

θθ

θ

θ

rErKrKtXzrErKrKztXztPX

rKrKezrKerKezX rArBrArBrArB

++=

++==

+++= −−−

(3.3.43)


.0),(),()(2 251 =++ zrErErtK zθθ Differentiating this equation first with respect to t

and then with respect to ,θ we get 0)(1 =rK and

),()(),(0),( 6555 rKrKrErE +=⇒= θθθθθ where )(5 rK and )(6 rK are functions

of integration. Substituting back the above values, we get

),()(),()(),( 75252 rKrzKzrErKzrEz +−=⇒−= where )(7 rK is a function of


( )).()()(),()()(

),,,(),()()(65237532

2142)()(3)()(0

rKrKrtKXrKrKzrKtXztPXrKrKezrKeX rArBrArB

++=+−=

=++= −−

θ

θθ (3.3.44)

116


.0)()(21)()(

)(21)()(

21),,(

4)(4)(2)(

2)(3)(3)(2

=−−−

⎟⎠⎞

⎜⎝⎛ −−−⎟

⎠⎞

⎜⎝⎛ −−

rKerArKerKez

rKeABzrKerKeABztP

rArr

rAr

rB

rBrrr

rBrBrrt θθθ

(3.3.45)





.0)()(21),( 3)(3)(6 =−⎟

⎠⎞


rBrBrrθθ (3.3.46)





equation (3.3.45) and solving we get,

.,)( 2

)(2

)(

22

)()(2

)(

13 ℜ∈+=

−−

−


Substituting all the above

information in equation (3.3.45) and differentiating twice with respect to ,z we get



.,)( 5

)(2

)(

52

)()(2

)(

32 ℜ∈+=

−−

−




.,)( 62

)(

62

)(2

)(

44 ℜ∈+=

−−−



(3.3.44) we get

117

).()(

),()(

),,(

,

65)(2

)(

52

)()(2

)(

33

75)(2

)(

22

)()(2

)(

12

7431

1

2)(

62

)(2

)(

42

)(

52

)(2

)(

32

)(

22

)(2

)(

10

rKrKetcdreectX

rKrKzectdreectX

zEtcctzctX


rBrArArBrA

rBrArArBrA

rArArArArArArArArA

+++=

+−+=

+++=

+++++=

−−

−

−−

−

−−−−−−−−−

∫

∫

∫∫∫

θ

θθ

θθ

(3.3.47)


.0)()(21)()()()(

21

)]()([21)]()([

21),(2

7)(7)(5)(5)(

2)(

22

)(2

)(

17

1

=++−−

−+−++ ∫−

rKerBrKerKezrKerBz


rBrr

rBr

rBrBr

rA

rr

rArA

rrθθ (3.3.48)


.0)]()([21)]()([

212 2

)(

22

)(2

)(

11 =−+−+ ∫− rA

rr

rArA










,)(0)()(21 2

)(

92

)(2

)(

755)(5)(

7

rBrBrB

rrBrB

r ecdreecrKrKerKeBc−−−

+=⇒=−− ∫ .9 ℜ∈c Now

substituting back all the above information in equation (3.3.48) we get

.,)(0)()()(21

102

)(

102

)(2

)(

877)(7)(

8 ℜ∈+−=⇒=++−−−


rrBrB

r


118

).(

,

),(

,

62)(

92

)(2

)(

7

)(2

)(

52

)()(2

)(

33

2)(

102

)(2

)(

82

)(

92

)(2

)(

72

118743

1

2)(

62

)(2

)(

42

)(

52

)(2

)(

30

rKecdreecetcdreectX

ecdreececzdreeczX

zKczctcctzX

ecdreececzdreeczX

rBrBrBrBrArArBrA

rBrBrBrBrBrB

rArArArArArA

++++=

+−−−=

++++=

+++=

−−−−

−−

−−−−−−

−−−−−−

∫∫

∫∫

∫∫

θθ

θθ

(3.3.49)


.0)()(21)()]()([

21

)]()([21)(22

6)(6)(2)(

5

2)(

2)(

311

73

=++−+

−+++ ∫−

rKerBrKeerBrAct

dreerBrActzKctc

rBrr

rBrA

rr

rArA

rrzθ

(3.3.50)


.0)]()([21)]()([

212 2

)(

52

)(2

)(

33 =−+−+ ∫− rA

rr

rArA




(3.3.50) and differentiating with respect to ,θ we get .07 =c Now substituting the

above information in equation (3.3.50) and differentiating twice with respect to ,z we

get .,,)(0)( 121112111111 ℜ∈+=⇒= ccczczKzKzz Substituting back these information

in equation (3.3.50) and differentiating the resulting equation with respect to ,z we

get ,)(0)()()(21 2

)(

132

)(2

)(

1166)(6)(

11

rBrBrB

rrBrB

r ecdreecrKrKerKerBc−−−

+−=⇒=++ ∫


,

,

,,

2)(

132

)(

92

)(2

)(

113

2)(

102

)(

92

)(2

)(

82

12118412

)(

62

)(2

)(

40

rBrBrBrB

rBrBrBrB

rArArA

ececdreecX

ecezcdreecX

czcctcXecdreecX

−−−−

−−−−

−−−

++−=

+−−=

+++=+=

∫

∫

∫

θ

θ

(3.3.51)

119

where .,,,,,,, 131211109864 ℜ∈cccccccc The line element for static cylindrically


),( 22)(22)(2 dzdedrdteds rBrA +++−= θ (3.3.52)


fields which are ,r∂∂ ,2

)(

te

rA

∂∂−

,2)(

θ∂∂− rB

e ,2)(

ze

rB

∂∂−

),(2)(

θθ

∂∂

−∂∂−

zz

erB

,)(1

trM

rt

∂∂

+∂∂

θθ

∂∂

+∂∂ )(2 rMr

and ,)(2

zrM

rz

∂∂

+∂∂ where

∫−−

= dreerMrArA

2)(

2)(

1 )(

and .)( 2)(

2)(

2 dreerMrBrB

∫−−

−= Killing vector fields in

general relativity are ,t∂∂ ,

θ∂∂

z∂∂ and ).(

θθ

∂∂

−∂∂ zz

It is evident that teleparallel

Killing vector fields are different and more in number to the Killing vector fields in

general relativity. Cases (III)(b) and (III)(c) can be solved exactly the same as in the

above case.

Case (IV):

In this case we have )()()( rCrBrA == and .0)( ≠rAr Now substituting equation

(3.3.12) in equation (3.3.6), we get

.0),,(),,( 13 =− zrPzrtPt θθ (3.3.53)




⇒= ),(),,( 11 zrEzrP θθ ),,(),(),,( 311 zrEzrEzrP +=θθ where ),(3 zrE is a function

of integration. Refreshing the system of equations (3.3.12) we get

).,,(),,(),(),,,(),,(),(

43212

21310

θ

θθ

rtPXzrEzrEtXztPXzrEzrEX

=+=

=+= (3.3.54)

120


.0),(),(),,( 314 =−− zrEzrErtP zzt θθ (3.3.55)




.0),(),(),( 314 =−− zrEzrErE zzθθ Differentiating the above equation with respect to

θ twice, we get ),()(),(0),( 2144 rKrKrErE +=⇒= θθθθθ where )(1 rK and

)(2 rK are functions of integration. Substituting back the above values in equation

(3.3.55) and solving we get )()(),( 311 rKrKzzrE += and

),()(),( 423 rKrKzzrE += where )(3 rK and )(4 rK are functions of integration.


( )( ) ( ) ).,()()(),,()()(

),,,(),()()()(52132312

2142310

θθ

θθ

rErKrKtXzrErKrKztXztPXrKreKzrKrKzX++=++=

=+++= (3.3.56)








).()()(),()()(),,,(),()()(

65237532

214230

rKrKrtKXrKrKzrKtXztPXrKrKzrKX

++=+−=

=++=

θ

θθ (3.3.57)


121

.0)()(21)()(

)()(21)()()(

21),,(

4)(4)(2)(

2)(3)(3)(2

=−−−

−−−

rKerArKerKez

rKerAzrKerKerAztP

rArr

rAr

rA

rArr

rArArt θθθ

(3.3.58)





.0)()()(21),( 3)(3)(6 =−− rKerKerAzE r

rArArθθ (3.3.59)





equation (3.3.59) and solving we get, .,)( 22

)(

22

)(2

)(

13 ℜ∈+=

−−−





.,)( 52

)(

52

)(2

)(

32 ℜ∈+=

−−−




.,)( 62

)(

62

)(2

)(

44 ℜ∈+=

−−−



(3.3.57) we get

122

).()(

),()(),,(

,

652)(

52

)(2

)(

33

752)(

22

)(2

)(

127

4311

2)(

62

)(2

)(

42

)(

52

)(2

)(

32

)(

22

)(2

)(

10

rKrKetcdreectX

rKrKzectdreectXzEtcctzctX


rArArA

rArArA

rArArArArArArArArA

+++=

+−+=+++=

+++++=

−−−

−−−

−−−−−−−−−

∫

∫

∫∫∫

θ

θθ

θθ

(3.3.60)


.0)()(21)()()()(

21),(2 7)(7)(5)(5)(7

1 =++−−+ rKerArKerKezrKerAzzEtc rArr

rAr

rArArθθ (3.3.61)

Differentiating equation (3.3.61) with respect to ,t we get .01 =c Substituting back

this information and differentiating with respect to ,θ we get






,)(0)()(21 2

)(

92

)(2

)(

755)(5)(

7

rArArA

rrArA

r ecdreecrKrKerKeAc−−−

+=⇒=−− ∫ .9 ℜ∈c Now


.,)(0)()()(21

102

)(

102

)(2

)(

877)(7)(

8 ℜ∈+−=⇒=++−−−

∫ cecdreecrKrKerKerAcrArArA

rrArA

r


).(

,

),(

,

62)(

92

)(2

)(

72

)(

52

)(2

)(

33

2)(

102

)(2

)(

82

)(

92

)(2

)(

72

)(

22

118743

1

2)(

62

)(2

)(

42

)(

22

)(

52

)(2

)(

30

rKecdreecetcdreectX

ecdreececzdreeczectX

zKczctcctzX

ecdreecececzdreeczX

rArArArArArA

rArArArArArArA

rArArArArArArA

++++=

+−−−=

++++=

++++=

−−−−−−

−−−−−−−

−−−−−−−

∫∫

∫∫

∫∫

θθ

θθ

θ

(3.3.62)


123

.0)()(21)()(22 6)(6)(11

73 =++++ rKerArKezKctc rArr

rAzθ (3.3.63)

Differentiating equation (3.3.63) with respect to t and θ respectively, we get

.073 == cc Now substituting the above information in equation (3.3.63) and




0)()()(21 6)(6)(

11 =++ rKerKerAc rrArA

r

,)( 2)(

132

)(2

)(

116

rArArA

ecdreecrK−−−

+−=⇒ ∫ .13 ℜ∈c Refreshing the system of equations

(3.3.62) we get

.

,

,,

2)(

132

)(

92

)(

52

)(2

)(

113

2)(

102

)(

92

)(

22

)(2

)(

82

12118412

)(

62

)(2

)(

42

)(

22

)(

50

rArArArArA

rArArArArA

rArArArArA

ececetcdreecX

ececzectdreecX

czcctcXecdreecececzX

−−−−−

−−−−−

−−−−−

+++−=

+−+−=

+++=+++=

∫

∫

∫

θ

θθ

(3.3.64)

where .,,,,,,,,, 13121110986542 ℜ∈cccccccccc The line element for static cylindrically


),( 222)(22 dzddtedrds rA ++−+= θ (3.3.65)

The above space-time admits ten linearly independent teleparallel Killing vector


)(

te

rA

∂∂−

,2)(

θ∂∂− rA

e ,2)(

ze

rA

∂∂−

),(2)(

θθ

∂∂

−∂∂−

zz

erA

),(2)(

θθ

∂∂

+∂∂−

tt

erA

),(2)(

zt

tze

rA

∂∂

+∂∂−

,)(t

rMr

t∂∂

+∂∂

θθ

∂∂

−∂∂ )(rMr

and

,)(z

rMr

z∂∂

−∂∂ where .)( 2

)(2

)(

∫−−

= dreerMrArA


124

relativity are ,t∂∂ ,

θ∂∂ ,

z∂∂ ),(

zt

tz

∂∂

+∂∂ )(

θθ

∂∂

+∂∂ tt

and ).(θ

θ∂∂

−∂∂ zz

It is

evident that teleparallel Killing vector fields are different and more in number to the


Case (V)(a):

In this case we have ,constant=A ),(rBB = )(rCC = and .CB = Now substituting


.0),,(),,( 13)( =− zrPzrtPe trB θθ (3.3.66)






).,,(),,(),(),,,(),,(),(

43212

21310

θ

θθ

rtPXzrEzrEtXztPXzrEzrEeX B

=+=

=+= (3.3.67)



trB θθ (3.3.68)




.0),(),(),( 31)(4)( =−− zrEzrEerEe zzrBrB θ Differentiating the above equation with




125

),()(),( 42)(3 rKrKezzrE rB += where )(3 rK and )(4 rK are functions of integration.


( )( )

( ) ).,()()(),,()()(),,,(

),()()()(

5213

231221

42)(3)(1)(0

θθ

θ

θ


rKrKezrKerKezX rBrBrB

++=

++==

+++=

(3.3.69)








( )).()()(),()()(

),,,(),()()(65237532

2142)(3)(0

rKrKrtKXrKrKzrKtXztPXrKrKezrKeX rBrB

++=+−=

=++=

θ

θθ (3.3.70)


.0)()()()()()()(),,( 42)(2)(3)(3)(2 =−−−−− rKrKezrKerBzrKerKerBztP rrrBrB

rrrBrB

rt θθθ (3.3.71)




respect to ,θ we get .0)()()(),( 3)(3)(6 =−− rKerKerBzE rrBrB

rθθ Now differentiating this

equation with respect to ,θ we get ),()(),(0),( 9866 zKzKzEzE +=⇒= θθθθθ

where )(8 zK and )(9 zK are functions of integration. Substituting back the above

value and differentiating with respect to ,z we get .,)(0)( 1188 ℜ∈=⇒= cczKzKz

Now substituting back this value in the above equation and solving we get,

.,)( 2)(

2)(

13 ℜ∈+= −− cecercrK rBrB Substituting all the above information in

126

equation (3.3.71) and differentiating twice with respect to ,z we get



.,)( 5)(

5)(

32 ℜ∈+= −− cecercrK rBrB Now substituting all the above information in

equation (3.3.71) and solving we get, .,)( 6644 ℜ∈+= ccrcrK Refreshing the


).()(),()(

),,(,65)(

5)(

3375)(

2)(

12

7431

1645321

0

rKrKetcectrXrKrKzecterctX

zEtcctzctXcrcczcrzccrXrBrBrBrB +++=+−+=

+++=+++++=−−−− θ

θθθθ (3.3.72)


.0)()(21)(

)()()(21)(

21)(

21),(2

7)(7)(

5)(5)(21

71

=++

−−−−+

rKerBrKe

rKezrKerBzrBctrBrctzEtc

rBrr

rB

rrBrB

rrrθθ

(3.3.73)


.0)(21)(

212 211 =−− rBctrBrctc rr Solving this equation and remember that in this

case ,0)( ≠rBr we get ,01 =c which on back substitution gives us .02 =c


respect to ,θ we get ),()(),(0),( 111077 zKzKzEzE +=⇒= θθθθθ where )(10 zK and

)(11 zK are functions of integration. Now substituting the above information in




0)()(21 5)(5)(

7 =−− rKerKeBc rrBrB

r ,)( 2)(

92

)(2

)(

75

rBrBrB

ecdreecrK−−−

+=⇒ ∫ .9 ℜ∈c

Now substituting back all the above information in equation (3.3.73) we get

127

0)()()(21 7)(7)(

8 =++ rKerKerBc rrBrB

r

.,)( 102

)(

102

)(2

)(

87 ℜ∈+−=⇒

−−−

∫ cecdreecrKrBrBrB


(3.3.72) we get

).(

,

),(,

62)(

92

)(2

)(

7)(

5)(

33

2)(

102

)(2

)(

82

)(

92

)(2

)(

72

118743

16453

0

rKecdreecetcerctX

ecdreececzdreeczX

zKczctcctzXccrczcrzX

rBrBrBrBrB

rBrBrBrBrBrB

++++=

+−−−=

++++=+++=

−−−−−

−−−−−−

∫

∫∫θθ

θθ

(3.3.74)


.0)()(21)()(

21)(

21)(22 6)(6)(

5311

73 =++−−++ rKerBrKerBctrBctrzKctc rBrr

rBrrzθ (3.3.75)


.0)(21)(

212 533 =−− rBcrBrcc rr Solving this equation and remember that in this case

,0)( ≠rBr we get .053 == cc Also differentiating equation (3.3.75) with respect to

,θ we get .07 =c Now substituting the above information in equation (3.3.75) and




,)(0)()()(21 2

)(

132

)(2

)(

1166)(6)(

11

rBrBrB

rrBrB

r ecdreecrKrKerKerBc−−−

+−=⇒=++ ∫ .13 ℜ∈c


,

,

,,

2)(

132

)(

92

)(2

)(

113

2)(

102

)(

92

)(2

)(

82

1211841

640

rBrBrBrB

rBrBrBrB

ececdreecX

ecezcdreecX

czcctcXcrcX

−−−−

−−−−

++−=

+−−=

+++=+=

∫

∫θ

θ

(3.3.76)

128



),( 22)(222 dzdedrdtds rB +++−= θ (3.3.77)


fields which are ,r∂∂ ,

t∂∂ ,2

)(

θ∂∂− rB

e ,2)(

ze

rB

∂∂−

),(2)(

θθ

∂∂

−∂∂−

zz

erB

,t

rr

t∂∂

+∂∂

θθ

∂∂

−∂∂ )(rMr

and ,)(z

rMr

z∂∂

−∂∂ where .)( 2

)(2

)(

dreerMrBrB

∫−−

= Killing vector

fields in general relativity are ,t∂∂ ,

θ∂∂

z∂∂ and ).(

θθ

∂∂

−∂∂ zz

It is evident that

only one teleparallel Killing vector field is the same as Killing vector field in general

relativity and teleparallel Killing vector fields are more in number to the Killing

vector fields in general relativity. Cases (V)(b) and (V)(c) can be solved exactly the


Case (VI)(a):

In this case we have )(rAA = and .constant== CB Now substituting equation

(3.3.12) in equation (3.3.6), we get

.0),,(),,( 1)(3 =− zrPezrtP rAt θθ (3.3.78)




⇒= − ),(),,( 11 zrEezrP Aθθ ),,(),(),,( 311 zrEzrEezrP A += −θθ where ),(3 zrE is a


).,,(),,(),(),,,(),,(),(

43212

21310

θ

θθ

rtPXzrEzrEtXztPXzrEzrEeX A

=+=

=+= −

(3.3.79)


129

.0),(),(),,( 3)(14 =−− zrEezrErtP zrA

zt θθ (3.3.80)




.0),(),(),( 3)(14 =−− zrEezrErE zrA

zθθ Differentiating the above equation with




),()(),( 42)(3 rKrKezzrE rA += − where )(3 rK and )(4 rK are functions of


( )( )

( ) ).,()()(),,()()(),,,(

),()()()(

5213

231221

42)(3)(1)(0

θθ

θ

θ


rKrKezrKerKezX rArArA

++=

++==

+++= −−−

(3.3.81)








( )).()()(),()()(

),,,(),()()(65237532

2142)(3)(0

rKrKrtKXrKrKzrKtXztPXrKrKezrKeX rArA

++=+−=

=++= −−

θ

θθ (3.3.82)


130

.0)()(21)(

)()(21)()(

21),,(

4)(4)(

22332

=−−

−+−+

rKerArKe

rKzrKAzrKrKAztP

rArr

rA

rrrrt θθθ (3.3.83)





.0)()(21),( 336 =−+ rKrKAzE rrθθ (3.3.84)






)(

22

)(2

)(

13 ℜ∈+= ∫

−

cecdreecrKrArArA




.,)( 52

)(

52

)(2

)(

32 ℜ∈+= ∫

−

cecdreecrKrArArA



.,)( 62

)(

62

)(2

)(

44 ℜ∈+=

−−−



(3.3.82) we get

131

).()(

),()(

),,(

,

652)(

52

)(2

)(

33

752)(

22

)(2

)(

12

7431

1

2)(

62

)(2

)(

42

)(

52

)(2

)(

32

)(

22

)(2

)(

10

rKrKetcdreectX

rKrKzectdreectX

zEtcctzctX


rArArA

rArArA

rArArArArArArArArA

+++=

+−+=

+++=

+++++=

∫

∫

∫∫∫

−

−

−−−−−−−−−

θ

θθ

θθ

(3.3.85)


.0)()()(21)(

21),(2 752

)(

22

)(2

)(

17

1 =+−+++ ∫−

rKrKzerActdreerActzEtc rr

rA

r

rArA

rθθ (3.3.86)


.0)(21)(

212 2

)(

22

)(2

)(

11 =++ ∫− rA

r

rArA

r erAcdreerAcc Solving this equation and

remember that in this case ,0)( ≠rAr we get .021 == cc Substituting the above

information in equation (3.3.86) and differentiating with respect to ,θ we get






.,)(0)( 99755

7 ℜ∈+=⇒=− ccrcrKrKc r Now substituting back all the above


.,)(0)( 1010877

8 ℜ∈+−=⇒=+ ccrcrKrKc r Refreshing the system of equations

(3.3.85) we get

132

).(

,

),(

,

697

2)(

52

)(2

)(

33

108972

118743

1

2)(

62

)(2

)(

42

)(

52

)(2

)(

30

rKcrcetcdreectX

crcczrczX

zKczctcctzX

ecdreececzdreeczX

rArArA

rArArArArArA

++++=

+−−−=

++++=

+++=

∫

∫∫

−

−−−−−−

θθ

θθ

(3.3.87)


.0)()(21)(

21)(22 62

)(

52

)(2

)(

311

73 =+++++ ∫−

rKerActdreerActzKctc r

rA

r

rArA

rzθ (3.3.88)


.0)(21)(

212 2

)(

52

)(2

)(

33 =++ ∫− rA

r

rArA

r erAcdreerAcc Solving this equation and

remember that in this case ,0)( ≠rAr we get .053 == cc Substituting the above

information in equation (3.3.88) and differentiating with respect to ,θ we get .07 =c

Now substituting the above information in equation (3.3.88) and differentiating twice

with respect to ,z we get .,,)(0)( 121112111111 ℜ∈+=⇒= ccczczKzKzz Substituting

back these information in equation (3.3.88) and differentiating the resulting equation

with respect to ,z we get .,)(0)( 13131166

11 ℜ∈+−=⇒=+ ccrcrKrKc r Refreshing


,,

,,

139113

10982

12118412

)(

62

)(2

)(

40

ccrcXczcrcX

czcctcXecdreecXrArArA

++−=+−−=

+++=+=−−−

∫θ

θ

(3.3.89)



),( 2222)(2 dzddrdteds rA +++−= θ (3.3.90)

133



)(

te

rA

∂∂−

,θ∂∂ ,

z∂∂ ),(

θθ

∂∂

−∂∂ zz

,)(t

rMr

t∂∂

+∂∂

θθ

∂∂

−∂∂ rr

and ,z

rr

z∂∂

−∂∂ where .)( 2

)(2

)(

∫−−

= dreerMrArA

Killing vector fields

in general relativity are ,t∂∂ ,

θ∂∂

z∂∂ and ).(

θθ

∂∂

−∂∂ zz

It is obvious that in this

case three teleparallel Killing vector fields are the same as Killing vector fields in

general relativity. Other teleparallel Killing vector fields are different than Killing

vector fields in general relativity. The number of teleparallel Killing vector fields are

more than Killing vector fields in general relativity. Cases (VI)(b) and (VI)(c) can be

solved exactly the same as in the above case.

3.4. Teleparallel Killing Vector Fields in Spatially

Homogeneous Rotating Space-Times

Consider spatially homogeneous rotating space-times in usual coordinates ),,,( zrt φ

(labeled by ),,,,( 3210 xxxx respectively) with the line element [39]

,)(2)( 22222 φφ ddtrBdzdrAdrdtds −+++−= (3.4.1)

where A and B are no where zero functions of r only. The above space-times

(3.4.1) admit at least three linearly independent Killing vector fields in general

relativity which are ,t∂∂

φ∂∂ and .

z∂∂ The tetrad components and its inverse can be

obtained by using the relation (1.3.4) as [70]

134

,

10000)()(0000100)(01

2

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+=

rBrA

rB

S aµ

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+−

=

1000

0)()(

1000010

0)()(

)(01

2

2

rBrA

rBrArB

Saµ (3.4.2)

Using (3.4.2) in (1.3.5) the corresponding non-vanishing Weitzenböck connections

are obtained as

,)(2

22

012 BA

BAAB+

′+′−=Γ ,

)(22

22

12 BABBA

+′+′

=Γ (3.4.3)

where dash denotes the derivative with respect to .r The non vanishing torsion

components by using equation (1.3.11) are [70]

,)(2

2221

0

BABAABT

+′−′

= ,)(2

2212

2

BABBAT

+′+′

= (3.4.4)


(1.3.17). One can write (1.3.17) explicitly using (3.4.1.) and (3.4.4) as follows

,0)( 0,2

0,0 =+ XrBX (3.4.5)

,0)()( 21,

21,

00,

1 =′−−− XrBXrBXX (3.4.6)

,0)()()( 2,2

2,0

0,2

0,0 =++− XrBXXrAXrB (3.4.7)

,0)( 3,2

3,0

0,3 =−− XrBXX (3.4.8)

,01,1 =X (3.4.9)

,02

)()()( 22,

11,

21,

0 =′

−−− XrAXXrAXrB (3.4.10)

,03,1

1,3 =+ XX (3.4.11)

,0)()( 2,0

2,2 =− XrBXrA (3.4.12)

135

,0)()( 3,2

3,0

2,3 =+− XrAXrBX (3.4.13)

,03,3 =X (3.4.14)

Integrating equations (3.4.9) and (3.4.14), we get

,),,(),,,( 2311 φφ rtEXztEX ==

where ),,(1 ztE φ and ),,(2 φrtE are functions of integrations which are to be

determined. Now differentiating equation (3.4.8) and (3.4.13) with respect to ,z we

get respectively

,0)( 33,2

33,0 =+ XrBX (3.4.15)

,0)()( 33,2

33,0 =− XrAXrB (3.4.16)

Multiplying (3.4.15) by A(r) and (3.4.16) by B(r) then adding and solving we get

),,,(),,( 430 φφ rtErtEzX += (3.4.17)

Substituting back (3.4.17) in equation (3.4.15) we get

),,,(),,( 652 φφ rtErtEzX += (3.4.18)

where ),,,(3 φrtE ),,,(4 φrtE ),,(5 φrtE and ),,(6 φrtE are functions of integration

which are to be determined. Our system of equations become

,),,(),,,(),,(),,,(),,,(),,(

23652

11430

φφφ

φφφ

rtEXrtErtEzXztEXrtErtEzX

=+=

=+= (3.4.19)

Now we have to solve the system (3.4.19) by using the remaining six equations.


respect to ,z we get ),,(),(),,(0),,( 2111 φφφφ tFtzFztEztEzz +=⇒= where

),(1 φtF and ),(2 φtF are functions of integration. Substituting back in (3.4.11) we

get ),,(),(),,(),(),,( 31212 φφφφφ tFtrFrtEtFrtEr +−=⇒−= where ),(3 φtF is a


136

).,(),(),,,(),,(),,(),(),,,(),,(

313652

211430

φφφφ

φφφφ

tFtrFXrtErtEzXtFtzFXrtErtEzX

+−=+=

+=+= (3.4.20)


respect to z and ,φ we get

),,(),,()(),,()( 435 rtFrtErBrtErA =− φφ (3.4.21)

where ),(4 rtF is a function of integration. Substituting back (3.4.21) in (3.4.12) we

get

),,(),,()(),,()( 546 rtFrtErBrtErA =− φφ (3.4.22)

where ),(5 rtF is the function of integration. Now considering equation (3.4.13) and

using equations (3.4.20) and (3.4.22) then differentiating the resulting equation with

respect to ,r we get .0),(),( 41 =− rtFtF rφφ Differentiating this equation with respect

to ,φ we get ),()(),(0),( 2111 tRtRtFtF +=⇒= φφφφφ where )(1 tR and )(2 tR are

functions of integration. Substituting back in the above equation we get

⇒=− 0),()( 41 rtFtR r ),()(),( 314 tRtRrrtF += where )(3 tR is the function of

integration. Substituting back all the above information in (3.4.13) we get

⇒=+ 0)(),( 33 tRtF φφ ),()(),( 433 tRtRtF +−= φφ where )(4 tR is the function of


( )( ) ).()()()(),,,(),,(

),,()()(),,,(),,(43213652

2211430

tRtRtRtRrXrtErtEzXtFtRtRzXrtErtEzX

+−+−=+=

++=+=

φφφφ

φφφφ (3.4.23)


respect to z we get ⇒=+ 0),,()(),,( 53 φφ rtErBrtE tt

),,(),,()(),,( 653 φφφ rFrtErBrtE +−= where ),(6 φrF is a function of integration.

Also considering (3.4.8) and using the above information and equation (3.4.23) then

differentiating with respect to φ we get .0),()()( 631 =−−− φφ rFtRtrR tt

137

Differentiating the above equation with respect to φ we get ⇒= 0),(6 φφφ rF

),()(),( 216 rGrGrF += φφ where )(1 rG and )(2 rG are functions of integration.


,)( 211 ctctR +−= 31

1 )( crcrG +−= and ,)( 433 ctctR +−= .,,, 4321 ℜ∈cccc Now

substituting back all the above information in (3.4.8) and solving we get

,)( 652 ctctR +−= 75

2 )( crcrG += and ,)( 874 ctctR += .,,, 8765 ℜ∈cccc Now


),,(),,()(),,( 764 φφφ rFrtErBrtE +−= where ),(7 φrF is the function of

integration. Refreshing the system of equations (3.4.23) by substituting all the above

information, we get

( )[ ]( )[ ]( )[ ] ( ) ,

),,,(),,(),,(

),,(),,()(),,()(

874365213

65226521

1

767531

50

ctcctcctcctcrX

rtErtEzXtFctcctczX

rFrtErBcrccrcrtErBzX

+++−−+−+−−=

+=++−+−=

+−++++−=

φφ

φφφφ

φφφφ

(3.4.24)

where .,,,,,,, 87654321 ℜ∈cccccccc Considering equation (3.4.12) and using equation

(3.4.24) then differentiating with respect to ,z we get

( )( )⇒++

= 3125 ),,( crc

BABrtE φφ ( ) ( ) ),,(),,( 8

3125 rtFcrc

BABrtE +++

= φφ where

),(8 rtF is a function of integration. Now substituting back the above value in

equation (3.4.12) we get ( ) ⇒+

= ),(),,( 72

6 φφ φφ rFBA

BrtE ( ) ),,(),(),,( 972

6 rtFrFBA

BrtE ++

= φφ

where ),(9 rtF is a function of integration. Refreshing the system of equations

(3.4.24) we get

138

( ) ( ) ( )( )[ ]

( ) ( ) ( )( )[ ] ( ) ,

),,(),(),(

),,(

),,(),(),(

874365213

972

8312

2

26521

1

97275

8312

0

ctcctcctcctcrX

rtFrFBA

BrtFcrcBA

BzX

tFctcctczX

rtBFrFBA

AcrcrtBFcrcBA

AzX

+++−−+−+−−=

++

+⎥⎦

⎤⎢⎣

⎡++

+=

++−+−=

−+

+⎥⎦

⎤⎢⎣

⎡++−+

+=

φφ

φφ

φφ

φφ

(3.4.25)

where .,,,,,,, 87654321 ℜ∈cccccccc Considering equation (3.4.7) and using equation

(3.4.25) then differentiating with respect to ,z we get ( )( )⇒++

= 3128 1),( crc

BArtFt

( )( ) ),(),( 3312

8 rGcrcBA

trtF +++

= where )(3 rG is a function of integration.

Substituting back the above value in equation (3.4.7) we get

.0),(),()( 792 =−+ φφ rFrtFBA t Differentiating the above equation with respect to ,t

we get ⇒= 0),(9 rtFtt ),()(),( 549 rGrtGrtF += where )(4 rG and )(5 rG are

functions of integration. Substituting back this value in the above equation we get

( ) ),()(),( 6427 rGrGBArF ++= φφ where )(6 rG is a function of integration.

Refreshing the system of equation (3.4.25) we get

( ) ( ) ( )

( ) ( )( )[ ]

( ) ( ) ( )

( )( )[ ] ( ) .

),()()()()(

)(

),,(

,)()()()()(

)()()(

874365213

54462

3312312

2

26521

1

54462

753

3123120

ctcctcctcctcrX

rGrtGrGrBrGBA

B

rGcrcBA

tcrcBA

BzX

tFctcctczX

rGrGtBrGrArGBA

A

crcrGrBcrctBArBcrc

BAAzX

+++−−+−+−−=

++++

+

⎥⎦

⎤⎢⎣

⎡++

+++

+=

++−+−=

+−++

+

⎥⎦

⎤⎢⎣

⎡++−+

+−+

+=

φφ

φ

φ

φφ

φ

φ

(3.4.26)

Now considering equation (3.4.13) and using equation (3.4.26) then differentiating

with respect to ,t we get ⇒=+ 031 ccr .031 == cc Also considering equation

(3.4.6) and using equation (3.4.26) then differentiating the resulting equation with

139

respect to ,z we get .05 =c Substituting all the above information in system of


[ ] ( ) ( )[ ]

( )[ ] .

),()()()()()(

),,(

,)()()()()()()(

874623

54462

32

262

1

544627

30

ctccccrX

rGrtGrGrBrGBA

BrGzX

tFcczX

rGrGtBrGrArGBA

AcrGrBzX

++−+−=

++++

+=

++=

+−++

++−=

φφ

φ

φφ

φ

(3.4.27)


respect to ,t we get ⇒= 0),(2 φtFtt ),()(),( 212 φφφ KtKtF += where )(1 φK and

)(2 φK are functions of integration. Now substituting the above information and

equation (3.4.27) in equation (3.4.6) and differentiating the resulting equation with

respect to φ twice we get ⇒= 0)(1 φφφK .,,)( 1091091 ℜ∈+= ccccK φφ Also

substituting (3.4.27) and all the above information in (3.4.10) and differentiating with

respect to ,φ we get ⇒= 0)(2 φφφK ℜ∈+= 121112112 ,,)( ccccK φφ and

( ) .0)()(2)()()(4 =′−′ rBrArBrArG In order to solve this equation we need to discuss

the following three possibilities:

(I) ( ) ,0)()(2)()( ≠′−′ rBrArBrA ,0)(4 =rG

(II) ( ) ,0)()(2)()( =′−′ rBrArBrA .0)(4 ≠rG

(III) ( ) ,0)()(2)()( =′−′ rBrArBrA .0)(4 =rG

We will discuss each case in turn. It is important to note that in case (III) we will

obtain the same teleparallel Killing vector fields as in case (II) with the reduction of

one teleparallel Killing vector fields generated from .)(4 rG Hence we will not

discuss case (III).

140

Case (I):

In this case we have 0)(4 =rG and ( ) .0)()(2)()( ≠′−′ rBrArBrA Substituting the

above information in (3.4.6) and differentiating with respect to ,φ we get .09 =c

Refreshing the above system of equations (3.4.27) we get

[ ] ( )[ ] ( )[ ] .

),()()(,

),()()()(

874623

562

3212111062

1

5627

30

ctccccrX

rGrGBA

BrGzXcctccczX

rGBrGBA

AcrGrBzX

++−+−=

++

+=++++=

−+

++−=

φφ

φφ (3.4.28)


respect to z and solving we get 02 =c and .,)( 152

153 ℜ∈+

= cBA

crG Also

considering equation (3.4.6) and using equation (3.4.28) then differentiating with

respect to z and solving we get .,)( 1313106 ℜ∈+= cccrrG Substituting all the

above information in (3.4.13) and differentiating w.r.t ,r we get

( ) .0221

7152=′−

+

′+′ cBcBABBA (3.4.29)

Substituting the above equation (3.4.29) in the remaining equations and solving, after

some calculation we reach to the following three possibilities:

(i) ,0157 == cc ,0)( ≠′ rA ,0)( ≠′ rB

(ii) ,015 =c ,07 ≠c ,0)( ≠′ rA ,0)( =′ rB

(iii) ,0)( =′ rA .0)( ≠′ rB


141

Case (I)(i):

In this case we have ,0157 == cc 0)( ≠′ rA and .0)( ≠′ rB Substituting all the above

information in (3.4.28) we get

( )

( ) .),()(

,),()(

8635

131022

121110615

131020

ccrXrGcrcBA

BX

ccctczXrGBcrcBA

AX

+−=+++

=

+++=−++

= φ (3.4.30)

Considering equation (3.4.10) and using equation (3.4.30), the final solution of

equations (3.4.5) to (3.4.14) is given as

,,1,, 863

212

2126

122212

0 ccrXcBA

XcczXccBA

BX +−=+

=+=++

−= (3.4.31)

where .,,,, 22211286 ℜ∈ccccc The line element for spatially homogeneous rotating

space-time is given in equation (3.4.1). The above space-time (3.4.1) admits five


),(12 t

BBA ∂

∂−

∂∂

+ φ ,

r∂∂

z∂∂ and .

zr

rz

∂∂

−∂∂ Killing vector fields in general

relativity are ,t∂∂

φ∂∂ and .

z∂∂ On comparison to the Killing vector fields in general

relativity we see that only two teleparallel Killing vector field are same in both the

theories while others are totally different.

Case (I)(ii):

In this case we have ,015 =c ,07 ≠c ,0)( ≠′ rA .0)( =′ rB Substituting back the

above information in (3.4.10) we get 0131110 === ccc and .7

4

cc

B −= The line

element for space-time (3.4.1) in this case becomes

142

.2)(7

422222 φφ ddtcc

dzdrAdrdtds ++++−= (3.4.32)

Solution of equations (3.4.5) to (3.4.14) is given as [70]

,,1

,,

87463

2122

2

1261

2472122

230

ctcccrXccA

X

cczXcczccA

cX

++−−=+

=

+=+++

−=

φ

(3.4.33)

where ,0,0,,,,,,,,, 7424232221128764 ≠≠ℜ∈ ccccccccccc 7

423 c

cc

−= and

.)( 2

7

422 c

cc

−= The above space-time (3.4.32) admits five linearly independent

teleparallel Killing vector fields which can be written ,t∂∂ ),(1

2322 t

ccA ∂

∂−

∂∂

+ φ

,r∂∂

z∂∂ and .

zr

rz

∂∂

−∂∂ Killing vector fields in general relativity are ,

t∂∂

φ∂∂ and

.z∂∂ In this case also two teleparallel Killing vector field are same as in general

relativity.

Case (I)(iii):

In this case we have 0)( ≠′ rB and .0)( =′ rA Equation

.0\,0)( 2323 ℜ∈=⇒=′ ccArA Substituting back the above information in

(3.4.29) we get 0157 == cc which on back substitution gives .04 =c Now

substituting the above information in (3.4.10) we get .0131110 === ccc The line

element (3.4.1) in this case becomes

,)(22223

222 φφ ddtrBdzdcdrdtds −+++−= (3.4.34)

Solution of equations (3.4.5) to (3.4.14) is given as [70]

143

,,,, 863

223

242126

125242

23

0 ccrXBc

cXcczXcc

Bc

BX +−=+

=+=++

−= (3.4.35)

where ).0(,,,,, 232524231286 ≠ℜ∈ ccccccc The above space-time (3.4.34) admits five


),(12

23t

BBc ∂

∂−

∂∂

+ φ ,

r∂∂ ,

z∂∂ and .

zr

rz

∂∂

−∂∂ Killing vector fields in general

relativity are ,t∂∂

φ∂∂ and .

z∂∂ In this case two teleparallel Killing vector field are

the same as in general relativity.

Case (II):

In this case we have 0)(4 ≠rG and ( ) .0)()(2)()( =′−′ rBrArBrA Equation

( ) ⇒=′−′ 0)()(2)()( rBrArBrA .)]([)( 2rBrA = Substituting all the above

information in equation (3.4.6) and solving we get 09 =c and

.0\,)( 13134 ℜ∈= cA

crG Refreshing the system of equations (3.4.27) we get

[ ][ ]

[ ] .

),(1)(211)(

,

),(1)(21)()(

874623

5132

613

32

121110621

513

6137

30

cctcccrX

rGctB

rGB

cB

rGzX

cctccczX

rGBctB

rGccrGrBzX

++−+−=

++++=

++++=

−−+++−=

φφ

φ

φφ

φ

(3.4.36)

Considering equation (3.4.6) and using equation (3.4.36) we get ⇒= 106 )( crGr

.,)( 1414106 ℜ∈+= cccrrG Now substituting all the above information in (3.4.10)

and differentiating w.r.t ,z get 02 =c and .,)( 152153 ℜ∈= c

Bc

rG Our system of

equations (3.4.36) becomes

144

.

),(1)(211

,

),(1)(21

87463

51321410132

152

12111061

5131410137

150

cctcrcX

rGctB

crcB

cBB

czX

cctczcX

rGBctB

crcccBc

zX

++−−=

+++++=

+++=

−−+++⎥⎦⎤

⎢⎣⎡ +−

=

φ

φ

φ

φ

(3.4.37)

Considering equation (3.4.13) and using equation (3.4.37) we get .0)( 7 =′ crB We

will discuss here the following three different possibilities:

(a) ,0)( ≠′ rB ,07 =c (b) ,0)( =′ rB ,07 ≠c (c) ,0)( =′ rB .07 =c

We will discuss each case in turn. It is important to note that in case (II)(c) we will

obtain the same teleparallel Killing vector fields as in case (II)(b) with the reduction

of one teleparallel Killing vector fields generated from .7c Hence we will not discuss

case (II)(c).

Case (II) (a):

In this case we have ,2BA = 0)( ≠′ rB and .07 =c Substituting back the above

information in (3.4.13) we get .2 415 cc = Also substituting the above values in (3.4.6)

we get .010 =c Now substituting the above information in (3.4.10) and differentiating

with respect to φ we get .0)( 13 =′ crB In this case both 0)( ≠′ rB and .013 ≠c This

case leads to contradiction and hence is not possible.

Case (II) (b):

In this case we have ,2BA = 0)( =′ rB and .07 ≠c Equation ⇒=′ 0)(rB

0\b ℜ∈=B which in turn implies .0\, 2 ℜ∈== baaA The line element for

spatially homogeneous rotating space-time in this case becomes

,222222 φφ ddtbdzdadrdtds −+++−= (3.4.38)

It is important to note that in this case all the torsion components become zero. The

above space-times admit ten linearly independent teleparallel Killing vector fields. In

145

this case the teleparallel Killing vector fields are same as in general relativity and are

given below

,

,111,

,

32413

82212929252722

64571

102212929252720

ccrctcX

czcba

zcba

bcba

btcba

rcba

rcba

bX

czcctcX

czcba

bzcba

acba

atcba

brcba

brcba

aX

+++=

++

−+

++

++

−+

−+

=

+++=

++

++

++

++

−+

++

=

φ

φ

φ

φ

(3.4.39)

where .,,,,,,,,, 10987654321 ℜ∈cccccccccc These Killing vector fields can be written

as ,t∂∂ ,

r∂∂ ,

φ∂∂ ,

z∂∂ ),(2 φ∂

∂+

∂∂

++

∂∂ b

ta

bar

rt ),(2 φ

φ∂∂

−∂∂

++

∂∂

tb

bar

r

,)()(12 ι

φφ∂∂

−+∂∂

−+

tbt

btaba

),(2 φ∂∂

+∂∂

++

∂∂ b

ta

baz

zt

)(2 φφ

∂∂

−∂∂

++

∂∂

tb

baz

z and .

zr

rz

∂∂

−∂∂

Examples

In the following we will discuss teleparallel Killing vector fields of some well known

spatially homogeneous rotating space-times. We are listing the results and details are

omitted. These results are obtained by the same procedure as adopted above in section

(3.4). These examples are as follows:

(1). Reboucas space-time:

If we choose )2cosh31()( 2 rrA +−= and ,2cosh2)( rrB = the above space-time

(3.4.1) becomes Reboucas space-time and takes the form [70]

.2cosh4)2cosh31( 222222 φφ ddtrdzdrdrdtds −++−+−= (3.4.40)

For the above space-time (3.4.40) the teleparallel Killing vector fields are given as

,,)2(cos,,)2coth2( 423

12

321

510 ccrXcrechXcczXccrX +−==+=+−= (3.4.41)

146

where .,,,, 54321 ℜ∈ccccc

(2). Som-Raychaudhuri space-time:

If we choose )1()( 22 rrrA −= and ,)( 2rrB = the above space-time (3.4.1) becomes

Som-Raychaudhuri space-time and takes the form [70]

.2)1( 22222222 φφ ddtrdzdrrdrdtds −+−++−= (3.4.42)


,,1,, 423

12

321

510 ccrXc

rXcczXccrX +−==+=+−= (3.4.43)


(3). Hoenselaers-Vishveshwara space-time:

If we choose )3)(cosh1(cosh21)( −−−= rrrA and ),1(cosh)( −= rrB the above

space-time (3.4.1) becomes Hoenselaers-Vishveshwara space-time and takes the form

.)1(cosh2)3)(cosh1(cosh21 22222 φφ ddtrdzdrrdrdtds −−+−−−+−= (3.4.44)


,,cos2,,2 423

12

321

510 ccrXcechrXcczXccX +−==+=+−= (3.4.45)


(4). Gödel-Friedmann space-time:

Choosing )sinh1(sinh)( 22 rrrA −= and ,sinh2)( 2 rrB = the above space-time

(3.4.1) becomes Gödel-Friedmann space-time and takes the form

.sinh22)sinh1(sinh 22222222 φφ ddtrdzdrrdrdtds −+−++−= (3.4.46)


147

,,cosh

cos,,tanh2 423

12

321

510 ccrXc

rechrXcczXccrX +−==+=+−= (3.4.47)


(5). Stationary Gödel space-time:

Choosing raerA 2

21)( −= and raerB =)( where .0\Ra∈ The above space-time

(3.4.1) becomes stationary Gödel space-time and takes the form

.221 222222 φφ ddtedzdedrdtds rara −+−+−= (3.4.48)


,,2,,2 423

12

321

510 ccrXceXcczXccX ar +−==+=+−= − (3.4.49)



In this chapter we investigated teleparallel Killing vector fields for Kantowski-Sachs,

Bianchi type III, static cylindrically symmetric and spatially homogeneous rotating

space-times using direct integration technique. Following results are obtained from

the above study:

(1) In Kantowski-Sachs and Bianchi type III space-times: It turns out that the above

space-times admit only 4 or 6 teleparallel Killing vector fields. The space-time admits

six teleparallel Killing vector fields only when it’s metric function )(tA becomes

constant. The results for teleparallel Killing vector fields when it admits four

teleparallel Killing vector field are given in equations (3.2.21), (3.2.27), (3.2.34) and

(3.2.41). The result when the above space-time admits six teleparallel Killing vector

fields is given in equation (3.2.48).

148

(2) In static cylindrically symmetric space-times: Different possibilities for the

existence of teleparallel Killing vector fields have been found by using direct

integration technique. It turns out that the above space-times admit 7, 8 or 10

teleparallel Killing vector fields. The above space-time admits eight teleparallel

Killing vector fields when it becomes static plane symmetric space-time. The only

case when static cylindrically symmetric space-times admit ten teleparallel Killing

vector fields is the space like version of the Friedmann Robertson Walker 0=K

model. When the space-time becomes Minkowski then all the torsion components

become zero and the teleparallel Lie derivative for the metric gives the same

equations as in general relativity, hence the Killing vector fields are same in both the

theories. The results for above space time when it admits seven teleparallel Killing

vector fields are given in equations (3.3.25) and (3.3.38). When the above space time

admits eight teleparallel Killing vector fields, results are given in equations (3.3.51),

(3.3.76) and (3.3.89). When the above space time admits ten teleparallel Killing

vector fields, result is given in equation (3.3.64).

(3) In spatially homogeneous rotating space-times: Teleparallel Killing vector fields

have been explored in the above space-times and it turns out that these space-times

admit 5 or 10 teleparallel Killing vector fields. The results for five teleparallel Killing

vector fields are given in equations (3.4.31), (3.4.33) and (3.4.35). Also the result for

ten teleparallel Killing vector fields is given in equation (3.4.39). We also

investigated teleparallel Killing vector fields for some special classes of spatially

homogeneous rotating space-times and all these space-times possess five teleparallel

Killing vector fields which are given in equations (3.4.41), (3.4.43), (3.4.45), (3.4.47)

and (3.4.49).

149

Chapter 4

Teleparallel Proper Homothetic Vector

Fields in Bianchi Type I, Non Static Plane

Symmetric and Static Cylindrically

Symmetric Space-Times

4.1. Introduction

This chapter is devoted to investigate teleparallel proper homothetic vector fields in

Bianchi type I, non static plane symmetric and static cylindrically symmetric space-times

by using direct integration technique. We have discussed each possibility for the

existence of teleparallel proper homothetic vector fields in the above space-times. It turns

out that all the above space-times possess proper teleparallel homothetic vector fields for

special choice of the metric functions. This chapter is organized as follows: In section

(4.2) teleparallel proper homothetic vector fields of Bianchi type I space-times are

investigated. In the next section (4.3) teleparallel proper homothetic vector fields in non

static plane symmetric space-times in the context of teleparallel theory have been

explored. In section (4.4) teleparallel proper homothetic vector fields of static

cylindrically symmetric space-times are explored. Last section (4.5) of the chapter is

devoted to a detailed summary of the work.

150

4.2. Teleparallel Proper Homothetic Vector Fields In

Bianchi Type I Space-Times

The line element for Bianchi type I space-time in the usual coordinate system ),,,( zyxt

(labeled by ),,,,( 3210 xxxx respectively) is given by [75, 78]

.2)(22)(22)(222 dzedyedxedtds tCtBtA +++−= (4.2.1)

where BA, and C are functions of t only. The tetrad components and its inverse, non-

vanishing Weitzenböck connections and the non vanishing torsion components for (4.2.1)

are given in equations (4.2.2), (4.2.3) and (4.2.4). Now using (4.2.1) and (4.2.4) in

(1.3.17) we get the teleparallel homothetic equations as follows:

,3,3

2,2

1,1

0,0 α==== XXXX (4.4.2)

,01,2)(2

2,1)(2 =+ XeXe tBtA (4.2.3)

,01,3)(2

3,1)(2 =+ XeXe tCtA (4.2.4)

,02,3)(2

3,2)(2 =+ XeXe tCtB (4.2.5)

,01)(20,

1)(21,

0 =−− • XAeXeX tAtA (4.2.6)

,02)(20,

2)(22,

0 =−− • XBeXeX tBtB (4.2.7)

.03)(20,

3)(23,

0 =−− • XCeXeX tCtC (4.2.8)


).,,(),,,(),,,(),,,(

4332

2110

yxtPzXzxtPyXzytPxXzyxPtX

+=+=

+=+=

αα

αα (4.2.9)

where ),,,(1 zyxP ),,,(2 zytP ),,(3 zxtP and ),,(4 yxtP are functions of integration

which are to be determined. In order to find solution for equations (4.4.2) to (4.2.8) we

will consider each possible form of the metric for Bianchi type I space-times and then

151

solve each possibility in turn. Following are the possible cases for the metric where the

above space-times admit teleparallel proper homothetic vector fields:

(I) )(),(),( tCCtBBtAA === and .,, CBCABA ≠≠≠

(II)(a) ),(),( tBBtAA == and .tan tconsC =

(II)(b) ),(),( tCCtAA == and .tan tconsB =

(II)(c) ),(),( tCCtBB == and .tan tconsA =

(III)(a) )(),(),( tCCtBBtAA === and ).()( tCtB =

(III)(b) )(),(),( tCCtBBtAA === and ).()( tCtA =

(III)(c) )(),(),( tCCtBBtAA === and ).()( tBtA =

(IV) )(),(),( tCCtBBtAA === and ).()()( tCtBtA ==

(V)(a) )(),(,tan tCCtBBtconsA === and ).()( tCtB =

(V)(b) )(,tan),( tCCtconsBtAA === and ).()( tCtA =

(V)(c) tconsCtBBtAA tan),(),( === and ).()( tBtA =

(VI)(a) )(tAA = and .tan tconsCB ==

(VI)(b) )(tBB = and .tan tconsCA ==

(VI)(c) )(tCC = and .tan tconsBA ==


Case (I):

In this case we have ),(tAA = ),(tBB = ),(tCC = ,BA ≠ CA ≠ and .CB ≠ Now


.0),,(),,( 2)(23)(2 =+ zytPezxtPe ytA

xtB (4.2.10)




152


),(3 ztE is a function of integration. Now refreshing the system of equations (4.2.9) we

get

).,,(),,(),(),,(),(),,,(

43212

31)(2)(2110

yxtPzXztEztxEyXztEztEeyxXzyxPtX tAtB

+=++=

+−=+= −

αα

αα (4.2.11)



xtC (4.2.12)





ztBtC Differentiating this equation with

respect to y twice we get ).()(),(0),( 2144 tKtyKytEytEyy +=⇒= Substituting back

the above value in equation (4.2.12) and solving, we get

)()(),( 31)(2)(21 tKtKezztE tBtC += − and ),()(),( 42)(2)(23 tKtKezztE tAtC +−= − where

),(1 tK ),(2 tK )(3 tK and )(4 tK are functions of integration. Substituting back the


).,()()(),,()()(

),()()()(),,,(

5213

231)(2)(22

42)(2)(23)(2)(21)(2)(21

10

ytEtxKtKyxzXztEtxKtKezxyX

tKtKeztKeytKezyxXzyxPtX

tBtC

tAtCtAtBtAtC

+++=

+++=

+−−−=

+=

−

−−−

α

α

α

α

(4.2.13)


.0),(),()(2 2)(25)(21)(2 =++ ztEeytEetKxe ztB

ytCtC (4.2.14)

Differentiating the above equation with respect to x we get .0)(1 =tK Substituting the

above value in (4.2.14) and differentiating the resulting equation with respect to ,y we

get ),()(),(0),( 6555 tKtKyytEytEyy +=⇒= where )(5 tK and )(6 tK are functions of

integration. Substituting the above values back in (4.2.14) we get

153

),()(),( 75)(2)(22 tKtKezztE tBtC +−= − where )(7 tK is a function of integration.


).()()(),()()(

),()()(),,,(

6523

75)(2)(232

42)(2)(23)(2)(21

10

tKtKytxKzXtKtKeztxKyX

tKtKeztKeyxXzyxPtX

tBtC

tAtCtAtB

+++=

+−+=

+−−=

+=

−

−−

α

α

α

α

(4.2.15)


.0)()()(),,()()(

)()]()(2[)()()]()(2[4)(2)(214)(22)(2

2)(23)(23)(2

=−−+−+

−++−

tKetAetxAzyxPtKetKez

tKetAtCztKeytKetAtBytA

ttA

txttA

ttC

tCttt

tBtBtt

α (4.2.16)

Differentiating (4.2.16) with respect to ,x we get .0)(),,( )(21 =− tAtxx etAzyxP α

Differentiating this equation with respect to ,x we get

),,(),(),(2

),,(0),,( 8762

11 zyEzyxEzyExzyxPzyxPxxx ++=⇒= where ),,(6 zyE

),(7 zyE and ),(8 zyE are functions of integration. Substituting back this value in the

above equation we get .0)(),( )(26 =− tAt etAzyE α Now differentiating this equation with

respect to ,y we get ),(),(0),( 866 zKzyEzyEy =⇒= where )(8 zK is a function of

integration. Now substituting back this value and differentiating with respect to ,z we get

.,)(0)( 1188 ℜ∈=⇒= cczKzKz Substituting back this value we get

.,20)( 221)(2)(2

1 ℜ∈+=⇒=− cctceetAc tAtAt α

α Substituting the above values back in

(4.2.16) and differentiating twice with respect to ,y we get

),()(),(0),( 10977 zKzKyzyEzyEyy +=⇒= where )(9 zK and )(10 zK are functions of

integration. Substituting back the above value in (4.2.16) and differentiating with respect

to ,z we get .,)(0)( 3399 ℜ∈=⇒= cczKzKz Substituting back the above value in

(4.2.16) and differentiating with respect to ,y we get

.)()()]()(2[ 33)(23)(2 ctKetKetAtB t

tBtBtt −=+− solving this equation we get

.,)( 4)(2)(

4)()(2)(


154


.,,)(0)( 65651010 ℜ∈+=⇒= ccczczKzKzz Substituting back in equation (4.2.16) and solve

after differentiating with respect to ,z we get

.,)( 7)(2)(

7)()(2)(

52 ℜ∈+−= −−− ∫ cecdteectK tCtAtAtCtA Now substituting all the above


.,)( 8)(

8)()(

64 ℜ∈+= −−− ∫ cecdteectK tAtAtA Refreshing the system of equation (4.2.15)

with the help of above information, we get

),()(

),()(

,

),,(2

65)(2)(7

)()(2)(5

3

75)(2)(2)(2)(4

)()(2)(3

2

)(8

)()(6

)(7

)()(5

)(4

)()(3

1

86531

20

tKtKyexcdteexczX

tKtKzeexcdteexcyX

ecdteec

ezcdteezceycdteeycxX

zyExcxzcxyccxtX

tCtAtAtCtA

tBtCtBtAtAtBtA

tAtAtA

tAtAtAtAtAtA

+++−=

+−+−=

++

−+−+=

+++++=

−−−

−−−−

−−−

−−−−−−

∫∫

∫∫∫

α

α

α

α

(4.2.17)

where .,,,,,, 8765431 ℜ∈ccccccc Now considering equation (4.2.10) and using equation

(4.2.17) we get

.0)()()()()()]()(2[

)()]()([)]()([),(27)(27)(25)(25)(2

)(2)(4

)()(3

83

=−−+−+

−−−−++ ∫ −

tKetBtKetKeztKetBtCz

etyBetBtAcxdteetBtAxczyExctB

tttB

ttCtC

tt

tBt

tAtt

tAtAtty α

(4.2.18)


.0)]()([)]()([2 )(4

)()(33 =−−−+ ∫ − tA

tttAtA

tt etBtAcdteetBtAcc Solving this equation and

remember that in this case ,0)()( ≠− tBtA tt we get .043 == cc Substituting the above

information in equation (4.2.18) and differentiating with respect to ,y we get

.0)(),( )(28 =− tBtyy etBzyE α Differentiating once again with respect to ,y we get

),()()(2

),(0),( 1312112

88 zKzyKzKyzyEzyEyyy ++=⇒= where ),(11 zK )(12 zK and

)(13 zK are functions of integration. Substituting back this value and differentiating the

resulting equation with respect to ,z we get .,)(0)( 991111 ℜ∈=⇒= cczKzKz Once

155

again substituting back we get ⇒=− 0)( )(29

tBt etBc α .,2

10109)(2 ℜ∈+= cctce tB

α Now

substituting the above information in equation (4.2.18) and differentiating with respect to

z twice, we get .,,)(0)( 121112111212 ℜ∈+=⇒= ccczczKzKzz Substituting back this value

in equation (4.2.18) and differentiating with respect to ,z we get

.0)()()]()(2[ 5)(25)(211 =+−+ tKetKetBtCc t

tCtCtt Solving this equation we get

.,)( 13)(2)(

13)()(2)(

115 ℜ∈+−= −−− ∫ cecdteectK tCtBtBtCtB Now substituting all the above


tBtBt

.,)( 14)(

14)()(

127 ℜ∈+= −−− ∫ cecdteectK tBtBtB Refreshing the system of equations (4.2.17)

we get

),(

,

,

),(22

6)(2)(13

)()(2)(11

)(2)(7

)()(2)(5

3

)(14

)()(12

)(13

)()(11

2

)(8

)()(6

)(7

)()(5

1

1312119

2

651

20

tKeycdteeycexcdteexczX

ecdteecezcdteezcyX

ecdteecezcdteezcxX

zKycyzccyxcxzccxtX

tCtBtBtCtBtCtAtAtCtA

tBtBtBtBtBtB

tAtAtAtAtAtA

++−+−=

++−−=

++−+=

+++++++=

−−−−−−

−−−−−−

−−−−−−

∫∫∫∫∫∫

α

α

α

α

(4.2.19)

where .,,,,,,,,, 14131211987651 ℜ∈cccccccccc Now considering equation (4.2.11) and using


.0)()()(

)())()(())()((

))()(())()(()(22

6)(2)(26

)(2)(13

)()(11

)(7

)()(5

13115

=−−

−−−−+

−−−+++

∫∫

−

−

tKetCetK

etzCetCtBycdteetCtByc

etCtAcxdteetCtAxczKycxc

tCt

tCt

tCt

tBtt

tBtBtt

tAtt

tAtAttz

α (4.2.20)


.0)]()([)]()([2 )(7

)()(55 =−−−+ ∫ − tA

tttAtA

tt etCtAcdteetCtAcc Solving this equation and

remember that in this case ,0)()( ≠− tCtA tt we get .075 == cc Substituting the above


.0)]()([)]()([2 )(13

)()(1111 =−−−+ ∫ − tB

tttBtB

tt etCtBcdteetCtBcc Solving this equation and

remember that in this case ,0)()( ≠− tCtB tt we get .01311 == cc Now substituting all the

156

above information in equation (4.2.20) and differentiating with respect to ,z we get

.0)()( )(213 =− tCtzz etCzK α Again differentiating this equation with respect to z and

solving we get .,,,2

)(0)( 171615171615

21313 ℜ∈++=⇒= cccczcczzKzKzzz Substituting

back in the above equation and solving we get ⇒=− 0)( )(215

tCt etCc α

.,2181815

)(2 ℜ∈+= cctce tC

α Now substituting the above information in equation (4.2.20)

we get ⇒=−− 0)()()( 6)(26)(216 tKetKetCc t

tCtCt .,)( 19

)(19

)()(16

6 ℜ∈+= −−− ∫ cecdteectK tCtCtC


,

,,

,222

)(19

)()(16

3

)(14

)()(12

2)(8

)()(6

1

171612615

2

9

2

1

20

tCtCtC

tBtBtBtAtAtA

ecdteeczX

ecdteecyXecdteecxX

czcycxcczcycxtX

−−−

−−−−−−

++=

++=++=

+++++++=

∫∫∫

α

αα

α

(4.2.21)

where .,,,,,,,,, 1917161514129861 ℜ∈cccccccccc The line element for Bianchi type I space-

times in this case takes the form

,)2()2()2( 21815

2109

221

22 dzctcdyctcdxctcdtds ++++++−=ααα

(4.2.22)

where ).,,,0,0,0(,,,,, 159151911591181510921 ccccccccccccccc ≠≠≠≠≠≠ℜ∈ The above

space-time admits eight linearly independent teleparallel homothetic vector fields in

which seven are teleparallel Killing vector fields given as ,t∂∂ ,)(

xe tA

∂∂− ,)(

ye tB

∂∂−

,)(

ze tC

∂∂− ,)(1

xtG

tx

∂∂

+∂∂

ytG

ty

∂∂

+∂∂ )(2 and ,)(3

ztG

tz

∂∂

+∂∂ where

,)( )()(1 ∫ −−= dteetG tAtA

∫ −−= dteetG tBtB )()(2 )( and ∫ −−= dteetG tCtC )()(3 )( and one is

proper teleparallel homothetic vector field. Proper teleparallel homothetic vector field

after subtracting teleparallel Killing vector fields from (4.2.21) is given as

,,,,222

32115

2

9

2

1

20 zXyXxXczcycxtX αααα ===+++= (4.2.23)

157

Case (II)(a):

In this case we have tCtBBtAA tancos),(),( === and ).()( tBtA ≠ Now substituting


.0),,(),,( 2)(23)(2 =+ zytPezxtPe ytA

xtB (4.2.24)






get

).,,(),,(),(),,(),(),,,(

43212

31)(2)(2110


+=++=

+−=+= −

αα

αα (4.2.25)


.0)],(),([),,( 31)(2)(2)(24 =+−+ − ztEztEeyeyxtP zztAtBtA

x (4.2.26)




.0),(),(),( 3)(21)(24 =+− ztEeztEeyytE ztA

ztB Differentiating this equation with respect to

y twice we get ).()(),(0),( 2144 tKtyKytEytEyy +=⇒= Substituting back the above

value in equation (4.2.26) and solving, we get )()(),( 31)(21 tKtKezztE tB += − and

),()(),( 42)(23 tKtKezztE tA +−= − where ),(1 tK ),(2 tK )(3 tK and )(4 tK are functions

of integration. Substituting back the above information in equation (4.2.25) we get

).,()()(),,()()(

),()()()(),,,(

5213

231)(22

42)(23)(2)(21)(21

10

ytEtxKtKyxzXztEtxKtKezxyX


tB

tAtAtBtA

+++=

+++=

+−−−=

+=

−

−−−

α

α

α

α

(4.2.27)

158


.0),(),()(2 2)(251 =++ ztEeytEtxK ztB

y (4.2.28)

Differentiating the above equation with respect to x we get .0)(1 =tK Substituting the

above value in (4.2.28) and differentiating the resulting equation with respect to ,y we

get ),()(),(0),( 6555 tKtKyytEytEyy +=⇒= where )(5 tK and )(6 tK are functions of


),()(),( 75)(22 tKtKezztE tB +−= − where )(7 tK is a function of integration. Refreshing


).()()(),()()(

),()()(),,,(

6523

75)(232

42)(23)(2)(21

10

tKtKytxKzXtKtKeztxKyX

tKtKeztKeyxXzyxPtX

tB

tAtAtB

+++=

+−+=

+−−=

+=

−

−−

α

α

α

α

(4.2.29)


.0)()()(),,()(

)()()()()()]()(2[4)(2)(214)(2

223)(23)(2

=−−+−

+−+−

tKetAetxAzyxPtKe

tKztKtAztKeytKetAtBytA

ttA

txttA

ttttBtB

tt

α (4.2.30)


Differentiating this equation with respect to ,x we get

),,(),(),(2

),,(0),,( 8762







.,20)( 221)(2)(2




159





.)()()]()(2[ 33)(23)(2 ctKetKetAtB t


.,)( 4)(2)(

4)()(2)(




after differentiating with respect to ,z we get .,)( 7)(

7)()(

52 ℜ∈+−= ∫ − cecdteectK tAtAtA

Now substituting all the above information in equation (4.2.30) and solving we get

.,)( 8)(

8)()(



),()(

),()(

,

),,(2

65)(7

)()(5

3

75)(2)(2)(4

)()(2)(3

2

)(8

)()(6

)(7

)()(5

)(4

)()(3

1

86531

20

tKtKyexcdteexczX

tKtKzeexcdteexcyX

ecdteecezcdteezceycdteeycxX

zyExcxzcxyccxtX

tAtAtA

tBtBtAtAtBtA

tAtAtAtAtAtAtAtAtA

+++−=

+−+−=

++−+−+=

+++++=

∫∫

∫∫∫

−

−−−−

−−−−−−−−−

α

α

α

α

(4.2.31)


(4.2.31) we get

.0)()()()()()(

)()]()([)]()([),(27)(27)(255

)(2)(4

)()(3

83

=−−+−

−−−−++ ∫ −

tKetBtKetKztKtBz


tttB

tt

tBt

tAtt

tAtAtty α

(4.2.32)


.0)]()([)]()([2 )(4

)()(33 =−−−+ ∫ − tA

tttAtA





160

),()()(2

),(0),( 1312112





tBt etBc α .,2

10109)(2 ℜ∈+= cctce tB

α Now




.0)()()( 5511 =+− tKtKtBc tt Solving this equation we get

.,)( 13)(

13)()(

115 ℜ∈+−= ∫ − cecdteectK tBtBtB Now substituting all the above information

in equation (4.2.32) we get ⇒=−− 0)()()( 7)(27)(212 tKetKetBc t

tBtBt

.,)( 14)(

14)()(


we get

),(

,

,

),(22

6)(13

)()(11

)(7

)()(5

3

)(14

)()(12

)(13

)()(11

2

)(8

)()(6

)(7

)()(5

1

1312119

2

651

20


ecdteecezcdteezcyX

ecdteecezcdteezcxX

zKycyzccyxcxzccxtX

tBtBtBtAtAtA

tBtBtBtBtBtB

tAtAtAtAtAtA

++−+−=

++−−=

++−+=

+++++++=

∫∫∫∫∫∫

−−

−−−−−−

−−−−−−

α

α

α

α

(4.2.33)



.0)()()(

)()()(226)(

13)()(

11

)(7

)()(5

13115

=−−+

−+++

∫∫

−

−

tKetBycdteetByc

etAcxdteetAxczKycxc

ttB

ttBtB

t

tAt

tAtAtz

(4.2.34)


.0)()(2 )(7

)()(55 =−+ ∫ − tA

ttAtA

t etAcdteetAcc Solving this equation and remember that in

this case ,0)( ≠tAt we get .075 == cc Substituting the above information in equation

161


.0)()(2 )(13

)()(1111 =−+ ∫ − tB

ttBtB

t etBcdteetBcc Solving this equation and remember that in

this case ,0)( ≠tBt we get .01311 == cc Now substituting all the above information in

equation (4.2.34) and differentiating with respect to ,z we get ⇒= 0)(13 zKzz

Differentiating this equation with respect to z and solving we get

.,,)( 1615161513 ℜ∈+= ccczczK Now substituting the above information in equation

(4.2.34) we get ⇒=− 0)(615 tKc t .,)( 171715

6 ℜ∈+= cctctK Refreshing the system of


,,

,,22

17153)(

14)()(

122

)(8

)()(6

116151269

2

1

20

ctczXecdteecyX

ecdteecxXczcycxccycxtX

tBtBtB

tAtAtA

++=++=

++=++++++=

−−−

−−−

∫∫

αα

αα (4.2.35)

where .,,,,,,,, 17161514129861 ℜ∈ccccccccc The line element for Bianchi type I space-times

in this case takes the form

,)2()2( 22109

221

22 dzdyctcdxctcdtds +++++−=αα

(4.2.36)

where ).,0,0(,,, 919110921 cccccccc ≠≠≠ℜ∈ The above space-time admits eight linearly

independent teleparallel homothetic vector fields in which seven are teleparallel Killing

vector fields given as ,t∂∂ ,)(

xe tA

∂∂− ,)(

ye tB

∂∂− ,

z∂∂ ,)(1

xtG

tx

∂∂

+∂∂

ytG

ty

∂∂

+∂∂ )(2

and ,z

tt

z∂∂


and ∫ −−= dteetG tBtB )()(2 )( and one is

proper teleparallel homothetic vector field. Proper teleparallel homothetic vector field

after subtracting teleparallel Killing vector fields from (4.2.35) is given as

,,,,22

3219

2

1

20 zXyXxXcycxtX αααα ===++= (4.2.37)

Cases (II)(b) and (II)(c) can be solved exactly the same as in the above case.

162

Case (III)(a):

In this case we have ),(),(),( tCCtBBtAA === ),()( tBtA ≠ )()( tCtA ≠ and

).()( tCtB = Now substituting equation (4.2.9) in equation (4.2.3), we get

.0),,(),,( 2)(23)(2 =+ zytPezxtPe ytA

xtB (4.2.38)






get

).,,(),,(),(),,(),(),,,(

43212

31)(2)(2110


+=++=

+−=+= −

αα

αα (4.2.39)


.0),(),(),,( 3)(21)(24)(2 =+− ztEeztEeyyxtPe ztA

ztB

xtB (4.2.40)





ztBtB Differentiating this equation with

respect to y twice we get ).()(),(0),( 2144 tKtyKytEytEyy +=⇒= Substituting back

the above value in equation (4.2.40) and solving, we get )()(),( 311 tKtKzztE += and

),()(),( 42)(2)(23 tKtKezztE tAtB +−= − where ),(1 tK ),(2 tK )(3 tK and )(4 tK are

functions of integration. Substituting back the above information in equation (4.2.39) we

get

163

).,()()(),,()()(

),()()()(),,,(

5213

2312

42)(2)(23)(2)(21)(2)(21

10

ytEtxKtKyxzXztEtxKtKzxyX


tAtBtAtBtAtB

+++=

+++=

+−−−=

+=−−−

α

α

α

α

(4.2.41)


.0),(),()(2 251 =++ ztEytEtxK zy (4.2.42)

Differentiating the above equation with respect to x we get .0)(1 =tK Now

differentiating the above equation with respect to ,y we get

),()(),(0),( 6555 tKtKyytEytEyy +=⇒= where )(5 tK and )(6 tK are functions of




).()()(),()()(

),()()(),,,(

6523

7532

42)(2)(23)(2)(21

10

tKtKytxKzXtKtKztxKyX

tKtKeztKeyxXzyxPtX

tAtBtAtB

+++=

+−+=

+−−=

+=−−

α

α

α

α

(4.2.43)


.0)()()(),,()()(

)()]()(2[)()()]()(2[4)(2)(214)(22)(2

2)(23)(23)(2

=−−+−+

−++−

tKetAetxAzyxPtKetKez

tKetAtBztKeytKetAtBytA

ttA

txttA

ttB

tBttt

tBtBtt

α (4.2.44)


Differentiating this equation again with respect to ,x we get

),,(),(),(2

),,(0),,( 8762






164


.,20)( 221)(2)(2








.)()()]()(2[ 33)(23)(2 ctKetKetAtB t


.,)( 4)(2)(

4)()(2)(




after differentiating with respect to ,z we get

.,)( 7)(2)(

7)()(2)(



.,)( 8)(

8)()(



),()(

),()(

,

),,(2

65)(2)(7

)()(2)(5

3

75)(2)(4

)()(2)(3

2

)(8

)()(6

)(7

)()(5

)(4

)()(3

1

86531

20

tKtKyexcdteexczX

tKtzKexcdteexcyX


zyExcxzcxyccxtX

tBtAtAtBtA

tBtAtAtBtA

tAtAtAtAtAtAtAtAtA

+++−=

+−+−=

++−+−+=

+++++=

−−−

−−−

−−−−−−−−−

∫∫

∫∫∫

α

α

α

α

(4.2.45)


(4.2.45) we get

.0)()()()()()(

)()]()([)]()([),(27)(27)(25)(25)(2

)(2)(4

)()(3

83

=−−++

−−−−++ ∫ −

tKetBtKetKeztKetBz


tttB

ttBtB

t

tBt

tAtt

tAtAtty α

(4.2.46)

165


.0)]()([)]()([2 )(4

)()(33 =−−−+ ∫ − tA

tttAtA





),()()(2

),(0),( 1312112





tBt etBc α .,2

10109)(2 ℜ∈+= cctce tB

α Now




.0)()()( 5)(25)(211 =++ tKetKetBc t

tBtBt Solving this equation we get

.,)( 13)(

13)()(

115 ℜ∈+−= −−− ∫ cecdteectK tBtBtB Now substituting all the above information


tBtBt

.,)( 14)(

14)()(


we get

),(

,

,

),(22

6)(13

)()(11

)(2)(7

)()(2)(5

3

)(14

)()(12

)(13

)()(11

2

)(8

)()(6

)(7

)()(5

1

1312119

2

651

20


ecdteecezcdteezcyX

ecdteecezcdteezcxX

zKycyzccyxcxzccxtX

tBtBtBtBtAtAtBtA

tBtBtBtBtBtB

tAtAtAtAtAtA

++−+−=

++−−=

++−+=

+++++++=

−−−−−−

−−−−−−

−−−−−−

∫∫∫∫∫∫

α

α

α

α

(4.2.47)



166

.0)()()()(

)]()([)]()([)(226)(2)(26)(2

)(7

)()(5

13115

=−−−

−−−+++ ∫ −

tKetBetKetzB

etBtAcxdteetBtAxczKycxctB

ttB

ttB

t

tAtt

tAtAttz

α (4.2.48)


.0)]()([)]()([2 )(7

)()(55 =−−−+ ∫ − tA

tttAtA



information in equation (4.2.48) and differentiating with respect to ,y we get .011 =c

Now substituting all the above information in equation (4.2.48) and differentiating with

respect to ,z we get ⇒=⇒=− 913)(213 )(0)()( czKetBzK zz

tBtzz α

.,,2

)( 161516159

213 ℜ∈++= ccczcczzK Now substituting the above information in

equation (4.2.48) we get ⇒=−− 0)()()( 6)(26)(215 tKetKetBc t

tBtBt

.,)( 17)(

17)()(


we get

,

,,

,)22

(2

)(17

)(13

)()(16

3

)(14

)(13

)()(12

2)(8

)()(6

1

16151269

22

1

20

tBtBtBtB

tBtBtBtBtAtAtA

eceycdteeczX

ecezcdteecyXecdteecxX

czcycxcczycxtX

−−−−

−−−−−−−

+++=

+−+=++=

+++++++=

∫∫∫

α

αα

α

(4.2.49)



),()2()2( 22109

221

22 dzdyctcdxctcdtds +++++−=αα

(4.2.50)

where ).,0,0(,,, 919110921 cccccccc ≠≠≠ℜ∈ The above space-time admits nine linearly

independent teleparallel homothetic vector fields in which eight are teleparallel Killing

vector fields given as ,t∂∂ ,)(

xe tA

∂∂− ,)(

ye tB

∂∂− ,)(

ze tB

∂∂− ),()(

yz

zye tB

∂∂

−∂∂−

,)(1

xtG

tx

∂∂

+∂∂

ytG

ty

∂∂

+∂∂ )(2 and ,)(2

ztG

tz

∂∂


and

167

∫ −−= dteetG tBtB )()(2 )( and one is proper teleparallel homothetic vector field. Proper

teleparallel homothetic vector field after subtracting teleparallel Killing vector fields from

(4.2.49) is given as

,,,,)22

(2

3219

22

1

20 zXyXxXczycxtX αααα ===+++= (4.2.51)

Cases (III)(b) and (III)(c) can be solved exactly the same as the above case.

Case (IV):

In this case we have )(),(),( tCCtBBtAA === and ).()()( tCtBtA == Now


.0),,(),,( 23 =+ zytPzxtP yx (4.2.52)




⇒−= ),(),,( 12 ztEzytPy ),,(),(),,( 312 ztEztEyzytP +−= where ),(3 ztE is a function

of integration. Now refreshing the system of equations (4.2.9) we get

).,,(),,(),(),,(),(),,,(

43212

31110

yxtPzXztEztxEyXztEztEyxXzyxPtX

+=++=

+−=+=

αα

αα (4.2.53)


.0),(),(),,( 314 =+− ztEztEyyxtP zzx (4.2.54)




.0),(),(),( 314 =+− ztEztEyytE zz Differentiating this equation with respect to y twice

we get ).()(),(0),( 2144 tKtyKytEytEyy +=⇒= Substituting back the above value in

equation (4.2.54) and solving, we get )()(),( 311 tKtKzztE += and

168

),()(),( 423 tKtzKztE +−= where ),(1 tK ),(2 tK )(3 tK and )(4 tK are functions of


).,()()(),,()()(

),()()()(),,,(

5213

2312

4231110


tKtzKtKytKzyxXzyxPtX

+++=

+++=

+−−−=+=

α

α

αα

(4.2.55)


.0),(),()(2 251 =++ ztEytEtxK zy (4.2.56)







).()()(),()()(),()()(),,,(

65237532

423110

tKtKytxKzXtKtKztxKyXtKtKztyKxXzyxPtX

+++=+−+=

+−−=+=

αα

αα (4.2.57)


.0)()()(),,()(

)()()()()()(4)(2)(214)(2

2)(22)(23)(23)(2

=−−+−

+++

tKetAetxAzyxPtKe

tKeztKetAztKeytKetaytA

ttA

txttA

ttAtA

tttAtA

t

α (4.2.58)


Differentiating again this equation again with respect to ,x we get

),,(),(),(2

),,(0),,( 8762






169


.,20)( 221)(2)(2







(4.2.58) and differentiating with respect to ,y we get .)()()( 33)(23)(2 ctKetKetA t

tAtAt −=+

solving this equation we get .,)( 4)(

4)()(

33 ℜ∈+−= −−− ∫ cecdteectK tAtAtA Now

substituting all the above information in (4.2.58) and solve after differentiating twice with

respect to ,z we get .,,)(0)( 65651010 ℜ∈+=⇒= ccczczKzKzz Substituting back in

equation (4.2.58) and solve after differentiating with respect to ,z we get

.,)( 7)(

7)()(

52 ℜ∈+−= −−− ∫ cecdteectK tAtAtA Now substituting all the above information in

equation (4.2.58) and solving we get .,)( 8)(

8)()(

64 ℜ∈+= −−− ∫ cecdteectK tAtAtA

Refreshing the system of equation (4.2.57) with the help of above information, we get

),()(

),()(

,

),,(2

65)(7

)()(5

3

75)(4

)()(3

2

)(8

)()(6

)(7

)()(5

)(4

)()(3

1

86531

20

tKtKyexcdteexczX

tKtzKexcdteexcyX


zyExcxzcxyccxtX

tAtAtA

tAtAtA

tAtAtAtAtAtAtAtAtA

+++−=

+−+−=

++−+−+=

+++++=

−−−

−−−

−−−−−−−−−

∫∫

∫∫∫

α

α

α

α

(4.2.59)


(4.2.59) we get

.0)()()(

)()()()(),(27)(27)(2

5)(25)(2)(283

=−−

++−+

tKetAtKe

tKeztKetzAetyAzyExctA

tttA

ttAtA

ttA

ty α (4.2.60)

Differentiating equation (4.2.60) with respect to ,x we get .03 =c Substituting the above


170

.0)(),( )(28 =− tBtyy etAzyE α Differentiating once again with respect to ,y we get

),()()(2

),(0),( 1312112


)(13 zK are functions of integration. Substituting back this value in the above equation,

we get .)(0)()( 111)(211 czKetAzK tA

t =⇒=−α Now substituting the above information in

equation (4.2.60) and differentiating with respect to z twice, we get

.,,)(0)( 111011101212 ℜ∈+=⇒= ccczczKzKzz Substituting back this value in equation

(4.2.60) and differentiating with respect to ,z we get .0)()()( 5)(25)(210 =++ tKetKetAc t

tAtAt

Solving this equation we get .,)( 12)(

12)()(

105 ℜ∈+−= −−− ∫ cecdteectK tAtAtA Now

substituting all the above information in equation (4.2.60) we get

⇒=−− 0)()()( 7)(27)(211 tKetKetAc t

tAtAt .,)( 13

)(13

)()(11



),(

,

,

),()22

(

6)(12

)()(10

)(7

)()(5

3

)(13

)()(11

)(12

)()(10

)(4

2

)(8

)()(6

)(7

)()(5

)(4

1

131110651

220


ecdteecezcdteezcexcyX

ecdteecezcdteezceycxX

zKycyzcxcxzccyxtX

tAtAtAtAtAtA

tAtAtAtAtAtAtA

tAtAtAtAtAtAtA

++−+−=

++−++=

++−+−=

+++++++=

−−−−−−

−−−−−−−

−−−−−−−

∫∫∫∫∫∫

α

α

α

α

(4.2.61)



.0)()()()()(22 6)(2)(26)(213105 =−−−++ tKetAetKetzAzKycxc tA

ttA

ttA

tz α (4.2.62)

Differentiating equation (4.2.62) with respect to x and y respectively, we get 05 =c and

.010 =c Now differentiating equation (4.2.62) with respect to ,z we get

⇒=⇒=− 113)(213 )(0)()( czKetAzK zz

tAtzz α .,,

2)( 151415141

213 ℜ∈++= ccczcczzK Now

substituting the above information in equation (4.2.62) we get

⇒=−− 0)()()( 6)(26)(214 tKetKetAc t

tAtAt .,)( 16

)(16

)()(14



171

,

,

,

,)222

(

)(16

)(12

)(7

)()(14

3

)(13

)(12

)(4

)()(11

2

)(8

)(7

)(4

)()(6

1

15141161

2220

tAtAtAtAtA

tAtAtAtAtA

tAtAtAtAtA

eceycexcdteeczX

ecezcexcdteecyX

ecezceycdteecxX

czcycxcczyxtX

−−−−−

−−−−−

−−−−−

++++=

+−++=

+−−+=

+++++++=

∫∫∫

α

α

α

α

(4.2.63)

where .,,,,,,,,, 1615141312118761 ℜ∈cccccccccc The line element for Bianchi type I space-

times in this case takes the form

),)(2( 22221

22 dzdydxctcdtds ++++−=α

(4.2.64)

where ).0(, 121 ≠ℜ∈ ccc The above space-time admits eleven linearly independent

teleparallel homothetic vector fields in which ten are teleparallel Killing vector fields

given as ,t∂∂ ,)(

xe tA

∂∂− ,)(

ye tA

∂∂− ,)(

ze tA

∂∂− ),()(

yz

zye tA

∂∂

−∂∂− ),()(

xy

yxe tA

∂∂

−∂∂−

),()(

xz

zxe tA

∂∂

−∂∂− ,)(

xtG

tx

∂∂

+∂∂

ytG

ty

∂∂

+∂∂ )( and ,)(

ztG

tz

∂∂

+∂∂ where

∫ −−= dteetG tAtA )()()(

and one is proper teleparallel homothetic vector field. Proper

teleparallel homothetic vector field after subtracting teleparallel Killing vector fields from

(4.2.63) is given as

,,,,)222

( 3211

2220 zXyXxXczyxtX αααα ===+++= (4.2.65)

Case (V)(a):

In this case we have )(),(,constant tCCtBBA === and ).()( tCtB = Now substituting


.0),,(),,( 23)(2 =+ zytPzxtPe yxtB (4.2.66)




172

⇒−= ),(),,( 1)(22 ztEezytP tBy ),,(),(),,( 31)(22 ztEztEeyzytP tB +−= where ),(3 ztE is a


).,,(),,(),(),,(),(),,,(

43212

31)(2110

yxtPzXztEztxEyXztEztEeyxXzyxPtX tB

+=++=

+−=+=

αα

αα (4.2.67)


.0),(),(),,( 31)(24)(2 =+− ztEztEeyyxtPe zztB

xtB (4.2.68)




.0),(),(),( 31)(24)(2 =+− ztEztEeyytEe zztBtB Differentiating this equation with respect to

y twice we get ).()(),(0),( 2144 tKtyKytEytEyy +=⇒= Substituting back the above

value in equation (4.2.68) and solving, we get )()(),( 311 tKtKzztE += and

),()(),( 42)(23 tKtKezztE tB +−= where ),(1 tK ),(2 tK )(3 tK and )(4 tK are functions of


).,()()(),,()()(

),()()()(),,,(

5213

2312

42)(23)(21)(21

10



tBtBtB

+++=

+++=

+−−−=

+=

α

α

α

α

(4.2.69)


.0),(),()(2 251 =++ ztEytEtxK zy (4.2.70)







173

).()()(),()()(

),()()(),,,(

6523

7532

42)(23)(21

10

tKtKytxKzXtKtKztxKyX

tKtKeztKeyxXzyxPtX

tBtB

+++=

+−+=

+−−=

+=

α

α

α

α

(4.2.71)


.0),,()(

)()()(2)()()(214

2)(22)(23)(23)(2

=+−

+++

zyxPtK

tKeztKetBztKeytKetBy

xt

ttBtB

tttBtB

t (4.2.72)

Differentiating (4.2.72) with respect to ,x we get ⇒= 0),,(1 zyxPxx

),,(),(),,( 761 zyEzyxEzyxP += where ),(6 zyE and ),(7 zyE are functions of

integration. Substituting back this value in equation (4.2.72) and differentiating the

resulting equation with respect to ,y we get .0),()()()(2 63)(23)(2 =++ zyEtKetKetB yttBtB

t

Now differentiating this equation with respect to ,y we get

),()(),(0),( 9866 zKzyKzyEzyEyy +=⇒= where )(8 zK and )(9 zK are functions of



⇒=++ 0)()()(2 13)(23)(2 ctKetKetB t

tBtBt .,)( 2

)(22

)(21

3 ℜ∈+−= −− cecetctK tBtB Substituting

the above values back in (4.2.72) and differentiating with respect to ,z we get

.0)()()(2)( 2)(22)(29 =++ tKetKetBzK ttBtB

tz Differentiating this equation with respect to ,z

we get ⇒= 0)(9 zKzz .,,)( 43439 ℜ∈+= ccczczK Substituting back this value in the

above equation we get ⇒=++ 0)()()(2 2)(22)(23 tKetKetBc t

tBtBt

.,)( 5)(2

5)(2

32 ℜ∈+−= −− cecetctK tBtB Now substituting all the above information in

equation (4.2.72) and solving we get .,)()( 6644

44 ℜ∈+=⇒= cctctKctKt Refreshing

the system of equation (4.2.71) with the help of above information, we get

),()(),()(

,),,(65)(2

5)(2

3375)(2

2)(2

12

64532117

4310

tKtKyexcextczXtKtzKexcextcyX

ctczcztcycytcxXzyExcxzcxyctXtBtBtBtB +++−=+−+−=

++−+−+=++++=−−−− αα

αα(4.2.73)

where .,,,, 65431 ℜ∈ccccc Now considering equation (4.2.10) and using equation (4.2.73)

we get

174

.0)()()()(

)()()()()(),(27)(27)(25)(2

5)(2)(221

71

=−−+

+−+−+

tKetBtKetKez

tKetBzetyBtBcxtBxtczyExctB

tttB

ttB

tBt

tBttty α

(4.2.74)

Differentiating equation (4.2.74) with respect to ,x we get .0)()(2 211 =+− tBctBtcc tt

Solving this equation and remember that in this case ,0)( ≠tBt we get .021 == cc

Substituting the above information in equation (4.2.74) and differentiating with respect to

,y we get .0)(),( )(27 =− tBtyy etBzyE α Differentiating once again with respect to ,y we

get ),()()(2

),(0),( 1211102





tBt etBc α .,2

887)(2 ℜ∈+= cctce tB

α Now




.0)()()( 5)(25)(29 =++ tKetKetBc t

tBtBt Solving this equation we get

.,)( 11)(

11)()(

95 ℜ∈+−= −−− ∫ cecdteectK tBtBtB Now substituting all the above information


tBtBt

.,)( 12)(

12)()(


we get

),(

,

,),(2

6)(11

)()(9

)(25

3

)(12

)()(10

)(11

)()(9

2

6453112

1097

2

430

tKeycdteeycexczX

ecdteecezcdteezcyX

ctczctzcxXzKycyzccyxcxzctX

tBtBtBtB

tBtBtBtBtBtB

++−+=

++−+=

++−+=++++++=

−−−−

−−−−−−

∫∫∫

α

α

αα

(4.2.75)

where .,,,,,,,, 121110976543 ℜ∈ccccccccc Now considering equation (4.2.11) and using


175

.0)(

)()()()()()(22)(2

36)(2)(26

512

93

=−

−−−+++tB

t

ttB

ttB

ttz

etzB

tBctxtKetBetKtBxczKycxc

α (4.2.76)

Differentiating equation (4.2.76) with respect to ,x we get .0)()(2 353 =−+ tBtctBcc tt

Solving this equation and remember that in this case ,0)( ≠tBt we get .053 == cc


,y we get .09 =c Now substituting all the above information in equation (4.2.76) and

differentiating with respect to ,z we get ⇒=⇒=− 712)(212 )(0)()( czKetBzK zz

tBtzz α

.,,2

)( 141314137

212 ℜ∈++= ccczcczzK Now substituting the above information in

equation (4.2.76) we get ⇒=−− 0)()()( 6)(26)(213 tKetKetBc t

tBtBt

.,)( 15)(

15)()(


we get

,

,

,,)22

(

)(15

)(11

)()(13

3

)(12

)(11

)()(10

2

641

14131047

220

tBtBtBtB

tBtBtBtB

eceycdteeczX

ecezcdteecyX

ctcxXczcycxcczytX

−−−−

−−−−

+++=

+−+=

++=++++++=

∫∫

α

α

αα

(4.2.77)

where .,,,,,,, 15141312111064 ℜ∈cccccccc The line element for Bianchi type I space-times in

this case takes the form

),()2( 2287

222 dzdyctcdxdtds ++++−=α

(4.2.78)

where ).0(, 787 ≠ℜ∈ ccc The above space-time admits nine linearly independent

teleparallel homothetic vector fields in which eight are teleparallel Killing vector fields

given as ,t∂∂ ,

x∂∂ ,)(

ye tB

∂∂− ,)(

ze tB

∂∂− ),()(

yz

zye tB

∂∂

−∂∂− ),(

xt

tx

∂∂

+∂∂

))((y

tGt

y∂∂

+∂∂ and ),)((

ztG

tz

∂∂

+∂∂ where ∫ −−= dteetG tBtB )()()( and one is proper

teleparallel homothetic vector field. Proper teleparallel homothetic vector field after

subtracting teleparallel Killing vector fields from (4.2.77) is given as

176

,,,,)22

( 3217

220 zXyXxXczytX αααα ===++= (4.2.79)

Cases (V)(b) and (V)(c) can be solved exactly the same as the above case.

Case (VI)(a):

In this case we have )(tAA = and constant.==CB Now substituting equation (4.2.9) in

equation (4.2.3), we get

.0),,(),,( 2)(23 =+ zytPezxtP ytA

x (4.2.80)




⇒−= − ),(),,( 1)(22 ztEezytP tAy ),,(),(),,( 31)(22 ztEztEeyzytP tA +−= − where ),(3 ztE is a


).,,(),,(),(),,(),(),,,(

43212

31)(2110

yxtPzXztEztxEyXztEztEeyxXzyxPtX tA

+=++=

+−=+= −

αα

αα (4.2.81)


.0),(),(),,( 3)(214 =+− ztEeztEyyxtP ztA

zx (4.2.82)




.0),(),(),( 3)(214 =+− ztEeztyEytE ztA

z Differentiating this equation with respect to y

twice we get ).()(),(0),( 2144 tKtyKytEytEyy +=⇒= Substituting back the above

value in equation (4.2.82) and solving, we get )()(),( 311 tKtKzztE += and

),()(),( 42)(23 tKtKezztE tA +−= − where ),(1 tK ),(2 tK )(3 tK and )(4 tK are functions


177

).,()()(),,()()(

),()()()(),,,(

5213

2312

42)(23)(21)(21

10



tAtAtA

+++=

+++=

+−−−=

+=−−−

α

α

α

α

(4.2.83)


.0),(),()(2 251 =++ ztEytEtxK zy (4.2.84)







).()()(),()()(),()()(),,,(

65237532

42)(23)(2110

tKtKytxKzXtKtKztxKyXtKtKeztKeyxXzyxPtX tAtA

+++=+−+=

+−−=+= −−

αα

αα (4.2.85)


.0)()()(

),,()()()()()()()(4)(2)(2

14)(22233

=−−

+−+−+−

tKetAetxA

zyxPtKetKztKtAztKytKtAytA

ttA

t

xttA

tttt

α (4.2.86)


Differentiating this equation again with respect to ,x we get

),,(),(),(2

),,(0),,( 8762







178

.,20)( 221)(2)(2







(4.2.86) and differentiating with respect to ,y we get .)()()( 333 ctKtKtA tt −=+− solving

this equation we get .,)( 4)(

4)()(

33 ℜ∈+−= ∫ − cecdteectK tAtAtA Now substituting all the

above information in (4.2.86) and solve after differentiating twice with respect to ,z we

get .,,)(0)( 65651010 ℜ∈+=⇒= ccczczKzKzz Substituting back in equation (4.2.86) and


.,)( 7)(

7)()(

52 ℜ∈+−= ∫ − cecdteectK tAtAtA Now substituting all the above information in

equation (4.2.86) and solving we get .,)( 8)(

8)()(


Refreshing the system of equation (4.2.85) with the help of above information, we get

),()(

),()(

,

),,(2

65)(7

)()(5

3

75)(4

)()(3

2

)(8

)()(6

)(7

)()(5

)(4

)()(3

1

86531

20

tKtKyexcdteexczX

tKtzKexcdteexcyX


zyExcxzcxyccxtX

tAtAtA

tAtAtA

tAtAtAtAtAtAtAtAtA

+++−=

+−+−=

++−+−+=

+++++=

∫∫

∫∫∫

−

−

−−−−−−−−−

α

α

α

α

(4.2.87)


(4.2.87) we get

.0)()()()(),(2 75)(4

)()(3

83 =−+−++ ∫ − tKtKzetAcxdteetAxczyExc tt

tAt

tAtAty (4.2.88)


.0)()(2 )(4

)()(33 =−+ ∫ − tA

ttAtA



(4.2.88) and differentiating with respect to ,y we get ⇒= 0),(8 zyEyy

179

),()(),( 12118 zKzyKzyE += where )(11 zK and )(12 zK are functions of integration.

Substituting back this value and differentiating the resulting equation twice with respect

to ,z we get .,,)(0)( 1091091111 ℜ∈+=⇒= ccczczKzKzz Once again substituting back we

get ⇒=+ 0)(59 tKc t .,)( 11119

5 ℜ∈+= cctctK Now substituting all the above

information in equation (4.2.88) we get ⇒=− 0)(710 tKc t .,)( 121210

7 ℜ∈+= cctctK


),(

,

,

),(2

6119

)(7

)()(5

3

12101192

)(8

)()(6

)(7

)()(5

1

12109651

20

tKyctcyexcdteexczX

ctczcztcyX

ecdteecezcdteezcxX

zKycyzcxcxzccxtX

tAtAtA

tAtAtAtAtAtA

++++−=

++−−=

++−+=

++++++=

∫

∫∫

−

−−−−−−

α

α

α

α

(4.2.89)

where .,,,,,,,, 121110987651 ℜ∈ccccccccc Now considering equation (4.2.11) and using


.0)()()()(22 6)(7

)()(5

1295 =−−+++ ∫ − tKetAcxdteetAxczKycxc t

tAt

tAtAtz (4.2.90)


.0)()(2 )(7

)()(55 =−+ ∫ − tA

ttAtA



(4.2.90) and differentiating with respect to ,y we get .09 =c Now substituting all the

above information in equation (4.2.90) and differentiating with respect to ,z we get

.,,)(0)( 141314131212 ℜ∈+=⇒= ccczczKzK zz Now substituting the above

information in equation (4.2.90) we get ⇒=− 0)(613 tKc t .,)( 151513

6 ℜ∈+= cctctK


,,

,,2

1511133

1211102

)(8

)()(6

114131061

20

cyctczXczctcyX

ecdteecxXczcycxccxtX tAtAtA

+++=+−+=

++=+++++= −−− ∫αα

αα (4.2.91)

180



),()2( 22221

22 dzdydxctcdtds ++++−=α

(4.2.92)



given as ,t∂∂ ,)(

xe tA

∂∂− ,

y∂∂ ,

z∂∂ ),(

yz

zy

∂∂

−∂∂ ,

yt

ty

∂∂

+∂∂

zt

tz

∂∂

+∂∂ and

,)(x

tGt

x∂∂

+∂∂ where ∫ −−= dteetG tAtA )()()(

and one is proper teleparallel homothetic

vector field. Proper teleparallel homothetic vector field after subtracting teleparallel

Killing vector fields from (4.2.91) is given as

,,,,2

3211

20 zXyXxXcxtX αααα ===+= (4.2.93)

Cases (VI)(b) and (VI)(c) can be solved exactly the same as the above case.

4.3. Teleparallel Proper Homothetic Vector Fields in

Non Static Plane Symmetric Space-Times

Consider plane symmetric non static space-times in usual coordinates ),,,( zyxt (labeled

by ),,,,( 3210 xxxx respectively) with the line element

),( 22),(222),(22 dzdyedxdteds xtBxtA +++−= (4.3.1)

where A and B are functions of t and x only. The above space-time admits five

teleparallel Killing vector fields which are [74]

,),(

te xtA

∂∂− ,

x∂∂ ,),(

ye xtB

∂∂− ,),(

ze xtB

∂∂− ).(),(

zy

yze xtB

∂∂

−∂∂− (4.3.2)

181

The tetrad components µaS and the inverse tetrad components µ

aS for the space-time

(4.3.1) can be obtained by using equation (1.3.4) as

),,,1,(diag ),(),(),( xtBxtBxtAa eeeS =µ ).,,1,(diag ),(),(),( xtBxtBxtAa eeeS −−−=µ (4.3.3)

The corresponding non-vanishing Weitzenböck connections for (4.3.3) are obtained by

using equation (1.3.5) as

,000 •=Γ A ,01

0 A′=Γ ,202 •=Γ B ,30

3 •=Γ B ,212 B′=Γ ,31

3 B′=Γ (4.3.4)

where “dot” represents the derivative with respect to t and “dash” denotes the derivative

with respect to .x The non vanishing torsion components are obtained by using equation

(1.3.11) as

,100 AT ′= ,02

2 •= BT ,033 •= BT ,12

2 BT ′= ,133 BT ′= (4.3.5)

A vector field X is said to be teleparallel homothetic vector field if it satisfies equation

(1.3.17). Using equation (1.3.17) explicitly by using (4.3.1) and (4.3.5) we get the

following equations:

,,000 α=+• XXA (4.3.6)

,,,, 33

22

11 α=== XXX (4.3.7)

,0,, 23

32 =+XX (4.3.8)

,0,, 0),(21

0),(20

1 =′−− XeAXeX xtAxtA (4.3.9)

,0,, 2),(21

2),(22

1 =′++ XeBXeX xtBxtB (4.3.10)

,0,, 3),(21

3),(23

1 =′++ XeBXeX xtBxtB (4.3.11)

,0,, 2),(22

0),(20

2),(2 =+− • XeBXeXe xtBxtAxtB (4.3.12)

.0,, 3),(23

0),(20

3),(2 =+− • XeBXeXe xtBxtAxtB (4.3.13)

Solving equation (4.3.6) and (4.3.7) by simple integration, we get

),,,(,),,(

),,,(),,,(4332

211),(),(),(0

yxtPzXzxtPyX

zytPxXzyxPedteeX xtAxtAxtA

+=+=

+=+= −− ∫αα

αα (4.3.14)

182

where ),,,(1 zyxP ),,,(2 zytP ),,(3 zxtP and ),,(4 yxtP are functions of integration and

we need to determine these functions. In order to find solution for equations (4.3.6) to

(4.3.13) we will consider each possible form of the metric for plane symmetric non static

space-times and then solve each possibility in turn. Following are the possible cases for

the metric where the above space-times admit teleparallel proper homotheric vector

fields:

(I) ),(tAA = ),,( xtBB = (II) ),(tAA = ),(tBB = ,BA ≠

(III) ),(tAA = ),(tBB = ,BA = (IV) ),(tAA = constant,=B

(V) ),,( xtAA = ),(xBB = (VI) ),,( xtAA = constant,=B

(VII) ),(tAA = ),(xBB = (VIII) ),(xAA = ),(xBB = ,BA ≠

(IX) ),(xAA = ),(xBB = ,BA = (X) ),(xAA = ,constant=B

(XI) ,constant=A ),,( xtBB = (XII) ,constant=A ),(tBB =

(XIII) ,constant=A ),(xBB =

It is important to note that when constant=A and costant=B the space-time becomes

Minkowski and gives the same proper homothetic vector fields as in general relativity.

We will solve equations from (4.3.8) to (4.3.13) for the above lited cases.

Case I:

In this case we have )(tAA = and ).,( xtBB = Considering equation (4.3.8) and using

equation (4.3.14) then differentiating with respect to ,z we get

),,(),(),,(0),,( 2133 xtKxtzKzxtPzxtPzz +=⇒= where ),(1 xtK and ),(2 xtK are


),,(),(),,(),(),,( 31414 xtKxtyKyxtPxtKyxtPy +−=⇒−= where ),(3 xtK is a function


).,(),(),,(),(

),,,(),,,(313212

211)()()(0

xtKxtyKzXxtKxtzKyX

zytPxXzyxPedteeX tAtAtA

+−=++=

+=+= −− ∫αα

αα (4.3.15)

183


respect to ,y we get ⇒=+ 0)(),,( 22 Bxyy exBzytP α ⇒= 0),,(2 zytPyyy

),,(),(),(2

),,( 6542

2 ztKztyKztKyzytP ++= where ),,(4 ztK ),(5 ztK and ),(6 ztK are


⇒=+ 0)(),( 24 Bx exBztK α ⇒= 0),(4 ztK z ),(),( 14 tDztK = where )(1 tD is a

function of integration. Once again substituting back we get ⇒=+ 0)()( 21 Bx exBtD α

),()(2 212 tDtxDe B +−

=α

where )(2 tD is a function of integration. Substituting back the

above values in (4.3.10) and solving after differentiating with respect to ,z we get

),(),( 125 tDztK = )(),( 3),(1 tDextK xtB−= and ),(),( 4),(2 tDextK xtB−= where ),(3 tD

)(4 tD and )(12 tD are functions of integration. Refreshing equation (4.3.15) we get

).,()(),()(

),,()()(2

),,,(

33),(34),(3),(2

61212

11)()()(0

xtKtDeyzXtDetDezyX

ztKtyDtDyxXzyxPedteeX

xtBxtBxtB

tAtAtA

+−=++=

+++=+=

−−−

−− ∫αα

αα (4.3.16)


⇒=+ 0)(),( 26 Bxzz exBztK α ⇒= )(),( 16 tDztK zz ),()()(

2),( 651

26 tDtzDtDzztK ++= where

)(5 tD and )(6 tD are functions of integration. Substituting back the above values in

equation (4.3.11) and solving we get 0)(5 =tD and ),(),( 73 tDextK B−= where )(7 tD is

the function of integration. Substituting all the above information in (4.3.16) we get

).()(),()(

),()()()22

(

),,,(

7),(3),(3

4),(3),(2

612122

1

1)()()(0

tDetDyezXtDetDzeyX

tDtyDtDzyxX

zyxPedteeX

xtBxtB

xtBxtB

tAtAtA

−−

−−

−−

+−=

++=

++++=

+= ∫

α

α

α

α

(4.3.17)

Considering equation (4.3.9) and using equation (4.3.17) then differentiating with respect

to x we get ),,(),(),,(0),,( 8711 zyKzyxKzyxPzyxPxx +=⇒= where ),(7 zyK and

184

),(8 zyK are functions of integration. Substituting back the above value in (4.3.9) and

solve after differentiating with respect to y twice we get ℜ∈= 111 ,)( cctD and

),()(),( 987 zDzyDzyK += where )(8 zD and )(9 zD are functions of integration. Once

again substituting back the above information in (4.3.9) and solve after differentiating

with respect to y and z respectively, we get ,,)( 2212 ℜ∈= cctD ,0)(8 =zD 3

9 )( czD =

and .,,)( 11311)(

36 ℜ∈+= ∫ cccdtectD tA Refreshing the above system of equations (4.3.17)

we get

),()(),()(

,)22

(

,),(

7),(3),(3

4),(3),(2

11)(

321

221

)(3

8)()()(0

tDetDeyzXtDetDezyX

cdtecycczyxX

exczyKedteeX

xtBxtB

xtBxtB

tA

tAtAtAtA

−−

−−

−−−

+−=

++=

+++++=

++=

∫

∫

α

α

α

α

(4.3.18)

where ).0(,, 1321 ≠ℜ∈ cccc Considering equation (4.3.12) and using equation (4.3.18)

then differentiating with respect to ,y we get ⇒=− 0)(),( 28 Btyy

A exBzyKe α

⇒= 0),(8 zyK yyy ),()()(2

),( 1311102

8 zDzyDzDyzyK ++= where ),(10 zD )(11 zD and

)(13 zD are functions of integration. Substituting back this value in the above equation we

get ⇒=− 0)( 210 Bt

A eBzDe α .,)(0)( 441010 ℜ∈=⇒= cczDzDz Now

).(),(2)()(2 2),(2212 tDextBtDtxDe txtB

tB =⇒+

−=

α Substituting back this value in the

above equation we get ⇒=− 0)(2

2)(4 tDec t

tA α .,2

)( 55)(42 ℜ∈+= ∫ ccdte

ctD tA

α

Substituting back all the above values in (4.3.12) and solve after differentiating first with

respect to x then with respect to ,z we get ,)( 63 ctD = ℜ∈= 767

4 ,,)( ccctD and

.0)(11 =zD Substituting back all the above information in (4.3.18) we get

185

),(,

,)22

(

,))(2

(

7),(6

),(37

),(6

),(2

11)(

321

221

)(3

134

2)()()(0

tDeceyzXcecezyX

cdtecycczyxX

exczDcyedteeX

xtBxtBxtBxtB

tA

tAtAtAtA

−−−−

−−−

+−=++=

+++++=

+++=

∫

∫

αα

α

α

(4.3.19)

where ).0,0(,,,,,, 4111764321 ≠≠ℜ∈ ccccccccc Considering equation (4.3.13) and using

equation (4.3.19) then differentiating with respect to ,z we get ⇒=− 0)(134 zDc zz

.,,2

)( 1091094

213 ℜ∈++= ccczcczzD Substituting back the above value in (4.3.13) and

solving we get ℜ∈= 887 ,)( cctD and .092 == cc Substituting all the above

information in (4.3.19), we get

,,

,)22

(

,)22

(

8),(

6),(3

7),(

6),(2

11)(

31

221

)(310

)(4

22)()()(0

ceceyzXcecezyX

cdtecczyxX

excceczyedteeX

xtBxtBxtBxtB

tA

tAtAtAtAtA

−−−−

−−−−

+−=++=

++++=

++++=

∫

∫

αα

α

α

(4.3.20)

where ).0,0(,,,,,,, 411110876431 ≠≠ℜ∈ cccccccccc After a suitable rescaling of ,t the

metric for non static plane symmetric space-time becomes

),)(22( 2241222 dzdytcxcdxdtds ++−

++−=αα

(4.3.21)

where ).0,0(, 4141 ≠≠ℜ∈ cccc The solution of equations (4.3.6) to (4.3.13) becomes

,,

,)22

(,)22

(

8),(

6),(3

7),(

6),(2

131

221

1034

220

ceceyzXcecezyX

ctcczyxXcxcczytX

xtBxtBxtBxtB −−−− +−=++=

++++=++++=

αα

αα (4.3.22)

where ).0,0(,,,,,,, 411110876431 ≠≠ℜ∈ cccccccccc The above space-time (4.3.21)

admits seven linearly independent teleparallel homothetic vector fields in which six are

teleparallel Killing vector fields given as ,t∂∂ ,

x∂∂ ),(

xt

tx

∂∂

+∂∂ ,),(

ye xtB

∂∂−

,),(

ze xtB

∂∂− )(),(

zy

yze xtB

∂∂

−∂∂− and one is teleparallel proper homothetic vector field.

186

Teleparallel proper homothetic vector field after subtracting teleparallel Killing vector

fields from (4.3.22) is [74]

.,,)22

(,)22

( 321

221

4

220 zXyXczyxXczytX αααα ==++=++= (4.3.23)

Case (II):

In this case we have ,),(),( BAtBBtAA ≠== In this case the line element (4.3.1) after

a suitable rescaling of t becomes

),()2( 2232

222 dzdyctcdxdtds ++++−=α

(4.3.24)

where ).0(, 232 ≠ℜ∈ ccc Solution of equations (4.3.6) to (4.3.13) is given by

.,

,,)22

(

10)(

5)(3

6)(

5)(2

1111

912

220

ceceyzXcecezyX

cctxXccxczytX

tBtBtBtB −−−− +−=++=

++=++++=

αα

αα (4.3.25)

where .,,,,, 11109651 ℜ∈cccccc The above space-time (4.3.24) admits seven linearly

independent teleparallel homothetic vector fields in which six are teleparallel Killing

vector fields given as ,t∂∂ ,

x∂∂ ),(

xt

tx

∂∂

+∂∂ ,)(

ye tB

∂∂− ,)(

ze tB

∂∂−

)()(

zy

yze tB

∂∂

−∂∂− and one is teleparallel proper homothetic vector field. Teleparallel

proper homothetic vector field after subtracting teleparallel Killing vector fields from

equation (4.3.25) is

.,,,)22

( 3212


The results listed above can be obtained by the same process as in case (I).

Case (III):

In this case we have ),(tAA = )(tBB = and .BA = In this case the line element (4.3.1)

becomes

187

),()( 22223

222 dzyddtctcdxds ++−++=α

(4.3.27)

where ).0(, 232 ≠ℜ∈ ccc Solution of equations (4.3.6) to (4.3.13) is given by

.

,,

,)22

()(2

11)()(

96)(3

7)(

5)(

6)(2

11

)(10

)(9

)(5

)(2

22

32

2

20

ceetcceyzX

cectecezyXcxX

ecezcecyeczyctcc

X

tAtAtA

tAtAtA

tAtAtAtA

−−−

−−−

−−−−

++−=

+++=+=

++++++=

α

αα

αα

(4.3.28)

where .,,,,,,,, 11109765321 ℜ∈ccccccccc The above space-time (4.3.27) admits eight

linearly independent teleparallel homothetic vector fields in which seven are teleparallel

Killing vector fields given as ,)(

te tA

∂∂− ,

x∂∂ ),()(

yt

tye tA

∂∂

+∂∂− ),()(

zt

tze tA

∂∂

+∂∂−

,)(

ye tA

∂∂− ,)(

ze tA

∂∂− )()(

zy

yze tA

∂∂

−∂∂− and one is teleparallel proper homothetic

vector field. Teleparallel proper homothetic vector field after subtracting teleparallel

Killing vector fields from system of equations (4.3.28) is

.,,,)22

()(2

321)(2

22

32

2

20 zXyXxXeczyctc

cX tA ααα

αα

===+++= − (4.3.29)

The results listed above can be obtained by the same process as in case (I).

Case (IV):

In this case we have )(tAA = and .constant=B The line element (4.3.1) after a suitable

rescaling of ,t y and z becomes

,22222 dzyddxdtds +++−= (4.3.30)

Teleparallel homothetic vector fields in this case are

.,

,,

76513

43212

108521

98630

cctcxcyzXctccxczyX

cctczcyxXccxczcytX

++−−=++++=

+++−=++++=

αα

αα (4.3.31)

where .,,,,,,,,, 10987654321 ℜ∈cccccccccc The above space-time (4.3.30) admits eleven

linearly independent teleparallel homothetic vector fields in which ten are teleparallel

188

Killing vector fields given as ,t∂∂ ,

x∂∂ ),(

xt

tx

∂∂

+∂∂ ),(

xy

yx

∂∂

−∂∂ ),(

yt

ty

∂∂

+∂∂

),(z

tt

z∂∂

+∂∂ ),(

zx

xz

∂∂

−∂∂ ,

y∂∂ ,

z∂∂ )(

zy

yz

∂∂

−∂∂ and one is teleparallel proper

homothetic vector field. Teleparallel proper homothetic vector field after subtracting

Killing vector fields from equation (4.3.31) is

.,,, 3210 zXyXxXtX αααα ==== (4.3.32)

Case (V):

In this case we have ),( xtAA = and ).(xBB = Considering equation (4.3.8) and using

equation (4.3.14) then differentiating with respect to ,z we get

),,(),(),,(0),,( 2133 xtKxtzKzxtPzxtPzz +=⇒= where ),(1 xtK and ),(2 xtK are


),,(),(),,(),(),,( 31414 xtKxtyKyxtPxtKyxtPy +−=⇒−= where ),(3 xtK is a function


).,(),(),,(),(

),,,(),,,(313212

211),(),(),(0

xtKxtyKzXxtKxtzKyX

zytPxXzyxPedteeX xtAxtAxtA

+−=++=

+=+= −− ∫αα

αα (4.3.33)

Considering equation (4.3.12) and using equation (4.3.33) then solve after differentiating

with respect to ,y we get ⇒= 0),,(1 zyxPyy ),,(),(),,( 541 zxKzxyKzyxP += where

),(4 zxK and ),(5 zxK are functions of integration. Substituting back the above value in

(4.3.12) and solve after differentiating with respect to ,z we get

),(),(0),( 144 xDzxKzxK z =⇒= where )(1 xD is a functions of integration. Substituting

back the above information in (4.3.12) and solving we get ),(),( 21 xDxtK = 0)(1 =xD

and ),(),( 32 xDxtK = where )(2 xD and )(3 xD are functions of integration. Refreshing


).,()(),()(

),,(),,(323322

215),(),(),(0

xtKxyDzXxDxzDyX

zytPxXzxKedteeX xtAxtAxtA

+−=++=

+=+= −− ∫αα

αα (4.3.34)

189


)(),( 45 xDzxK = and ),(),( 53 xDxtK = where )(3 xD and )(5 xD are functions of

integration. Now substituting the above information and equation (4.3.34) in (4.3.10) and

solve after differentiating twice with respect to ,y we get

),,(),(),(2

),,( 8762

2 ztKztyKztKyzytP ++= where ),,(6 ztK ),(7 ztK and ),(8 ztK

are functions of integration. Substituting back the above value and solving we get

),(),( 66 tDztK = where )(6 tD is a function of integration. Now substituting the above

information in (4.3.10) and differentiating with respect to ,y then solving we get

ℜ∈= 116 ,)( cctD and ⇒=+ 0)( 2

1B

x exBc α ,22

12 cxce B +−

=α

).0(, 121 ≠ℜ∈ ccc

Now substituting all the above information in equation (4.3.34) we get

).(,

),,(2

),(

5)(3

3)(4

)(3

2

81

214),(),(),(0

xDeyczXecezcyX

ztKcyxXxDedteeX

xBxBxB

xtAxtAxtA

+−=++=

++=+=

−−−

−− ∫αα

αα (4.3.35)

Considering equation (4.3.11) and using equation (4.3.35) then solving after

differentiating with respect to ,z we get ),()(2

),( 21201

28 tDtzDczztK ++= where

)(20 tD and )(21 tD are functions of integration. Substituting back the above value in

equation (4.3.11) and solving, we get ℜ∈= −5

)(5

5 ,)( cecxD xB and .0)(20 =tD


.,

),()22

(),(

)(5

)(3

3)(4

)(3

2

211

2214),(),(),(0

xBxBxBxB

xtAxtAxtA

eceyczXecezcyX

tDczyxXxDedteeX

−−−−

−−

+−=++=

+++=+= ∫αα

αα (4.3.36)


.0)(),()( 4),(),(21 =−− ∫ xDedtextAetD xAxtA

xxtA

t α In order to solve this equation we take

844 )(0)( cxDxDx =⇒= and .,,,)()( 87676

216

21 ℜ∈+=⇒= cccctctDctDt

Substituting back these values in the above equation we get

190

⇒=− ∫ 0),( ),(),(6 dtextAec xtA

xxtAα .ln),( 6

tcx

xtAα

= Finally, the metric of non static

plane symmetric space-time in this case takes the form

( ).2 222

12262 dzdycxc

dxdtt

xcds +⎟

⎠⎞

⎜⎝⎛ +−

++⎟⎟⎠

⎞⎜⎜⎝

⎛−=

αα (4.3.37)

Teleparallel homothetic vector fields in this case are

.,

,)22

(,2

)(5

)(3

3)(4

)(3

2

71

22

61

8),(0

xBxBxBxB

xtA

eceyczXecezcyX

cczytcxXcetX

−−−−

−

+−=++=

++++=+=

αα

αα (4.3.38)

where ).0,0(,,,,,,, 6187654321 >≠ℜ∈ cccccccccc The above space-time (4.3.37) admits

six linearly independent teleparallel homothetic vector fields in which five are teleparallel

Killing vector fields given as ,),(

te xtA

∂∂− ,

x∂∂ ,)(

ye xB

∂∂− ,)(

ze xB

∂∂−

)()(

zy

yze xB

∂∂



equation (4.3.38) is given as

.,,)22

(,2 321

22

610 zXyXczytcxXtX αααα ==+++== (4.3.39)

Case (VI):

In this case we have ),( xtAA = and constant.=B The metric for non static plane

symmetric space time after a suitable rescaling of y and z becomes

.222262 dzdydxdtt

xcds +++⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

α (4.3.40)

Solving equations (4.3.6) to (4.3.13) by the same procedure as explained above with the

help of equation (4.3.14), teleparallel homothetic vector fields become

.,

,,2

5413

3212

87421

6),(0

cxcyczXcxczcyX

ctczccyxXcetX xtA

+−−=+++=

+++−=+= −

αα

αα (4.3.41)

191

where .,,,,,,, 87654321 ℜ∈cccccccc The above space-time (4.3.40) admits eight linearly

independent teleparallel homothetic vector fields in which seven are teleparallel Killing

vector fields given as ,),(

te xtA

∂∂− ,

x∂∂ ),(

xy

yx

∂∂

−∂∂ ),(

zx

xz

∂∂

−∂∂ ,

y∂∂ ,

z∂∂

)(z

yy

z∂∂

−∂∂ and one is teleparallel proper homothetic vector field. Teleparallel proper

homothetic vector field after subtracting Killing vector fields from equation (4.3.41) is

given as

.,,,2 327

10 zXyXctxXtX αααα ==+== (4.3.42)

Case (VII):

In this case we have )(tAA = and ).(xBB = Considering equation (4.3.8) and using

equation (4.3.14) then solving after differentiating with respect to ,z we get

),(),(),,( 213 xtKxtzKzxtP += and ),,(),(),,( 314 xtKxtyKyxtP +−= where ),,(1 xtK

),(2 xtK and ),(3 xtK are functions of integration. Refreshing the system of equations

(4.3.14) we get

).,(),(),,(),(

),,,(),,,(313212

211)()()(0

xtKxtyKzXxtKxtzKyX

zytPxXzyxPedteeX tAtAtA

+−=++=

+=+= −− ∫αα

αα (4.3.43)


differentiating twice with respect to ,y we get ),(),(),,( 541 zxKzxyKzyxP += where

),(4 zxK and ),(5 zxK are functions of integration. Substituting back the above value in

(4.3.12) and solving after differentiating with respect to ,z we get ),(),( 14 xDzxK =

where )(1 xD is a functions of integration. Substituting back the above values in (4.3.12)

and solving we get ),(),( 21 xDxtK = 0)(1 =xD and ),(),( 32 xDxtK = where )(2 xD

and )(3 xD are functions of integration. Refreshing (4.3.43) we get

).,()(),()(

),,,(),,(323322

215)()()(0

xtKxyDzXxDxzDyX

zytPxXzxKedteeX tAtAtA

+−=++=

+=+= −− ∫αα

αα (4.3.44)

192

Considering (4.3.13) and using equation (4.3.44) then solving we get )(),( 45 xDzxK =

and ),(),( 53 xDxtK = where )(4 xD and )(5 xD are functions of integration. Now

substituting the above information and equation (4.3.44) in (4.3.10) and solve after


),,(),(),(2

),,( 8762

2 ztKztyKztKyzytP ++= where ),,(6 ztK ),(7 ztK and ),(8 ztK

are functions of integration. Substituting back and solving we get ),(),( 66 tDztK =

where )(6 tD is a function of integration. Now substituting the above information in

(4.3.10) and differentiating with respect to ,y then solving we get ℜ∈= 116 ,)( cctD

which on back substitution gives ⇒=+ 0)( 21

Bx exBc α

,22

12 cxce B +−

=α

).0(, 121 ≠ℜ∈ ccc Now substituting back the above values in

(4.3.10) and solve after differentiating with respect to ,z we get ),(),( 77 tDztK = where

)(7 tD is a function of integration and ,)( 32 BecxD −= ℜ∈= −

4343 ,,)( ccectD B and

.0)(7 =tD Refreshing (4.3.44) we get

).(,

),,(2

),(

5)(3

3)(4

)(3

2

81

214)()()(0

xDeyczXecezcyX

ztKcyxXxDedteeX

xBxBxB

tAtAtA

+−=++=

++=+=

−−−

−− ∫αα

αα (4.3.45)


differentiating with respect to ,z we get ),()(2

),( 21201

28 tDtzDczztK ++= where

)(20 tD and )(21 tD are functions of integration. Substituting back we get

ℜ∈= −55

5 ,)( cecxD B and .0)(20 =tD Refreshing the system of equations (4.3.45) we

get

.,

),()22

(),(

)(5

)(3

3)(4

)(3

2

211

2214)()()(0

xBxBxBxB

tAtAtA

eceyczXecezcyX

tDczyxXxDedteeX

−−−−

−−

+−=++=

+++=+= ∫αα

αα (4.3.46)

193


874 )( cxcxD += and .,,,)( 8766

)(7

21 ℜ∈+= ∫ ccccdtectD tA Substituting the above

information in equation (4.3.46), we get

.,

,)22

(

),(

)(5

)(3

3)(4

)(3

2

6)(

71

221

87)()()(0

xBxBxBxB

tA

tAtAtA

eceyczXecezcyX

cdtecczyxX

cxcedteeX

−−−−

−−

+−=++=

++++=

++=

∫

∫

αα

α

α

(4.3.47)

The space-time (4.3.1) in this case after a suitable rescaling of ,t becomes

( ).2 2221

222 dzdycxcdxdtds +⎟⎠⎞

⎜⎝⎛ +−

++−=α

(4.3.48)

Teleparallel homothetic vector fields for the above space-time (4.3.48) becomes

.,

,)22

(,

)(5

)(3

3)(4

)(3

2

671

221

870

xBxBxBxB eceyczXecezcyX

ctcczyxXcxctX

−−−− +−=++=

++++=++=

αα

αα (4.3.49)

where ).0(,,,,,,, 187654321 ≠ℜ∈ ccccccccc The above space-time (4.3.48) admits seven

linearly independent teleparallel homothetic vector fields in which six are teleparallel


x∂∂ ,)(

ye xB

∂∂− ,)(

ze xB

∂∂− ),()(

zy

yze xB

∂∂

−∂∂−

xt

tx

∂∂

+∂∂ and one is teleparallel proper homothetic vector field. Teleparallel proper

homothetic vector field after subtracting teleparallel Killing vector fields from equation

(4.3.49) is [74]

.,,)22

(, 321

2210 zXyXczyxXtX αααα ==++== (4.3.50)

Case (VIII):

In this case we have ),(xAA = )(xBB = and .BA ≠ Considering eqution (4.3.8) and

using equation (4.3.14) then solve after differentiating with respect to ,z we get

),(),(),,( 213 xtKxtzKzxtP += and ),,(),(),,( 314 xtKxtyKyxtP +−= where ),,(1 xtK

194

),(2 xtK and ),(3 xtK are functions of integration. Refreshing the system of equations

(4.3.14) we get

).,(),(),,(),(),,,(),,,(

313212

211)(0

xtKxtyKzXxtKxtzKyXzytPxXzyxPetX xA

+−=++=

+=+= −

αα

αα (4.3.51)

Considering equation (4.3.9) and using equation (4.3.51) then solving after differentiating

twice with respect to ,t we get ),(),(),(2

),,( 6542

2 zyKzytKzyKtzytP ++= where

),,(4 zyK ),(5 zyK and ),(6 zyK are functions of integration. Substituting back the

above value in (4.3.9) and solving after differentiating with respect to ,y we get

),(),( 14 zDzyK = where )(1 zD is a functions of integration. Substituting back the above

values in (4.3.9) and solving we get 11 )( czD = and ,2

21)(2 cxce xA +=

α

).0(, 121 ≠ℜ∈ ccc Substituting all the above information in equation (4.3.9) and solving

we get 0),(5 =zyK and ),,(),,( 7)(1 zyKezyxP xA−= where ),(7 zyK is a function of

integration. Now substituting back the above information in (4.3.51), we get

).,(),(),,(),(

),,(2

),,(

313212

61

217)(0

xtKxtyKzXxtKxtzKyX

zyKctxXzyKetX xA

+−=++=

++=+= −

αα

αα (4.3.52)


with respect to ,y we get ),()()(2

),( 4322

6 zDzyDzDyzyK ++= where ),(2 zD )(3 zD

and )(4 zD are functions of integration. Substituting back the abvoe value in (4.3.10) and

solve after differentiating with respect to y and z we get .,)( 332 ℜ∈= cczD

Substituting back all the above information in equation (4.3.10) and solving we get

,243

)(2 cxce xB +−

=α

).0(, 343 ≠ℜ∈ ccc Now substituting all the above values in (4.3.9)

and solving we get ,0)(3 =zD )(),( 7)(1 tDextK xB−= and ),(),( 8)(2 tDextK xB−= where

)(7 tD and )(8 tD are functions of integration. Refreshing the system of equations

(4.3.52) we get

195

).,()(,)()(

),(22

),,(

3)(738)(72

43

2

1

217)(0

xtKetyDzXetDetzDyX

zDcyctxXzyKetX

xBBxB

xA

+−=++=

+++=+=

−−−

−

αα

αα(4.3.53)

Considering equation (4.3.11) and using (4.3.53) then solve after differentiating the

resulting equation with respect to z twice we get .,,2

)( 1555153

24 ℜ∈++= ccczcczzD

Substituting back the above value in (4.3.11) we get 015 =c and ),(),( 9)(3 tDextK xB−=

where )(9 tD is a function of integration. Substituting all the above information in

(4.3.53) we get

.)()(,)()(

,)22

(2

),,(

)(9)(738)(72

53

22

1

217)(0

xBxBBxB

xA

etDetyDzXetDetzDyX

cczyctxXzyKetX

−−−−

−

+−=++=

++++=+=

αα

αα (4.3.54)


twice with respect to ,y we get ),()()(2

),( 1211102

7 zDzyDzDyzyK ++= where ),(10 zD

)(11 zD and )(12 zD are functions of integration. Substituting back the above value and

solving we get .0)(10 =zD Now substituting the above information in (4.3.12) and

solving we get ,)( 87 ctD = ℜ∈= 989

8 ,,)( ccctD and .0)(11 =zD Also substituting the

above values and equation (4.3.54) in equation (4.3.13) we get 109 )( ctD = and

.,,)( 11101112 ℜ∈= ccczD Now refreshing the system of equations (4.3.54) the

teleparallel homothetic vector fields are given as [74]

.,

,)22

(2

,

)(10

)(8

39

)(8

2

53

22

1

21

11)(0

xBxBBxB

xA

ececyzXececzyX

cczyctxXcetX

−−−−

−

+−=++=

++++=+=

αα

αα (4.3.55)

where ).,0,0(,,,,, 313111109,8531 ccccccccccc ≠≠≠ℜ∈ The line element (4.3.1) in this

case takes the form

),()2

()2

( 224

3222

12 zdydcxc

dxdtcxc

ds ++−+++−=αα

(4.3.56)

196

where ).,0,0(,,, 31314321 cccccccc ≠≠≠ℜ∈ The above space-time (4.3.56) admits six

linearly independent teleparallel homothetic vector fields in which five are teleparallel


te xA

∂∂− ,

x∂∂ ,)(

ye xB

∂∂− ,)(

ze xB

∂∂−

)()(

zy

yze xB

∂∂




.,,)22

(2

, 323

22

1

210 zXyXczyctxXtX αααα ==+++== (4.3.57)

Case (IX):

In this case we have ),(xAA = )(xBB = and .BA = The line element (4.3.1) takes the

form

),)(2

( 2223

222 dzdydtcxc

dxds ++−+−+=α

(4.3.58)

where ).0(, 232 ≠ℜ∈ ccc Teleparallel homothetic vector fields are given as

.

,,)2t

22(

,

11)()(

96)(3

8)(

7)(

6)(2

52

2221

)(10

)(9

)(7

0

ceetcceyzX

cectecezyXcczyxX

ececzecytX

xAxAxA

xAxAxA

xAxAxA

−−−

−−−

−−−

++−=

+++=+−++=

+++=

α

αα

α

(4.3.59)

where ).0(,,,,,,, 21110987652 ≠ℜ∈ ccccccccc The above space-time (4.3.58) admits eight



te xA

∂∂− ,

x∂∂ ),()(

yt

tye xA

∂∂

+∂∂− ),()(

zt

tze xA

∂∂

+∂∂−

,)(

ye xA

∂∂− ,)(

ze xA

∂∂− )()(

zy

yze xA

∂∂

−∂∂− and one is teleparallel proper homothetic

vector field. Teleparallel proper homothetic vector field after subtracting teleparallel

Killing vector fields from equation (4.3.59) is

197

.,,)2t

22(, 32

2

22210 zXyXczyxXtX αααα ==−++== (4.3.60)

The result listed above can be obtained by the same procedure as adopted above in case

(VIII).

Case (X):

In this case we have )(xAA = and .constant=B The line element (4.3.1) after a suitable

rescaling of y and z becomes

,)2

( 22222

12 dzyddxdtcxc

ds ++++−=α

(4.3.61)

where ).0(, 121 ≠ℜ∈ ccc Solution of equations (4.3.6) to (4.3.13) is given by [74]

.,

,2

,

9453

6352

10431

21

8)(0

ccxcyzXccxczyX

cczcyctxXcetX xA

+−−=+−+=

++++=+= −

αα

αα (4.3.62)

where ).0(,,,,,,, 1109865431 ≠ℜ∈ ccccccccc The above space-time (4.3.61) admits eight



te xA

∂∂− ,

x∂∂ ),(

yx

xy

∂∂

−∂∂ ),(

zx

xz

∂∂

−∂∂ ,

y∂∂

,z∂∂ )(

zy

yz

∂∂

−∂∂ and one is teleparallel proper homothetic vector field. Teleparallel



.,,2t, 32

1

210 zXyXcxXtX αααα ==+== (4.3.63)


(VIII).

Case (XI):

In this case we have constant=A and ).,( xtBB = The metric for non static plan

symmetric space-times in this case becomes

198

),)(22

( 2241222 dzdytc

xc

dxdtds ++−

++−=αα

(4.3.64)

where ).0,0(, 4141 ≠≠ℜ∈ cccc The solution of equations (4.3.6) to (4.3.13) becomes

[74]

,,

,)22

(,)22

(

8),(

6),(3

7),(

6),(2

1131

221

1034

220

ceceyzXcecezyX

ctcczyxXcxcczytX

xtBxtBxtBxtB −−−− +−=++=

++++=++++=

αα

αα (4.3.65)

where ).0,0(,,,,,,, 411110876431 ≠≠ℜ∈ cccccccccc The above space-time (4.3.64)

admits seven linearly independent teleparallel homothetic vector fields in which six are

teleparallel Killing vector fields given as ,t∂∂ ,

x∂∂ ),(

xt

tx

∂∂

+∂∂ ,),(

ye xtB

∂∂−

,),(

ze xtB

∂∂− )(),(

zy

yze xtB

∂∂

−∂∂− and one is teleparallel proper homothetic vector field.

Teleparallel proper homothetic vector field after subtracting teleparallel Killing vector

fields from (4.3.65) is

.,,)22

(,)22

( 321

221

4

220 zXyXczyxXczytX αααα ==++=++= (4.3.66)


(I).

Case (XII):

In this case we have constant=A and ).(tBB = The line element for the metric (4.3.1)

takes the form

),)(2

( 228

7222 dzdyctc

dxdtds ++−

++−=α

(4.3.67)

where ).0(, 787 ≠ℜ∈ ccc The solution of equations (4.3.6) to (4.3.13) becomes [74]

,,

,,)22

(

3)(

1)(3

2)(

1)(2

651

457

220

ceceyzXcecezyX

ctcxXcxcczytX

tBtBtBtB −−−− +−=++=

++=++++=

αα

αα (4.3.68)

199

where ).0(,,,,,, 77654321 ≠ℜ∈ cccccccc The above space-time (4.3.67) admits seven



x∂∂ ),(

xt

tx

∂∂

+∂∂ ,)(

ye tB

∂∂− ,)(

ze tB

∂∂−

)()(

zy

yze tB

∂∂


proper homothetic vector field after subtracting Killing vector fields from (4.3.68) is

given as

.,,,)22

( 3217



(II).

Case (XIII):

In this case we have constant=A and ).(xBB = The line element for the metric (4.3.1)

takes the form

),)(2

( 224

3222 dzdycxc

dxdtds ++−

++−=α

(4.3.70)

where ).0(, 343 ≠ℜ∈ ccc Teleparallel homothetic vector fields in this case are given as

,,

,)22

(,

10)(

6)(3

7)(

6)(2

913

221

210

ceceyzXcecezyX

ctcczyxXcxctX

xBxBxBxB −−−− +−=++=

++++=++=

αα

αα (4.3.71)

where ).0(,,,,,, 310976321 ≠ℜ∈ cccccccc The above space-time (4.3.70) admits seven



x∂∂ ),(

xt

tx

∂∂

+∂∂ ,)(

ye xB

∂∂− ,)(

ze xB

∂∂−

)()(

zy

yze xB

∂∂


200

proper homothetic vector field after subtracting Killing vector fields from (4.3.71) is

given as

.,,)22

(, 323

2210 zXyXczyxXtX αααα ==++== (4.3.72)


(VII).

4.4. Teleparallel Proper Homothetic Vector Fields in

Static Cylindrically Symmetric Space-Times




where ,A B and C are functions of r only. The above space-time admits minimum

seven linearly independent teleparallel Killing vector fields which are ,r∂∂ ,2

)(

te

rA

∂∂−

,2)(

θ∂∂− rB

e ,2)(

ze

rC

∂∂−

,)(1

trM

rt

∂∂

+∂∂

θθ

∂∂

+∂∂ )(2 rMr

and ,)(3

zrM

rz

∂∂

+∂∂ where

,)( 2)(

2)(

1 ∫−−

= dreerMrArA

dreerMrBrB

∫−−

−= 2)(

2)(

2 )( and .)( 2)(

2)(

3 ∫−−

−= dreerMrCrC

The

tetrad components and its inverse, non-vanishing Weitzenböck connections and the non

vanishing torsion components for (4.4.1) are given in equations (3.3.2), (3.3.3) and

(3.3.4). Now using (4.4.1) and (3.3.4) in (1.3.17) we get the teleparallel homothetic

equations as follows:

,3,3

2,2

1,1

0,0 α==== XXXX (4.4.2)

,02,0)(

0,2)( =− XeXe rArB (4.4.3)

,03,0)(

0,3)( =− XeXe rArC (4.4.4)

201

,03,2)(

2,3)( =+ XeXe rBrC (4.4.5)

,02

0)(1,

0)(0,

1 =′

−− XeAXeX rArA (4.4.6)

,02

2)(2,

11,

2)( =′

++ XeBXXe rBrB (4.4.7)

,02

3)(3,

11,

3)( =′

++ XeCXXe rCrC (4.4.8)


),,,(),,,(),,,(),,,(

4332

2110

θαθα

θαθα

rtPzXzrtPXztPrXzrPtX

+=+=

+=+= (4.4.9)

where ),,,(1 zrP θ ),,,(2 ztP θ ),,(3 zrtP and ),,(4 θrtP are functions of integration

which are to be determined. In order to find solution for equations (4.4.2) to (4.4.8) we

will consider each possible form of the metric for static cylindrically symmetric space-

times and then solve each possibility in turn. Following are the possible cases for the

metric where the above space-times admit teleparallel proper homothetic vector fields:

(I) )(),(),( rCCrBBrAA === and .,, CBCABA ≠≠≠

(II)(a) ),(),( rBBrAA == and .tan tconsC =

(II)(b) ),(),( rCCrAA == and .tan tconsB =

(II)(c) ),(),( rCCrBB == and .tan tconsA =

(III)(a) )(),(),( rCCrBBrAA === and ).()( rCrB =

(III)(b) )(),(),( rCCrBBrAA === and ).()( rCrA =

(III)(c) )(),(),( rCCrBBrAA === and ).()( rBrA =

(IV) )(),(),( rCCrBBrAA === and ).()()( rCrBrA ==

(V)(a) )(),(,tan rCCrBBtconsA === and ).()( rCrB =

(V)(b) )(,tan),( rCCtconsBrAA === and ).()( rCrA =

(V)(c) tconsCrBBrAA tan),(),( === and ).()( rBrA =

202

(VI)(a) )(rAA = and .tan tconsCB ==

(VI)(b) )(rBB = and .tan tconsCA ==

(VI)(c) )(rCC = and .tan tconsBA ==


Case (I):

In this case we have ),(rAA = ),(rBB = ),(rCC = ,BA ≠ CA ≠ and .CB ≠ Now


.0),,(),,( 1)(3)( =− zrPezrtPe rAt

rB θθ (4.4.10)




⇒= − ),(),,( 11 zrEezrP ABθθ ),,(),(),,( 311 zrEzrEezrP AB += −θθ where ),(3 zrE is a


).,,(),,(),(),,,(),,(),(

43212

21310

θαθα

θαθα

rtPzXzrEzrEtXztPrXzrEzrEetX AB

+=++=

+=++= −

(4.4.11)



trC θθ (4.4.12)




.0)],(),([),( 31)()()(4)( =+− − zrEzrEeerEe zzrArBrArC θθ Differentiating the above equation

with respect to θ twice, we get ),()(),(0),( 2144 rKrKrErE +=⇒= θθθθθ where

)(1 rK and )(2 rK are functions of integration. Substituting back the above value in

equation (4.4.12) and solving we get )()(),( 31)()(1 rKrKezzrE rBrC += − and

203

),()(),( 42)()(3 rKrKezzrE rArC += − where )(3 rK and )(4 rK are functions of


( )( )

( ) ).,()()(),,()()(),,,(

),()()()(

5213

231)()(221

42)()(3)()(1)()(0

θθα

θαθα

θα

rErKrKtzXzrErKrKeztXztPrX

rKrKezrKerKeztXrBrC

rArCrArBrArC

+++=

+++=+=

++++=−

−−−

(4.4.13)



Differentiating the above equation first with respect to t and then with respect to ,θ we

get 0)(1 =rK and ),()(),(0),( 6555 rKrKrErE +=⇒= θθθθθ where )(5 rK and

)(6 rK are functions of integration. Substituting back the above values in (4.4.14) we get

),()(),()(),( 75)()(25)()(2 rKrKezzrErKezrE rBrCrBrCz +−=⇒−= −− where )(7 rK is a

function of integration. Substituting all the above information in (4.4.13) we get

( )

).()()(),()()(),,,(

),()()(

6523

75)()(3221

42)()(3)()(0

rKrKrtKzXrKrKezrKtXztPrX

rKrKezrKetXrBrC

rArCrArB

+++=

+−+=+=

+++=−

−−

θα

θαθα

θα

(4.4.15)


.0)()(21)(

21)()(

)(21)()(

21),,(

4)()(4)(2)(

2)(3)(3)(2

=−−−−

⎟⎠⎞

⎜⎝⎛ −−−⎟

⎠⎞

⎜⎝⎛ −−

rKerAertArKerKez

rKeACzrKerKeABztP

rAr

rArr

rAr

rC

rCrrr

rBrBrrt

α

θθθ (4.4.16)

Differentiating equation (4.4.16) with respect to t we get .0)(21),,( )(2 =− rA

rtt erAztP αθ

Differentiating this equation with respect to ,t we get ⇒= 0),,(2 ztPttt θ

),,(),(),(2

),,( 8762

2 zEztEzEtztP θθθθ ++= where ),,(6 zE θ ),(7 zE θ and ),(8 zE θ are


.0)(21),( )(6 =− rA

r erAzE αθ Differentiating this equation with respect to ,θ we get

204

),(),(0),( 866 zKzEzE =⇒= θθθ where )(8 zK is a function of integration. Substituting back

in the above equation and differentiating with respect to ,z we get

.,)(0)( 1188 ℜ∈=⇒= cczKzKz Once again substituting back and solving we get

.,20)(21

221)()(

1 ℜ∈+=⇒=− ccrceerAc rArAr α

α Substituting all the above information in

equation (4.4.16) and differentiating with respect to ,θ we get

.0)()(21),( 3)(3)(7 =−⎟

⎠⎞


rBrBrrθθ Differentiating this equation with respect to

,θ we get ⇒=0),(7 zE θθθ ),()(),( 1097 zKzKzE +=θθ where )(9 zK and )(10 zK are

functions of integration. Substituting back the above value and differentiating with

respect to ,z we get .,)(0)( 3399 ℜ∈=⇒= cczKzKz Now substituting back this value

and solving we get, .,)( 4

)(2

)(

42

)()(2

)(

33 ℜ∈+=

−−

−


Substituting all the

above information in equation (4.4.16) and differentiating twice with respect to ,z we get



.,)( 7

)(2

)(

72

)()(2

)(

52 ℜ∈+=

−−

−

∫ cecdreecrKrCrArArCrA



.,)( 82

)(

82

)(2

)(

64 ℜ∈+=

−−−


Refreshing the system of equations (4.4.15)

we get

).()(

),()(

),,(2

,

65)(2

)(

72

)()(2

)(

53

75)()()(2

)(

42

)()(2

)(

32

86531

21

2)(

82

)(2

)(

62

)(

72

)(2

)(

52

)(

42

)(2

)(

30

rKrKetcdreectzX

rKrKezectdreectX

zEtcctzctctrX

ecdreececzdreeczecdreectX

rCrArArCrA

rBrCrBrArArBrA

rArArArArArArArArA

++++=

+−++=

+++++=

++++++=

−−

−

−−−

−

−−−−−−−−−

∫

∫

∫∫∫

θα

θα

θθα

θθα

(4.4.17)


205

.0)()(21)(

21

)()()()](21)([

)]()([21)]()([

21),(2

7)()(

7)(5)(5)(

2)(

42

)(2

)(

38

3

=++

+−−−

−+−++ ∫−

rKerBerB

rKerKezrKerBrCz


rBr

rBr

rrB

rrCrC

rr

rA

rr

rArA

rr

θα

θθ

(4.4.18)


.0)]()([21)]()([

212 2

)(

42

)(2

)(

33 =−+−+ ∫− rA

rr

rArA


equation and remember that in this case ,0)()( ≠− rBrA rr we get .021 == cc

Substituting the above information in equation (4.4.18) and differentiating twice with

respect to ,θ we get ),()()(2

),(0),( 1312112

88 zKzKzKzEzE ++=⇒= θθθθθθθ where

),(11 zK )(12 zK and )(13 zK are functions of integration. Now substituting back this

information and differentiating with respect to ,z we get

.,)(0)( 991111 ℜ∈=⇒= cczKzKz Substituting back these information in equation

(4.4.18) and differentiating the resulting equation with respect to ,θ we get

.,20)(21

10109)()(

9 ℜ∈+−=⇒=+ ccrceerBc rBrBr α

α Substituting back the above information

in equation (4.4.18) and differentiating the resulting equation with respect to ,z we get

.0)()(21)( 5)(5)(12 =−⎟

⎠⎞

⎜⎝⎛ −− rKerKeBCzK r

rCrCrrz Differentiating this equation with respect to ,z

we get .,,)(0)( 121112111212 ℜ∈+=⇒= ccczczKzKzz Substituting back this value in the above

equation and solving we get ,)()(

2)(

132

)()(2

)(

115 rCrBrBrCrB

ecdreecrK−

−−

+= ∫ .13 ℜ∈c Now


.,)(0)()()(21

142

)(

142

)(2

)(

1277)(7)(

12 ℜ∈+−=⇒=++−−−


rrBrB

r


206

).(

,

),(22

,

6)(2

)(

132

)()(2

)(

11

)(2

)(

72

)()(2

)(

53

2)(

142

)(2

)(

122

)(

132

)(2

)(

112

131211659

2

1

21

2)(

82

)(2

)(

62

)(

72

)(2

)(

50

rKecdreecetcdreectzX

ecdreececzdreeczX

zKczctcctzcctrX

ecdreececzdreecztX

rCrBrBrCrBrCrArArCrA

rBrBrBrBrBrB

rArArArArArA

+++++=

+−−−=

+++++++=

++++=

−−

−−−

−

−−−−−−

−−−−−−

∫∫

∫∫

∫∫

θθα

θα

θθθα

α

(4.4.19)


.0)()(21)(

21)]()([

21

)()]()([21)]()([

21

)]()([21)(22

6)()(2)(

13

6)(2)(

2)(

112

)(

7

2)(

2)(

513

115

=++−+

+−+−+

−+++

∫

∫−

−

rKerCerzCerCrBc

rKedreerCrBcerCrAct

dreerCrActzKctc

rCr

rCr

rB

rr

rrC

rBrB

rr

rA

rr

rArA

rrz

αθ

θ

θ

(4.4.20)


.0)]()([21)]()([

212 2

)(

72

)(2

)(

55 =−+−+ ∫− rA

rr

rArA

rr erCrAcdreerCrAcc Solving this

equation and remember that in this case ,0)()( ≠− rCrA rr we get .075 == cc


,θ we get .0)]()([21)]()([

212 2

)(

132

)(2

)(

1111 =−+−+ ∫− rB

rr

rBrB

rr erCrBcdreerCrBcc

Solving this equation and remember that in this case ,0)()( ≠− rCrB rr we get

.01311 == cc Now substituting the above information in equation (4.4.20) and


.,,,2

)(0)( 171615171615

21313 ℜ∈++=⇒= cccczcczzKzKzzz Substituting back these

information in equation (4.4.20) and differentiating the resulting equation with respect to

,z we get .,20)(21

181815)()(

15 ℜ∈+−=⇒=+ ccrceerCc rCrCr α

α Substituting back this value

in equation (4.4.20) and solving we get ⇒=++ 0)()()(21 6)(6)(

16 rKerKerCc rrCrC

r

207

,)( 2)(

192

)(2

)(

166

rCrCrC

ecdreecrK−−−

+−= ∫ .19 ℜ∈c Refreshing the system of equations (4.4.19)

we get

,,

,222

,

2)(

192

)(2

)(

1632

)(

142

)(2

)(

122

171612615

2

9

2

1

21

2)(

82

)(2

)(

60

rCrCrCrBrBrB

rArArA

ecdreeczXecdreecX

czcctcczcctrX

ecdreectX

−−−−−−

−−−

+−=+−=

+++++++=

++=

∫∫

∫

αθα

θθα

α

(4.4.21)



,)2()2()2( 21815

2109

2221

2 dzcrcdcrcdrdtcrcds +−++−+++−=α

θαα

(4.4.22)

where ).,,(,,,,, 15915191181510921 cccccccccccc ≠≠≠ℜ∈ The above space-time admits eight


Killing vector fields given as ,r∂∂ ,2

)(

te

rA

∂∂−

,2)(

θ∂∂− rB

e ,2)(

ze

rC

∂∂−

,)(1

trM

rt

∂∂

+∂∂

θθ

∂∂

+∂∂ )(2 rMr

and ,)(3

zrM

rz

∂∂

+∂∂ where ,)( 2

)(2

)(1 ∫

−−

= dreerMrArA

dreerMrBrB

∫−−

−= 2)(

2)(

2 )( and .)( 2)(

2)(

3 ∫−−

−= dreerMrCrC

One teleparallel proper

homothetic vector field is given as

,,,222

, 3215

2

9

2

1

210 zXXczcctrXtX αθαθαα ==+++== (4.4.23)

Case (II)(a):

In this case we have ),(rAA = ),(rBB = constant=C and .BA ≠ Now substituting


.0),,(),,( 1)(3)( =− zrPezrtPe rAt

rB θθ (4.4.24)

208






).,,(),,(),(),,,(),,(),(

43212

21310

θαθα

θαθα


+=++=

+=++= −

(4.4.25)


.0)],(),([),,( 31)()()(4 =+− − zrEzrEeertP zzrArBrA

t θθ (4.4.26)




.0)],(),([),( 31)()()(4 =+− − zrEzrEeerE zzrArBrA θθ Differentiating the above equation with

respect to θ twice, we get ),()(),(0),( 2144 rKrKrErE +=⇒= θθθθθ where )(1 rK and

)(2 rK are functions of integration. Substituting back the above value in equation (4.4.26)

and solving we get )()(),( 31)(1 rKrKezzrE rB += − and ),()(),( 42)(3 rKrKezzrE rA += −

where )(3 rK and )(4 rK are functions of integration. Substituting all the above


( )( )

( ) ).,()()(),,()()(),,,(

),()()()(

5213

231)(221

42)(3)()(1)(0

θθα

θαθα

θα

rErKrKtzXzrErKrKeztXztPrX

rKrKezrKerKeztXrB

rArArBrA

+++=

+++=+=

++++=−

−−−

(4.4.27)


.0),(),()(2 2)(51 =++ zrEerErtK zrBθθ (4.4.28)




209

),()(),()(),( 75)(25)(2 rKrKezzrErKezrE rBrBz +−=⇒−= −− where )(7 rK is a function

of integration. Substituting all the above information in (4.4.27) we get

( )

).()()(),()()(),,,(

),()()(

6523

75)(3221

42)(3)()(0

rKrKrtKzXrKrKezrKtXztPrX

rKrKezrKetXrB

rArArB

+++=

+−+=+=

+++=−

−−

θα

θαθα

θα

(4.4.29)


.0)()(21)(

21)()(

)(21)()(

21),,(

4)()(4)(2

23)(3)(2

=−−−−

+−⎟⎠⎞

⎜⎝⎛ −−

rKerAertArKerKz

rKAzrKerKeABztP

rAr

rArr

rAr

rrrBrB

rrt

α

θθθ (4.4.30)


rtt erAztP αθ


),,(),(),(2

),,( 8762



.0)(21),( )(6 =− rA





.,20)(21

221)()(




.0)()(21),( 3)(3)(7 =−⎟

⎠⎞






210


)(2

)(

42

)()(2

)(

33 ℜ∈+=

−−

−






.,)( 72

)(

72

)(2

)(

52 ℜ∈+= ∫

−

cecdreecrKrArArA

Now substituting all the above information in


)(

82

)(2

)(

64 ℜ∈+=

−−−



).()(

),()(

),,(2

,

652)(

72

)(2

)(

53

75)()(2

)(

42

)()(2

)(

32

86531

21

2)(

82

)(2

)(

62

)(

72

)(2

)(

52

)(

42

)(2

)(

30

rKrKetcdreectzX

rKrKezectdreectX

zEtcctzctctrX


rArArA

rBrBrArArBrA

rArArArArArArArArA

++++=

+−++=

+++++=

++++++=

∫

∫

∫∫∫

−

−−−

−

−−−−−−−−−

θα

θα

θθα

θθα

(4.4.31)


.0)()(21)(

21)()()()(

21

)]()([21)]()([

21),(2

7)()(7)(55

2)(

42

)(2

)(

38

3

=+++−+

−+−++ ∫−

rKerBerBrKerKzrKrzB


rBr

rBrr

rBrr

rA

rr

rArA

rr

θα

θθ (4.4.32)


.0)]()([21)]()([

212 2

)(

42

)(2

)(

33 =−+−+ ∫− rA

rr

rArA





),(0),( 1312112




211



.,20)(21

10109)()(




.0)()(21)( 5512 =−+ rKrKBzK rrz Differentiating this equation with respect to ,z we get

.,,)(0)( 121112111212 ℜ∈+=⇒= ccczczKzKzz Substituting back this value in the above

equation and solving we get ,)( 2)(

132

)(2

)(

115

rBrBrB

ecdreecrK += ∫−

.13 ℜ∈c Now substituting

back all the above information in equation (4.4.32) we get

.,)(0)()()(21

142

)(

142

)(2

)(

1277)(7)(

12 ℜ∈+−=⇒=++−−−


rrBrB

r


).(

,

),(22

,

62)(

132

)(2

)(

112

)(

72

)(2

)(

53

2)(

142

)(2

)(

122

)(

132

)(2

)(

112

131211659

2

1

21

2)(

82

)(2

)(

62

)(

72

)(2

)(

50


ecdreececzdreeczX

zKczctcctzcctrX

ecdreececzdreecztX

rBrBrBrArArA

rBrBrBrBrBrB

rArArArArArA

+++++=

+−−−=

+++++++=

++++=

∫∫

∫∫

∫∫

−−

−−−−−−

−−−−−−

θθα

θα

θθθα

α

(4.4.33)


.0)(21)()(

21

)(21)(

21)(22

2)(

1362

)(2

)(

11

2)(

72

)(2

)(

513

115

=+++

++++

∫

∫−

−

rB

rr

rBrB

r

rA

r

rArA

rz

erBcrKdreerBc

erActdreerActzKctc

θθ

θ

(4.4.34)


.0)(21)(

212 2

)(

72

)(2

)(

55 =++ ∫− rA

r

rArA

r erAcdreerAcc Solving this equation and remember

that in this case ,0)( ≠rAr we get .075 == cc Substituting the above information in


212

.0)(21)(

212 2

)(

132

)(2

)(

1111 =++ ∫− rB

r

rBrB

r erBcdreerBcc Solving this equation and remember

that in this case ,0)( ≠rBr we get .01311 == cc Now substituting the above information

in equation (4.4.34) and differentiating twice with respect to ,z we get

.,,)(0)( 161516151313 ℜ∈+=⇒= ccczczKzKzz Substituting back and solving we get

.,)( 1717156 ℜ∈+−= ccrcrK Refreshing the system of equations (4.4.33) we get

,,

,22

,

171532

)(

142

)(2

)(

122

16151269

2

1

21

2)(

82

)(2

)(

60

crczXecdreecX

czcctccctrX

ecdreectX

rBrBrB

rArArA

+−=+−=

++++++=

++=

−−−

−−−

∫

∫

αθα

θθα

α

(4.4.35)

where .,,,,,,,, 17161514129861 ℜ∈ccccccccc The line element for static cylindrically


,)2()2( 22109

2221

2 dzdcrcdrdtcrcds ++−+++−= θαα

(4.4.36)

where ).(,,, 9110921 cccccc ≠ℜ∈ The above space-time admits eight linearly independent

teleparallel homothetic vector fields in which seven are teleparallel Killing vector fields

given as ,r∂∂ ,2

)(

te

rA

∂∂−

,2)(

θ∂∂− rB

e ,z∂∂ ,

zr

rz

∂∂

−∂∂

trM

rt

∂∂

+∂∂ )(1 and

θθ

∂∂

+∂∂ )(2 rMr

where ∫−−

= dreerMrArA

2)(

2)(

1 )(

and .)( 2)(

2)(

2 dreerMrBrB

∫−−

−= One

teleparallel proper homothetic vector field is given as

,,,22

, 329

2

1

210 zXXcctrXtX αθαθαα ==++== (4.4.37)

The cases (II)(b) and (II)(c) can be solved exactly the same as the above case.

Case (III)(a):

In this case we have ),(rAA = ),(rBB = ),(rCC = ,BA ≠ CA ≠ and .CB = Now


213

.0),,(),,( 1)(3)( =− zrPezrtPe rAt

rB θθ (4.4.38)






).,,(),,(),(),,,(),,(),(

43212

21310

θαθα

θαθα


+=++=

+=++= −

(4.4.39)



trB θθ (4.4.40)




.0)],(),([),( 31)()()(4)( =+− − zrEzrEeerEe zzrArBrArB θθ Differentiating the above equation

with respect to θ twice, we get ),()(),(0),( 2144 rKrKrErE +=⇒= θθθθθ where

)(1 rK and )(2 rK are functions of integration. Substituting back the above value in

equation (4.4.40) and solving we get )()(),( 311 rKrKzzrE += and

),()(),( 42)()(3 rKrKezzrE rArB += − where )(3 rK and )(4 rK are functions of integration.


( )( )

( ) ).,()()(),,()()(),,,(

),()()()(

5213

231221

42)()(3)()(1)()(0

θθα

θαθα

θα

rErKrKtzXzrErKrKztXztPrX

rKrKezrKerKeztX rArBrArBrArB

+++=

+++=+=

++++= −−−

(4.4.41)


.0),(),()(2 251 =++ zrErErtK zθθ (4.4.42)



214


),()(),()(),( 75252 rKrKzzrErKzrEz +−=⇒−= where )(7 rK is a function of


( )

).()()(),()()(),,,(

),()()(

6523

753221

42)()(3)()(0

rKrKrtKzXrKrKzrKtXztPrX

rKrKezrKetX rArBrArB

+++=

+−+=+=

+++= −−

θα

θαθα

θα

(4.4.43)


.0)()(21)(

21)()(

)(21)()(

21),,(

4)()(4)(2)(

2)(3)(3)(2

=−−−−

⎟⎠⎞

⎜⎝⎛ −−−⎟

⎠⎞

⎜⎝⎛ −−

rKerAertArKerKez

rKeABzrKerKeABztP

rAr

rArr

rAr

rB

rBrrr

rBrBrrt

α

θθθ (4.4.44)


rtt erAztP αθ


),,(),(),(2

),,( 8762



.0)(21),( )(6 =− rA





.,20)(21

221)()(




.0)()(21),( 3)(3)(7 =−⎟

⎠⎞





215



)(2

)(

42

)()(2

)(

33 ℜ∈+=

−−

−






.,)( 7

)(2

)(

72

)()(2

)(

52 ℜ∈+=

−−

−




.,)( 82

)(

82

)(2

)(

64 ℜ∈+=

−−−


Refreshing the system of equations (4.4.43)

we get

).()(

),()(

),,(2

,

65)(2

)(

72

)()(2

)(

53

75)(2

)(

42

)()(

2)(

32

86531

21

2)(

82

)(2

)(

62

)(

72

)(2

)(

52

)(

42

)(2

)(

30

rKrKetcdreectzX

rKrKzectdreectX

zEtcctzctctrX


rBrArArBrA

rBrArA

rBrA

rArArArArArArArArA

++++=

+−++=

+++++=

++++++=

−−

−

−−

−

−−−−−−−−−

∫

∫

∫∫∫

θα

θα

θθα

θθα

(4.4.45)


.0)()(21)(

21)()()()(

21

)]()([21)]()([

21),(2

7)()(7)(5)(5)(

2)(

42

)(2

)(

38

3

=+++−−

−+−++ ∫−

rKerBerBrKerKezrKerBz


rBr

rBrr

rBr

rBrBr

rA

rr

rArA

rr

θα

θθ (4.4.46)


.0)]()([21)]()([

212 2

)(

42

)(2

)(

33 =−+−+ ∫− rA

rr

rArA





),(0),( 1312112


216





.,20)(21

10109)()(




.0)()(21)( 5)(5)(12 =−− rKerKeBzK r

rBrBrz Differentiating this equation with respect to ,z we

get .,,)(0)( 121112111212 ℜ∈+=⇒= ccczczKzKzz Substituting back this value in the above

equation and solving we get ,)( 2)(

132

)(2

)(

115

rBrBrB

ecdreecrK−−−

+= ∫ .13 ℜ∈c Now


.,)(0)()()(21

142

)(

142

)(2

)(

1277)(7)(

12 ℜ∈+−=⇒=++−−−


rrBrB

r


).(

,

),(22

,

62)(

132

)(2

)(

11

)(2

)(

72

)()(2

)(

53

2)(

142

)(2

)(

122

)(

132

)(2

)(

112

131211659

2

1

21

2)(

82

)(2

)(

62

)(

72

)(2

)(

50


ecdreececzdreeczX

zKczctcctzcctrX

ecdreececzdreecztX

rBrBrBrBrArArBrA

rBrBrBrBrBrB

rArArArArArA

+++++=

+−−−=

+++++++=

++++=

−−−−

−−

−−−−−−

−−−−−−

∫∫

∫∫

∫∫

θθα

θα

θθθα

α

(4.4.47)


.0)()(21)(

21)]()([

21

)()]()([21)(22

6)()(2)(

7

6)(2)(

2)(

513

115

=++−+

+−+++ ∫−

rKerBerzBerBrAct

rKedreerBrActzKctc

rBr

rBr

rA

rr

rrB

rArA

rrz

α

θ

(4.4.48)


.0)]()([21)]()([

212 2

)(

72

)(2

)(

55 =−+−+ ∫− rA

rr

rArA


217



,θ we get .011 =c Now substituting the above information in equation (4.4.48) and

differentiating with respect to ,z we get ⇒−= 913 )( czKzz

.,,2

)( 161516159

213 ℜ∈++−= ccczcczzK Substituting back this value in equation (4.4.48) and

solving we get ⇒=++ 0)()()(21 6)(6)(

15 rKerKerBc rrBrB

r ,)( 2)(

172

)(2

)(

156

rBrBrB

ecdreecrK−−−

+−= ∫


,

,

,)22

(2

,

2)(

172

)(

132

)(2

)(

153

2)(

142

)(

132

)(2

)(

122

16151269

22

1

21

2)(

82

)(2

)(

60

rBrBrBrB

rBrBrBrB

rArArA

ececdreeczX

ececzdreecX

czcctcczctrX

ecdreectX

−−−−

−−−−

−−−

++−=

+−−=

++++−++=

++=

∫

∫

∫

θα

θα

θθα

α

(4.4.49)



),()2()2( 22109

2221

2 dzdcrcdrdtcrcds ++−+++−= θαα

(4.4.50)

where ).(,,, 9110921 cccccc ≠ℜ∈ The above space-time admits nine linearly independent



)(

te

rA

∂∂−

,2)(

θ∂∂− rB

e ,2)(

ze

rB

∂∂−

),(2)(

θθ

∂∂

−∂∂−

zz

erB

,)(1

trM

rt

∂∂

+∂∂

θθ

∂∂

+∂∂ )(2 rMr

and ,)(2

zrM

rz

∂∂

+∂∂ where ∫

−−

= dreerMrArA

2)(

2)(

1 )(

and

.)( 2)(

2)(

2 dreerMrBrB

∫−−

−= One teleparallel proper homothetic vector field is given as

218

,,,)22

(2

, 329

22

1

210 zXXczctrXtX αθαθαα ==−++== (4.4.51)

The cases (III)(b) and (III)(c) can be solved exactly the same as the above case.

Case (IV):

In this case we have ),(rAA = ),(rBB = )(rCC = and .CBA == Now substituting


.0),,(),,( 13 =− zrPzrtPt θθ (4.4.52)






).,,(),,(),(),,,(),,(),(

43212

21310

θαθα

θαθα

rtPzXzrEzrEtXztPrXzrEzrEtX

+=++=

+=++= (4.4.53)


.0),(),(),,( 314 =−− zrEzrErtP zzt θθ (4.4.54)




.0),(),(),( 314 =−− zrEzrErE zzθθ Differentiating the above equation with respect to θ

twice, we get ),()(),(0),( 2144 rKrKrErE +=⇒= θθθθθ where )(1 rK and )(2 rK are

functions of integration. Substituting back the above value in equation (4.4.54) and

solving we get )()(),( 311 rKrzKzrE += and ),()(),( 423 rKrKzzrE += where )(3 rK

and )(4 rK are functions of integration. Substituting all the above information in (4.4.53)

we get

219

( )( )

( ) ).,()()(),,()()(),,,(

),()()()(

5213

231221

42310

θθα

θαθα

θα


rKrKzrKrKztX

+++=

+++=+=

++++=

(4.4.55)


.0),(),()(2 251 =++ zrErErtK zθθ (4.4.56)






).()()(),()()(),,,(),()()(

65237532

214230

rKrKrtKzXrKrKzrKtXztPrXrKrKzrKtX

+++=+−+=

+=+++=

θαθα

θαθα (4.4.57)


.0)()(21)(

21)(

)()()(21)()()(

21),,(

4)()(4)(

2)(2)(3)(3)(2

=−−−

−−−−

rKerAertArKe

rKezrKerzArKerKerAztP

rAr

rArr

rA

rrArA

rrrArA

rt

α

θθθ (4.4.58)


rtt erAztP αθ


),,(),(),(2

),,( 8762



.0)(21),( )(6 =− rA





220

.,20)(21

221)()(




.0)()()(21),( 3)(3)(7 =−− rKerKerAzE r

rArArθθ Differentiating this equation with respect to ,θ

we get ⇒=0),(7 zE θθθ ),()(),( 1097 zKzKzE +=θθ where )(9 zK and )(10 zK are




)(

42

)(2

)(

33 ℜ∈+=

−−−






.,)( 72

)(

72

)(2

)(

52 ℜ∈+=

−−−


Now substituting all the above information

in equation (4.4.58) and solving we get, .,)( 82

)(

82

)(2

)(

64 ℜ∈+=

−−−



).()(

),()(

),,(2

,

652)(

72

)(2

)(

53

752)(

42

)(2

)(

32

86531

21

2)(

82

)(2

)(

62

)(

72

)(2

)(

52

)(

42

)(2

)(

30

rKrKetcdreectzX

rKrKzectdreectX

zEtcctzctctrX


rArArA

rArArA

rArArArArArArArArA

++++=

+−++=

+++++=

++++++=

−−−

−−−

−−−−−−−−−

∫

∫

∫∫∫

θα

θα

θθα

θθα

(4.4.59)


.0)()(21)(

21

)()()()(21),(2

7)()(

7)(5)(5)(83

=++

+−−+

rKerAerA

rKerKezrKerzAzEtc

rAr

rAr

rrA

rrArA

r

θα

θθ

(4.4.60)

221

Differentiating equation (4.4.60) with respect to ,t we get .03 =c Substituting the above

information in equation (4.4.60) and differentiating with respect to ,θ we get

⇒=+⇒=+ 0),(0)(21),( 1

8)(8 czEerAzE rAr θαθ θθθθ ),()(

2),( 1211

1

28 zKzKczE ++−= θθθ

where )(11 zK and )(12 zK are functions of integration. Substituting back the above

information in equation (4.4.60) and differentiating the resulting equation with respect to

,z we get .0)()()(21)( 5)(5)(11 =−− rKerKerAzK r

rArArz Differentiating this equation with

respect to ,z we get .,,)(0)( 1091091211 ℜ∈+=⇒= ccczczKzKzz Substituting back this value

in the above equation and solving we get ,)( 2)(

132

)(2

)(

95

rArArA

ecdreecrK−−−

+= ∫ .13 ℜ∈c

Now substituting back all the above information in equation (4.4.60) we get

.,)(0)()()(21

142

)(

142

)(2

)(

1077)(7)(

10 ℜ∈+−=⇒=++−−−

∫ cecdreecrKrKerKerAcrArArA

rrArA

r


).(

,

),()22

(

,

62)(

132

)(2

)(

92

)(

72

)(2

)(

53

2)(

142

)(2

)(

102

)(

132

)(2

)(

92

)(

42

12109651

221

2)(

82

)(2

)(

62

)(

72

)(2

)(

52

)(

40



zKczctcctzctrX


rArArArArArA

rArArArArArArA

rArArArArArArA

+++++=

+−−−+=

+++++−+=

+++++=

−−−−−−

−−−−−−−

−−−−−−−

∫∫

∫∫

∫∫

θθα

θα

θθθα

θα

(4.4.61)


.0)()(21)(

21)()(22 6)()(6)(12

95 =+++++ rKerAerzArKezKctc rAr

rArr

rAz αθ (4.4.62)

Differentiating equation (4.4.62) with respect to ,t we get .05 =c Substituting the above

information in equation (4.4.62) and differentiating with respect to ,θ we get .09 =c

Now substituting the above information in equation (4.4.62) and differentiating with

respect to ,z we get ⇒−=⇒=+ 112)(12 )(0)(

21)( czKerAzK zz

rArzz α

222

.,,2

)( 161516151

212 ℜ∈++−= ccczcczzK Substituting back this value in equation (4.4.62)

and solving we get ⇒=++ 0)()()(21 6)(6)(

15 rKerKerAc rrArA

r ,)( 2)(

172

)(2

)(

156

rArArA

ecdreecrK−−−

+−= ∫


.

,

,)222

(

,

2)(

172

)(

132

)(

72

)(2

)(

153

2)(

142

)(

132

)(

42

)(2

)(

102

16151061

2221

2)(

82

)(

72

)(

42

)(2

)(

60

rArArArArA

rArArArArA

rArArArArA

ececetcdreeczX

ececzectdreecX

czcctccztrX

ececzecdreectX

−−−−−

−−−−−

−−−−−

+++−=

+−+−=

++++−−+=

++++=

∫

∫

∫

θα

θα

θθα

θα

(4.4.63)

where .,,,,,,,,,, 17161514131087641 ℜ∈ccccccccccc The line element for static cylindrically


,)()2( 222221

2 drdzddtcrcds +++−+= θα (4.4.64)

where ).0(, 121 ≠ℜ∈ ccc The above space-time admits eleven linearly independent

teleparallel homothetic vector fields in which ten are teleparallel Killing vector fields


)(

te

rA

∂∂−

,2)(

θ∂∂− rA

e ,2)(

ze

rA

∂∂−

),(2)(

θθ

∂∂

+∂∂−

tt

erA

),(2)(

zt

tze

rA

∂∂

+∂∂−

),(2)(

θθ

∂∂

−∂∂−

zz

erA

,)(t

rMr

t∂∂

+∂∂

θθ

∂∂

−∂∂ )(rMr

and z

rMr

z∂∂

−∂∂ )( where

.)( 2)(

2)(

∫−−

= dreerMrArA

One teleparallel proper homothetic vector field is given as

,,,)222

(, 321

22210 zXXcztrXtX αθαθαα ==−−+== (4.4.65)

Case (V)(a):

In this case we have ,constant=A ),(rBB = )(rCC = and .CB = Now substituting


223

.0),,(),,( 13)( =− zrPzrtPe trB θθ (4.4.66)






).,,(),,(),(),,,(),,(),(

43212

21310

θαθα

θαθα

rtPzXzrEzrEtXztPrXzrEzrEetX B

+=++=

+=++= (4.4.67)



trB θθ (4.4.68)




.0),(),(),( 31)(4)( =+− zrEzrEerEe zzrBrB θθ Differentiating the above equation with

respect to θ twice, we get ),()(),(0),( 2144 rKrKrErE +=⇒= θθθθθ where )(1 rK and

)(2 rK are functions of integration. Substituting back the above value in equation (4.4.68)

and solving we get )()(),( 311 rKrKzzrE += and ),()(),( 42)(3 rKrKezzrE rB += where

)(3 rK and )(4 rK are functions of integration. Substituting all the above information in

(4.4.67) we get

( )( )

( ) ).,()()(),,()()(),,,(

),()()()(

5213

231221

42)(3)(1)(0

θθα

θαθα

θα


rKrKezrKerKeztX rBrBrB

+++=

+++=+=

++++=

(4.4.69)


.0),(),()(2 251 =++ zrErErtK zθθ (4.4.70)



224




).()()(),()()(),,,(

),()()(

6523

753221

42)(3)(0

rKrKrtKzXrKrKzrKtXztPrX

rKrKezrKetX rBrB

+++=

+−+=+=

+++=

θα

θαθα

θα

(4.4.71)


.0)()(

)()()()()(),,(42)(

2)(3)(3)(2

=−−

−−−

rKrKez

rKerBzrKerKerBztP

rrrB

rBrr

rBrBrt θθθ

(4.4.72)


),,(),(),,( 762 zEztEztP θθθ += where ),(6 zE θ and ),(7 zE θ are functions of


to ,θ we get .0)()()(),( 3)(3)(6 =−− rKerKerBzE rrBrB

rθθ Differentiating this equation with

respect to ,θ we get ),()(),(0),( 9866 zKzKzEzE +=⇒= θθθθθ where )(8 zK and )(9 zK are

function of integration. Substituting back in the above equation and differentiating with

respect to ,z we get .,)(0)( 1188 ℜ∈=⇒= cczKzKz Once again substituting back and

solving we get ⇒=−− 0)()()( 3)(3)(1 rKerKerBc r

rBrBr .,)( 2

)(2

)(1

3 ℜ∈+= −− cecercrK rBrB

Substituting all the above information in equation (4.4.72) and solve after differentiating

with respect to ,z we get .0)()()()( 2)(2)(9 =−− rKerKerBzK rrBrB

rz Now differentiating this

equation with respect to ,z we get .,,)(0)( 434399 ℜ∈+=⇒= ccczczKzKzz

⇒=−− 0)()()( 2)(2)(3 rKerKerBc r

rBrBr .,)( 5

)(5

)(3

2 ℜ∈+= −− cecercrK rBrB Now substituting all

the above information in equation (4.4.72) and solving we get, .,)( 6644 ℜ∈+= ccrcrK


).()(

),()(

),,(,

65)(5

)(3

3

75)(2

)(1

2

7431

1654321

0

rKrKetcerctzX

rKrKzecterctX

zEtctzcctrXcczcrcrzccrtX

rBrB

rBrB

++++=

+−++=

++++=++++++=

−−

−−

θα

θα

θθαθθα

(4.4.73)


225

.0)()(21)(

21)(

)()()(21)(

21)(

21),(2

7)()(7)(

5)(5)(21

71

=+++

−−−−+

rKerBerBrKe

rKezrKerBzrBctrBrctzEtc

rBr

rBrr

rB

rrBrB

rrr

θα

θθ

(4.4.74)


.0)(21)(


,0)( ≠rBr we get .021 == cc Substituting the above information in equation (4.4.74) and

differentiating with respect to ,θ we get .0)(21),( )(7 =+ rB

r erBzE αθθθ Now

differentiating this equation with respect to ,θ we get ⇒= 0),(7 zE θθθθ

),()()(2

),( 1211102

7 zKzKzKzE ++= θθθ where ),(11 zK )(12 zK and )(12 zK are

functions of integration. Substituting back the above information in equation (4.4.74) and


.,)(0)( 771010 ℜ∈=⇒= cczKzKz Substituting back this value and solving we get

⇒=+ 0)(21 )(

7rB

r erBc α .,2887

2)(

ℜ∈+−= ccrcerB

α Substituting back this value in

equation (4.4.74) and differentiating the resulting equation twice with respect to ,z we

get .,,)(0)( 1091091111 ℜ∈+=⇒= ccczczKzKzz Substituting back this value and solving we

get .,)( 112

)(

112

)(2

)(

95 ℜ∈+=

−−−




.,)( 122

)(

122

)(2

)(

107 ℜ∈+−=

−−−



(4.4.73) we get

).(

,

),(2

,

62)(

112

)(2

)(

9)(

5)(

33

2)(

122

)(2

)(

102

)(

112

)(2

)(

92

12109437

21

64530

rKecdreecetcerctzX

ecdreececzdreeczX

zKczctcctzcrXcrcczcrztX

rBrBrBrBrB

rBrBrBrBrBrB

+++++=

+−−−=

++++++=++++=

−−−−−

−−−−−−

∫

∫∫θθα

θα

θθθαα

(4.4.75)

226


.0)()(21)(

21

)(21)()(

21)(22

6)()(

56)(

312

93

=++

−+−++

rKerBerzB

rBctrKerBrctzKctc

rBr

rBr

rrrB

rz

α

θ

(4.4.76)


.0)(21)(


,0)( ≠rBr we get .053 == cc Substituting the above information in equation (4.4.76) and

differentiating with respect to ,θ we get .09 =c Now substituting the above information

in equation (4.4.76) and differentiating with respect to ,z we get ⇒−= 712 )( czKzz

.,,2

)( 141314137

212 ℜ∈++−= ccczcczzK Substituting back this value in equation (4.4.76)

and solving we get ⇒=++ 0)()()(21 6)(6)(

13 rKerKerBc rrBrB

r ,)( 2)(

152

)(2

)(

136

rBrBrB

ecdreecrK−−−

+−= ∫


.

,

,)22

(,

2)(

152

)(

112

)(2

)(

133

2)(

122

)(

112

)(2

)(

102

14131047

221

640

rBrBrBrB

rBrBrBrB

ececdreeczX

ececzdreecX

czcctcczrXcrctX

−−−−

−−−−

++−=

+−−=

++++−+=++=

∫

∫θα

θα

θθαα

(4.4.77)



),()2( 2287

222 dzdcrcdrdtds ++−++−= θα

(4.4.78)



given as ,r∂∂ ,

t∂∂ ,2

)(

θ∂∂− rB

e ,2)(

ze

rB

∂∂−

),(2)(

θθ

∂∂

−∂∂−

zz

erB

,t

rr

t∂∂

+∂∂

227

θθ

∂∂

+∂∂ )(rMr

and ,)(z

rMr

z∂∂

+∂∂ where

.)( 2)(

2)(

dreerMrBrB

∫−−

−= One teleparallel

proper homothetic vector field is given as

,,,)22

(, 327

2210 zXXczrXtX αθαθαα ==−+== (4.4.79)

The cases (V)(b) and (V)(c) can be solved exactly the same as the above case.

Case (VI)(a):

In this case we have )(rAA = and .constant== CB Now substituting equation (4.4.9)

in equation (4.4.3), we get

.0),,(),,( 1)(3 =− zrPezrtP rAt θθ (4.4.80)




⇒= − ),(),,( 11 zrEezrP Aθθ ),,(),(),,( 311 zrEzrEezrP A += −θθ where ),(3 zrE is a


).,,(),,(),(),,,(),,(),(

43212

21310

θαθα

θαθα

rtPzXzrEzrEtXztPrXzrEzrEetX A

+=++=

+=++= −

(4.4.81)


.0),(),(),,( 3)(14 =−− zrEezrErtP zrA

zt θθ (4.4.82)




.0),(),(),( 3)(14 =−− zrEezrErE zrA

zθθ Differentiating the above equation with respect to

θ twice, we get ),()(),(0),( 2144 rKrKrErE +=⇒= θθθθθ where )(1 rK and )(2 rK

are functions of integration. Substituting back the above value in equation (4.4.82) and

solving we get )()(),( 311 rKrKzzrE += and ),()(),( 42)(3 rKrKezzrE rA += − where

228

)(3 rK and )(4 rK are functions of integration. Substituting all the above information in

(4.4.81) we get

( )( )

( ) ).,()()(),,()()(),,,(),()()()(

5213

231221

42)(3)(1)(0

θθα

θαθα

θα

rErKrKtzXzrErKrKztXztPrXrKrKezrKerKeztX rArArA

+++=

+++=+=

++++= −−−

(4.4.83)


.0),(),()(2 251 =++ zrErErtK zθθ (4.4.84)






( )).()()(),()()(),,,(),()()(

65237532

2142)(3)(0

rKrKrtKzXrKrKzrKtXztPrXrKrKezrKetX rArA

+++=+−+=

+=+++= −−

θαθα

θαθα (4.4.85)


.0)()(21)(

21

)()()()(21)()()(

21),,(

4)()(

4)(22332

=−−

−−+−+

rKerAertA

rKerKzrKrzArKrKrAztP

rAr

rAr

rrA

rrrrt

α

θθθ (4.4.86)


rtt erAztP αθ


),,(),(),(2

),,( 8762



.0)(21),( )(6 =− rA



229



.,20)(21

221)()(




.0)()()(21),( 337 =−+ rKrKrAzE rrθθ Differentiating this equation with respect to ,θ we get

⇒=0),(7 zE θθθ ),()(),( 1097 zKzKzE +=θθ where )(9 zK and )(10 zK are functions of

integration. Substituting back the above value and differentiating with respect to ,z we

get .,)(0)( 3399 ℜ∈=⇒= cczKzKz Now substituting back this value and solving we get,

.,)( 42

)(

42

)(2

)(

33 ℜ∈+= ∫

−

cecdreecrKrArArA

Substituting all the above information in




.,)( 72

)(

72

)(2

)(

52 ℜ∈+= ∫

−

cecdreecrKrArArA

Now substituting all the above information in


)(

82

)(2

)(

64 ℜ∈+=

−−−



).()(

),()(

),,(2

,

652)(

72

)(2

)(

53

752)(

42

)(2

)(

32

86531

21

2)(

82

)(2

)(

62

)(

72

)(2

)(

52

)(

42

)(2

)(

30

rKrKetcdreectzX

rKrKzectdreectX

zEtcctzctctrX


rArArA

rArArA

rArArArArArArArArA

++++=

+−++=

+++++=

++++++=

∫

∫

∫∫∫

−

−

−−−−−−−−−

θα

θα

θθα

θθα

(4.4.87)


.0)()()(21)(

21),(2 752

)(

42

)(2

)(

38

3 =+−+++ ∫−

rKrKzerActdreerActzEtc rr

rA

r

rArA

rθθ (4.4.88)

230


.0)(21)(

212 2

)(

42

)(2

)(

33 =++ ∫− rA

r

rArA




),()(),(0),( 121188 zKzKzEzE +=⇒= θθθθθ where )(11 zK and )(12 zK are functions of

integration. Substituting back the above information in equation (4.4.88) and

differentiating the resulting equation with respect to ,z we get .0)()( 511 =− rKzK rz

Differentiating this equation with respect to ,z we get

.,,)(0)( 121112111111 ℜ∈+=⇒= ccczczKzKzz Substituting back this value in the above equation

and solving we get ,)( 13115 crcrK += .13 ℜ∈c Now substituting back all the above

information in equation (4.4.88) we get .,)(0)( 14141277

12 ℜ∈+−=⇒=+ ccrcrKrKc r


).(

,

),(2

,

61311

2)(

72

)(2

)(

53

141213112

121211651

21

2)(

82

)(2

)(

62

)(

72

)(2

)(

50

rKcrcetcdreectzX

crczcrczX

zKczctcctzctrX

ecdreececzdreecztX

rArArA

rArArArArArA

+++++=

+−−−=

++++++=

++++=

∫

∫∫

−

−−−−−−

θθα

θα

θθα

α

(4.4.89)


.0)(21)()(

21)(22 2

)(

762

)(2

)(

512

115 =+++++ ∫− rA

rr

rArA

rz erActrKdreerActzKctc θ (4.4.90)


.0)(21)(

212 2

)(

72

)(2

)(

55 =++ ∫− rA

r

rArA



equation (4.4.90) and differentiating with respect to ,θ we get .011 =c Now substituting

the above information in equation (4.4.90) and differentiating with respect to ,z we get

231

⇒= 0)(12 zKzz .,,)( 1615161512 ℜ∈+= ccczczK Substituting back this value in equation

(4.4.90) and solving we get ⇒=+ 0)(615 rKc r ,)( 1715

6 crcrK +−= .17 ℜ∈c Refreshing the


.,

,2

,

1713153

1413122

16151261

212

)(

82

)(2

)(

60

ccrczXczcrcX

czcctcctrXecdreectXrArArA

++−=+−−=

+++++=++=−−−

∫θαθα

θαα

(4.4.91)



),()2( 222221

2 dzddrdtcrcds ++++−= θα

(4.4.92)




)(

te

rA

∂∂−

,θ∂∂ ,

z∂∂ ),(

θθ

∂∂

−∂∂ zz

,)(t

rMr

t∂∂

+∂∂ )(

θθ

∂∂

−∂∂ rr

and

),(z

rr

z∂∂

−∂∂ where .)( 2

)(2

)(

∫−−

= dreerMrArA

One teleparallel proper homothetic vector

field is given as

,,,2

, 321

210 zXXctrXtX αθααα ==+== (4.4.93)

The cases (VI)(b) and (VI)(c) can be solved exactly the same as the above case.


In this chapter we investigated teleparallel proper homothetic vector fields for Bianchi

type I, non static plane symmetric and static cylindrically symmetric space-times by

using direct integration technique. It follows that all of the above space-times admit

teleparallel proper homothetic vector fields for special classes of the metric. In all the

above space-times we have shown that the dimension of the teleparallel proper

232

homothetic vector field is one like in general relativity. Following results are obtained

from the above study:

(1) In Bianchi type I space-times: Teleparallel homothetic vector fields have been

explored and it is shown that the above space-times admit 8, 9 or 11 teleparallel

homothetic vector fields. In all the cases when the above space-times admit teleparallel

homothetic vector fields it admits only one proper teleparallel homothetic vector fields

for a special class of metric. When the space-time becomes Minkowski then all the

torsion components become zero and the teleparallel Lie derivative for the metric gives

the same equations as in general relativity, hence the homothetic vector fields are same in

both the theories.

The results for above space time when it admits eight teleparallel homothetic vector

fields, teleparallel proper homothetic vector fields are given in equations (4.2.23) and

(4.2.37). When the above space-time admits nine teleparallel homothetic vector fields,

teleparallel proper homothetic vector fields are given in equations (4.2.51), (4.2.79) and

(4.2.93). When the above space-time admits eleven teleparallel homothetic vector fields,

teleparallel proper homothetic vector field is given in equation (4.2.65).

(2) In non static plane symmetric space-times: Teleparallel homothetic vector fields have

been explored and it is shown that the above space-times admit 6, 7, 8 or 11 teleparallel

homothetic vector fields. The above space-time admits eleven teleparallel homothetic

vector fields only when the space-time becomes Minkowski and all the torsion

components become zero. In all the cases when the above space-times admit teleparallel

homothetic vector fields it admits only one proper teleparallel homothetic vector fields

for a special class of metric. When the above space-time admits six teleparallel

homothetic vector fields, results for teleparallel proper homothetic vector fields are given

in equations (4.3.39) and (4.3.57). When the above space-time admits seven teleparallel

homothetic vector fields, results for teleparallel proper homothetic vector fields are given

in equations (4.3.23), (4.3.26), (4.3.50), (4.3.66), (4.3.69) and (4.3.72). When the above

space-time admits eight teleparallel homothetic vector fields, results for teleparallel

proper homothetic vector fields are given in equations (4.3.29), (4.3.42), (4.3.60) and

233

(4.3.63). The result for teleparallel proper homothetic vector field when the space-time

admits eleven teleparallel homothetic vector fields is given in equation (4.3.32).

(3) In static cylindrically symmetric space-times: Teleparallel homothetic vector fields

have been explored and it turns out that the above space-times admit 8, 9 or 11

teleparallel homothetic vector fields. In all the cases when the above space-time admits

teleparallel homothetic vector fields, it possesses only one teleparallel proper homothetic

vector fields. The above space-times admit teleparallel homothetic vector fields for a

special class of the metric. When the space-time becomes Minkowski then all the torsion

components become zero and hence admit the same homothetic vector fields as in

general relativity. When the above space-time admits eight teleparallel homothetic vector

fields, results for teleparallel proper homothetic vector fields are given in equations

(4.4.23) and (4.4.37). When the above space-time admits nine teleparallel homothetic

vector fields, results for teleparallel proper homothetic vector fields are given in

equations (4.4.51), (4.4.79) and (4.4.93). When the above space-time admits eleven

teleparallel homothetic vector fields, teleparallel proper homothetic vector field is given

in equation (4.4.65).

234

Chapter 5

Proper Conformal Vector Fields in Non

Conformally Flat Non Static Cylindrically

Symmetric, Kantowski-Sachs and Bianchi

Type III Space-Times

5.1. Introduction

This chapter is devoted to find proper conformal vector fields in non conformally flat

non static cylindrically symmetric, Kantowski-Sachs and Bianchi type III space-times

using direct integration technique. All the above space-times possess proper

conformal vector fields for special class of space-times. It is important to note that

here the classification of non static cylindrically symmetric space-times will also

cover non static plane symmetric, static cylindrically symmetric and static plane

symmetric space-times [42]. This chapter is organized as follows. In section (5.2)

proper conformal vector fields of non conformally flat non static cylindrically

symmetric space-times are investigated. In the next section (5.3) proper conformal

vector fields in non conformally flat Kantowski-Sachs and Bianchi type III space-

times are investigated. In section (5.4) a detailed summary of the work is given.

235

5.2. Proper Conformal Vector Fields in Non

Conformally Flat Non Static Cylindrically


The most general form of the metric for non static cylindrically symmetric space-time

is given by [78]

,2),(2),(22),(2 dzededrdteds rtCrtBrtA +++−= θ (5.2.1)

where ,A B and C are functions of rt and only. The above space-time admits two

independent Killing vector fields which are θ∂∂ and .

z∂∂ If in the above space-times

,CB = it becomes non static plane symmetric space-time and possess three Killing

vector fields which are ,θ∂∂

z∂∂ and ).(

zz

∂∂

−∂∂ θθ

A vector field X is said to be a

conformal vector field if it satisfies equation (1.2.4). One can write (1.2.4) explicitly

using (5.2.1)

,2 0,010 η=+′+• XXAXA (5.2.2)

,01,0),(

0,1 =− XeX rtA (5.2.3)

,02,0),(

0,2),( =− XeXe rtArtB (5.2.4)

,03,0),(

0,3),( =− XeXe rtArtC (5.2.5)

,2 1,1 η=X (5.2.6)

,02,1

1,2),( =+ XXe rtB (5.2.7)

,03,1

1,3),( =+ XXe rtC (5.2.8)

236

,2 2,210 η=+′+• XXBXB (5.2.9)

,03,2),(

2,3),( =+ XeXe rtBrtC (5.2.10)

,2 3,310 η=+′+• XXCXC (5.2.11)

where dot denotes the derivative with respect to t and dash denotes the derivative

with respect to .r Here it follows that [42] ).,( rtηη = Equations (5.2.3), (5.2.6),

(5.2.7) and (5.2.8) give

( )

),,,(),,(

),,,(),,(),,,(),(21

),,,(),,(),(21

4),(13

3),(1211

2),(1),(0

ztKdreztKX

ztKdreztKXztKdrrtX

ztKdreztKdrdrrteX

rtCz

rtB

rtAtt

rtA

θθ

θθθη

θθη

θ

+−=

+−=+=

++=

∫∫∫

∫∫ ∫

−

−

−−

(5.2.12)

where ),,(),,,(),,,( 321 ztKztKztK θθθ and ),,(4 ztK θ are functions of integration.

In order to determine ),,(),,,(),,,( 321 ztKztKztK θθθ and ),,(4 ztK θ we need to

integrate the remaining six equations. Considering equation (5.2.11) and using

equation (5.2.21) then solve after differentiating with respect to z we get

),,(),,(0),,( 111 θθθ tEztKztK z =⇒= where ),(1 θtE is a function of integration.

Substituting back the above value in (5.2.11) we get

0),,(2),,(),( 42 =+ ztKztKrtC zzzt θθ (5.2.13)

Differentiating the above equation with respect to ,r we get .0),,(),( 2 =ztKrtC zrt θ

In order to solve this equation we will discuss three possibilities:

(I) ,0),( ≠rtCrt ,0),,(2 =ztK z θ (II) ,0),( =rtCrt ,0),,(2 ≠ztK z θ

(III) ,0),( =rtCrt .0),,(2 =ztK z θ


237

Case I:

In this case we have 0),( ≠rtCrt and .0),,(2 =ztK z θ Equation ⇒= 0),,(2 ztK z θ

),,(),,( 22 θθ tEztK = where ),(2 θtE is a function of integration. Substituting back

the above value in (5.2.13) and solving we get ),,(),(),,( 434 θθθ tEtzEztK +=

where ),(3 θtE and ),(4 θtE are functions of integration. Substituting all the above


( )

),,(),(

,),,(),,(,),(),(21

,),(),(),(21

433

31211

210

θθ

θθθη

θθη

θ

tEtzEX

ztKdreztEXtEdrrtX

tEdretEdrdrrteX

B

Att

A

+=

+−=+=

++=

∫∫

∫ ∫∫−

−−

(5.2.14)

Now considering equation (5.2.9) and using equation (5.2.14) then solve after

differentiating with respect to ,θ we get ),(),(0),( 111 tFtEtE =⇒= θθθ where

)(1 tF is a function of integration. Substituting back the above value in (5.2.9) and

differentiating with respect to ,r we get .0),(),( 2 =θθ tErtBrt To solve this equation

we need to discuss three different possibilities:

(a) ,0),( ≠rtBrt ,0),(2 =θθ tE (b) ,0),( =rtBrt ,0),(2 ≠θθ tE

(c) ,0),( =rtBrt .0),(2 =θθ tE


Case I (a):

In this case we have 0),( ≠rtBrt and .0),(2 =θθ tE Equation

),(),(0),( 222 tFtEtE =⇒= θθθ where )(2 tF is a function of integration.

Substituting back the above value in (5.2.9) we get

),,(),(),,(0),,( 6533 ztEztEztKztK +=⇒= θθθθθ where ),(5 ztE and ),(6 ztE are

238

functions of integration. Substituting all the above information in the system of


( )

),,(),(

),,(),(,)(),(21

,)()(),(21

433

65211

210

θθ

θη

η

tEtzEX

ztEztEXtFdrrtX

tFdretFdrdrrteX Att

A

+=

+=+=

++=

∫

∫ ∫∫ −−

(5.2.15)


respect to ,θ we get ⇒= 0),(5 ztEt ),(),( 35 zFztE = where )(3 zF is a function of

integration. Substituting back the above value in (5.2.4) and solving we get

),(),(0),( 466 zFztEztEt =⇒= where )(4 zF is a function of integration.

Substituting the above information in (5.2.15) we get

( )

).,(),(),()(

,)(),(21

,)()(),(21

433432

11

210

θθθ

η

η

tEtzEXzFzFX

tFdrrtX

tFdretFdrdrrteX Att

A

+=+=

+=

++=

∫

∫ ∫∫ −−

(5.2.16)


differentiating with respect to ,z we get )(),( 53 θθ FtE = and ),(),( 64 θθ FtE =

where )(5 θF and )(6 θF are functions of integration. Substituting back all the above


( )

).()(),()(

),(),(21

,)()(),(21

653432

11

210

θθθ

η

η

FzFXzFzFX

tFdrrtX

tFdretFdrdrrteX Att

A

+=+=

+=

++=

∫

∫ ∫∫ −−

(5.2.17)

Now subtracting equation (5.2.9) from (5.2.11) we get

.0))()((2)()( 3510 =−+−+− zFFXBCXBC rrtt θ Now considering this equation

and using equation (5.2.17) then suppose ,0)( =− tt BC we get

239

( ).)()(2)(),(21)( 531 θη FzFtFdrrtBC rr −=⎥⎦

⎤⎢⎣⎡ +− ∫ Differentiating the above

equation with respect to ,t we get .0)(),(21)( 1 =⎥⎦

⎤⎢⎣⎡ +− ∫ tFdrrtBC ttrr η Solution of

this equation involves three different possibilities:

(i) ,0)( ≠− rr BC ,0)(),(21 1 =⎥⎦

⎤⎢⎣⎡ +∫ tFdrrt ttη

(ii) ,0)( =− rr BC ,0)(),(21 1 ≠⎥⎦


(iii) ,0)( =− rr BC .0)(),(21 1 =⎥⎦



Case I (a) (i):

In this case we have 0)( ≠− rr BC and .0)(),(21 1 =⎥⎦

⎤⎢⎣⎡ +∫ tFdrrt ttη Equation

⇒=⎥⎦⎤

⎢⎣⎡ +∫ 0)(),(21 1 tFdrrt ttη )(rηη = and .,)( 11

1 ℜ∈= cctF Refreshing the


).()(),()(

,)(21),(

653432

1120

θθθ

η

FzFXzFzFX

cdrrXtFX

+=+=

+== ∫ (5.2.18)


differentiating with respect to z and ,θ we get 3255 )(0)( ccFF +=⇒= θθθθθ and

⇒= 0)(3 zFzz ,)( 543 czczF += .,,, 5432 ℜ∈cccc Substituting back the above values

in (5.2.10) and differentiating with respect to ,z we get 02 =c and ⇒= 0)(4 zFzz

.,,)( 76764 ℜ∈+= ccczczF Now substituting back all the above values in (5.2.10)

240

we get 986 )( ccF += θθ and .0864 === ccc Substituting all the above information

in (5.2.18) we get

,,,)(21),( 93

375

21

120 czcXccXcdrrXtFX +=+=+== ∫ θη (5.2.19)

where .,,,, 97531 ℜ∈ccccc Considering equation (5.2.2) and using equation (5.2.19)

then differentiating with respect to t we get

.0)(2)(21),()(),()(),( 2

122 =+⎟

⎠⎞

⎜⎝⎛ +++ ∫ tFcdrrrtAtFrtAtFrtA ttrttttt η (5.2.20)

In order to solve (5.2.20) we discuss different possibilities. First, suppose 0),( =rtAt

then ⇒= 0)(2 tFtt .,,)( 111011102 ℜ∈+= ccctctF Substituting back all the above

information in (5.2.2) we get ⇒=+⎥⎦⎤

⎢⎣⎡ +∫ )(2)(21

101 rccdrrAr ηη

( ) .)(

21

2)(ln)(1

10

2

1 ∫∫

∫+

−+=cdrr

drccdrrrAη

η Now differentiating equation

(5.2.11) with respect to ,t we get

.0)(21),(),()(),( 1101110 =⎥⎦

⎤⎢⎣⎡ ++++ ∫ cdrrrtCrtCcctcrtC rtttt η In order to solve this

equation consider 0),( =rtCtt then from the above equation we get

⇒−=⎥⎦⎤

⎢⎣⎡ +∫ ),()(21),( 101 rtCccdrrrtC trt η ⇒∫=

− drrvctt etPrtC

)(10)(),(

),()(),( 1)(10 rKetPtrCdrrvc+∫=

− where )(1 rK is a function of integration and

.)(

21

1)(1∫ +

=cdrr

rVη

In this case ⇒= ),(),( rtBrtC tt ⇒∫=− drrvc

tt etPrtB)(10)(),(

),()(),( 2)(10 rKetPtrBdrrvc+∫=

− where )(2 rK is a function of integration. Now

differentiate the value of ),( trC with respect to r and substitute that value in (5.2.11)

241

then differentiating the resulting equation with respect to t and solving we get

.,,)( 13121312 ℜ∈+= ccctctP Substituting back the above value in equation (5.2.11)

and solving we get ⇒=+⎥⎦⎤

⎢⎣⎡ ++∫ ∫

−)(2)(

21)( 31

1)(14

10 rccdrrrKec rdrrVc

ηη

( ) ,)()(2)(ln)()(

143

2

11 10∫ ∫∫ ∫−−+=

−drerVcdrrVccdrrrK

drrVcη where

.01310121114 ≠−= ccccc Similarly substituting the above information in (5.2.9) and

solving we get ( ) .)()(2)(ln)()(

145

2

12 10∫ ∫∫ ∫−−+=


drrVcη Now

substituting the above information in equation (5.2.19) we get the conformal vector

fields [42]

933

752

11

11100 ,,)(

21, czcXccXcdrrXctcX +=+=+=+= ∫ θη (5.2.21)

The line element for non static cylindrically symmetric space-time in this case is

given as

,2),(2),(22)(2 dzededrdteds rtCrtBrA +++−= θ (5.2.22)

where

∫−= ,)(2)(ln)( 102 drrVcrUrA

∫∫ ∫−−+∫+=−−

,)()(2)(ln)(),()(

1452)(

13121010 drerVcdrrVcrUectcrtB

drrVcdrrVc

∫∫ ∫−−+∫+=−−

,)()(2)(ln)(),()(

1432)(

13121010 drerVcdrrVcrUectcrtC

drrVcdrrVc

,)(

1)(rU

rV = ,)(21)( 1∫ += cdrrrU η 53 cc ≠ and .01310121114 ≠−= ccccc Proper

conformal vector field after subtracting Killing vector fields is given as

( ).,),(, 351110 zccrUctcX θ+= (5.2.23)

We now consider another possibility for the solution of equation (5.2.20). Let in this

case ⇒= 0)(2 tFtt ,,,)( 76762 ℜ∈+= ccctctF 0),( =rtAtt and .0),( ≠rtArt

242

Substituting all the above information in equation (5.2.20) we get

⇒=⎟⎠⎞

⎜⎝⎛ ++ ∫ 0)(

21),(),( 16 cdrrrtAcrtA rtt η ),()(),( 1)(6 rEetgrtA

drrVc+∫=

− where

)(tg and )(1 rE are functions of integration and .)(

21

1)(1∫ +

=cdrr

rVη

Substituting

back the above information in (5.2.2) and solve after differentiating with respect to t

we get .,,)( 9898 ℜ∈+= ccctctg Substituting back the above value in (5.2.2) and

solving we get ( ) .)()(2)(ln)()(

106

2

11 6∫ ∫∫ ∫−−+=

−drerVcdrrVccdrrrE

drrVcη

Refreshing the value of ),( rtA we get

∫∫ ∫−−+∫+=−−

,)()(2)(ln)(),()(

1062)(

9866 drerVcdrrVcrUectcrtA

drrVcdrrVc where

,0968710 ≠−= ccccc )(

1)(rU

rV = and .)(21)( 1∫ += cdrrrU η Similarly solving

equations (5.2.9) and (5.2.11) we get

∫∫ ∫−−+∫+=−−

,)()(2)(ln)(),()(

1332)(


drrVcdrrVc

∫∫ ∫−−+∫+=−−

,)()(2)(ln)(),()(

1312)(

121166 drerVcdrrVcrUectcrtC

drrVcdrrVc

,012611713 ≠−= ccccc .,, 311636 cccccc ≠≠≠ The conformal vector field in this

case is given as

.,,)(21, 21

343

21

176

0 czcXccXcdrrXctcX +=+=+=+= ∫ θη (5.2.24)

The line element for non static cylindrically symmetric space-times becomes

,2),(2),(22),(2 dzededrdteds rtCrtBrtA +++−= θ (5.2.25)

where the metric functions are given above. Proper conformal vector field after

subtracting Killing vector fields from (5.2.24) is given as

( ).,),(, 1376 zccrUctcX θ+= (5.2.26)

243

Case I (a) (ii):

In this case we have 0)( =− rr BC and .0)(),(21 1 ≠⎥⎦

⎤⎢⎣⎡ +∫ tFdrrt ttη To solve

equations (5.2.2), (5.2.8) and (5.2.11) we must take .0)( =rη In this case no proper

conformal vector fields exists.

Case I (a) (iii):

In this case we have 0)( =− rr BC and .0)(),(21 1 =⎥⎦


⇒=− 0)( rr BC ),(),( rtBrtC rr = and equation ⇒=⎥⎦⎤

⎢⎣⎡ +∫ 0)(),(21 1 tFdrrt ttη

),(rηη = .,)( 111 ℜ∈= cctF The system of equations (5.2.17) becomes

).()(),()(

,)(21),(

653432

1120

θθθ

η

FzFXzFzFX

cdrrXtFX

+=+=

+== ∫ (5.2.27)

Now subtracting equation (5.2.11) from equation (5.2.9) and using the fact that in this

case ),(),( rtBrtC rr = and ),(),( rtBrtC tt = we get ⇒=− 0)()( 35 zFF θ

.,)()( 2235 ℜ∈== cczFF θ Substituting back the above values in equation (5.2.9)

and solving we get 546 )( ccF += θθ and .,,,)( 65464

4 ℜ∈+−= cccczczF

Substituting all the above information in equation (5.2.27), we get

5423

6422

1120

,

)(21),(

cczcXcczcX

cdrrXtFX

++=+−=

+== ∫θθ

η (5.2.28)


respect to ,t we get

.0)(2)(21)()( 2

122 =+⎥⎦

⎤⎢⎣⎡ +++ ∫ tFcdrrAtFAtFA ttrttttt η (5.2.29)

244

In order to solve this equation we consider different possibilities. First we take

⇒= 0),( rtAt ).(rAA = Substituting the above information in equation (5.2.29) we

get .,,)(0)( 878722 ℜ∈+=⇒= ccctctFtFtt Substituting back the above value in

(5.2.2) and solving we get ( ) ∫∫ −+= ,)(2)(ln)( 7

2

1 drrVccdrrrA η where

.)(

21

1)(1∫ +

=cdrr

rVη

Now substituting the above information (5.2.28) we get

5423

6422

11

870

,

)(21,

cczcXcczcX

cdrrXctcX

++=+−=

+=+= ∫θθ

η (5.2.30)


respect to ,t we get .0)(21)( 1787 =⎟

⎠⎞

⎜⎝⎛ ++++ ∫ cdrrBcBctcB rtttt η Substituting

0),( =rtBtt in the above equation we get ⇒∫=− drrvc

tt etgB)(7)(

),()(),( 1)(7 rKetgrtBdrrvc+∫=

− where )(1 rK is a function of integration. Now

differentiating the above value twice with respect to t and remember that 0),( =rtBtt

we get ⇒=∫=−

0)(),()(7 drrvc

tttt etgrtB .,,)( 109109 ℜ∈+= ccctctg Differentiating

the value of ),( rtB with respect to r and solving we get

( ) .)()(2)(ln)()(

112

2

11 7∫ ∫∫ ∫−−+=


drrVcη Finally the line

element for non static cylindrically symmetric space-times becomes

),( 22),(22)(2 dzdedrdteds rtBrA +++−= θ (5.2.31)

where

∫−= ,)(2)(ln)( 72 drrVcrUrA

∫∫ ∫−−+∫+=−−

,)()(2)(ln)(),()(

1122)(


drrVcdrrVc

245

,)(

1)(rU

rV = ,)(21)( 1∫ += cdrrrU η 72 cc ≠ and .7108911 ccccc −= Conformal

vector case in this case is given in equation (5.2.30). Proper conformal vector field

can be obtained by subtracting Killing vector fields from (5.2.30) as

( ).,),(, 2287 zccrUctcX θ+= (5.2.32)

Also for solving equation (5.2.29) we suppose that 0),( =rtAtt and ⇒= 0)(2 tFtt

.,,)( 87872 ℜ∈+= ccctctF Substituting back the above values in (5.2.29) and

solving we get ),()(),( 3)(7 rKetfrtAdrrvc+∫=

− where )(tf and )(3 rK are functions

of integration. Differentiating the above value twice with respect to ,t we get

⇒== 0)(),( tfrtA tttt .,,)( 13121312 ℜ∈+= ccctctf Similarly differentiating the

above value with respect to r and solving we get

∫∫ ∫−−+∫+=−−

,)()(2)(ln)(),()(

1472)(


drrVcdrrVc where

,)(

1)(rU

rV = ∫ += 1)(21)( cdrrrU η and .71381214 ccccc −= Solution of equations

(5.2.2) to (5.2.11) becomes [42]

.,

,)(21,

5423

6422

11

870

cczcXcczcX

cdrrXctcX

++=+−=

+=+= ∫θθ

η (5.2.33)

The line element for non static cylindrically symmetric space-time in this case

becomes non static plan symmetric space-times and is given as

),( 22),(22),(2 dzdedrdteds rtBrtA +++−= θ (5.2.34)

where

∫∫ ∫−−+∫+=−−

,)()(2)(ln)(),()(

1472)(


drrVcdrrVc

∫∫ ∫−−+∫+=−−

,)()(2)(ln)(),()(

1122)(


drrVcdrrVc

246

,)(

1)(rU

rV = ,)(21)( 1∫ += cdrrrU η 72 cc ≠ ,7108911 ccccc −= 71381214 ccccc −=

and .1411 cc ≠ Proper conformal vector fields for (5.2.34) can be obtained by

subtracting Killing vector fields form (5.2.33) as

( ).,),(, 2287 zccrUctcX θ+= (5.2.35)

Case I (b):

In this case we have 0),( =rtB tr and .0),(2 ≠θθ tE Substituting 0),( =rtB tr in

(5.2.9) with )(),( 11 tFtE =θ and solve after differentiating with respect to θ and ,z

we get ),,(),(),(),,( 8763 θθθ tEztEztEztK ++= where ),,(6 ztE ),(7 ztE and

),(8 θtE are functions of integration. Substituting back all the above information in


( )

).,(),(

),,(),(),(,)(),(21

,),()(),(21

433

876211

210

θθ

θθη

θη

tEtzEX

tEztEztEXtFdrrtX

tEdretFdrdrrteX Att

A

+=

++=+=

++=

∫

∫∫∫ −−

(5.2.36)


differentiating with respect to z and ,θ we get ),()(),( 326 zFtFztE += where

)(2 tF and )(3 zF are functions of integration. Substituting back the above value in

(5.2.4) and solving we get ),()(),( 547 zFtFztE += where )(4 tF and )(5 zF are

functions of integration. Substituting back all the above information in (5.2.4) and

solving after differentiating twice with respect to ,θ we get

)()()(),( 8768 θθθ FtFtFtE ++= and ),()()(21),( 1110922 tFtFtFtE ++= θθθ

where ),(6 tF ),(7 tF ),(8 θF ),(9 tF )(10 tF and )(11 tF are functions of integration.

Refreshing the system of equations (5.2.36) with the help of above information we

get

247

( )

( )).,(),(

),()()()()()()(

,)(),(21

,)()()(21)(),(

21

433

87654322

11

11109210

θθ

θθθ

η

θθη

tEtzEXFtFtFzFtFzFtFX

tFdrrtX

tFtFtFdretFdrdrrteX Att

A

+=

++++++=

+=

++++=

∫

∫∫∫ −−

(5.2.37)


differentiating twice with respect to z then with respect to ,θ we get

.,,,21)( 321321

23 ℜ∈++= cccczcczzF Substituting back the above value we get

.,,,21)( 654654

25 ℜ∈++= cccczcczzF Substituting back the above information in

(5.2.10) and solve after differentiating first with respect to z then with respect to ,θ

we get 01 =c and ),()()(21),( 14131223 tFtFtFtE ++= θθθ where ),(12 tF )(13 tF

and )(14 tF are functions of integration. Substituting back these values and solving

we get 04 =c and .0)()( 1312 == tFtF Substituting the above information in (5.2.10)

and solving we get ,052 == cc ),(),( 164 tFtE =θ ),(),( 143 tFtE =θ 65 )( czF = and

.)( 33 czF = Substituting all the above information in (5.2.37) we get

( )

( )).()(

),()()()()(

,)(),(21

,)()()(21)(),(

21

16143

8766

43

22

11

11109210

tFtzFX

FtFtFctFctFX

tFdrrtX

tFtFtFdretFdrdrrteX Att

A

+=

++++++=

+=

++++=

∫

∫∫∫ −−

θθθ

η

θθη

(5.2.38)


differentiating with respect to ,θ we get 0)(9 =tF and ).()( 62 tFtF tt −= Substituting

back the above values we get 0)(10 =tF and ).()( 74 tFtF tt −= Substituting all the

248

above values in the equation ⇒++= )()()(21),( 1110922 tFtFtFtE θθθ .0),(2 =θθ tE

Which is a contradiction to the fact that .0),(2 ≠θθ tE Hence this case is not possible.

Case I (c):

In this case we have 0),( =rtB tr and .0),(2 =θθ tE Equation


Substituting the above information in (5.2.9) and solve after differentiating with

respect to ,θ we get ),,(),(),,( 653 ztEztEztK += θθ where ),(5 ztE and ),(6 ztE

are functions of integration. Substituting all the above information in (5.2.14) we get

( )

).,(),(

),,(),(,)(),(21

,)()(),(21

433

65211

210

θθ

θη

η

tEtzEX

ztEztEXtFdrrtX

tFdretFdrdrrteX Att

A

+=

+=+=

++=

∫

∫∫∫ −−

(5.2.39)


)(),( 35 zFztE = and ),(),( 46 zFztE = where )(3 zF and )(4 zF are functions of

integration. Also considering equation (5.2.5) and using equation (5.2.39) then

solving we get )(),( 53 θθ FtE = and ),(),( 64 θθ FtE = where )(5 θF and )(6 θF are

functions of integration. Substituting the above values in equation (5.2.39) we get

( )

).()(

),()(,)(),(21

,)()(),(21

653

43211

210

θθ

θη

η

FzFX

zFzFXtFdrrtX

tFdretFdrdrrteX Att

A

+=

+=+=

++=

∫

∫∫∫ −−

(5.2.40)


respect to ,r we get ( )( ) .0)()( 65 =+− θθ θθ FzFBC rr In this case we have

( ) ⇒≠− 0rr BC ( ) ⇒=+ 0)()( 65 θθ θθ FzF 15 )( cF =θ and .,,)( 212

6 ℜ∈= cccF θ

249

Substituting back the above information in (5.2.10) we get 33 )( czF = and

.,,)( 4344 ℜ∈= ccczF Refreshing the system of equations (5.2.40) we get

( ).,,)(),(

21

,)()(),(21

213

43211

210

czcXccXtFdrrtX

tFdretFdrdrrteX Att

A

+=+=+=

++=

∫

∫∫∫ −−

θη

η (5.2.41)

Now subtracting equation (5.2.9) from equation (5.2.11) and solving after

differentiating with respect to ,r we get .0),( =rtη Which means that in this case no

proper conformal vector field exist.

Case II:

In this case we have 0),,(2 ≠ztK z θ and we suppose .0),( =rtCt Substituting the

above value in equation (5.2.13) we get ⇒= 0),,(4 ztK zz θ

),,(),(),,( 324 θθθ tEtzEztK += where ),(2 θtE and ),(3 θtE are functions of

integration. Substituting the above information in equation (5.2.12) we get

( )

).,(),(),,,(),(

),,(),(21

),,,(),(),(21

3233),(12

11

2),(1),(0

θθθθ

θη

θθη

θ tEtzEXztKdretEX

tEdrrtX

ztKdretEdrdrrteX

rtB

rtAtt

rtA

+=+−=

+=

++=

∫∫

∫∫ ∫

−

−−

(5.2.42)


differentiating with respect to θ we get ),(),(0),( 111 tFtEtE =⇒= θθθ where )(1 tF

is a function of integration and .0),,(2),,(),( 32 =+ ztKztKrtBt θθ θθθ Differentiating

the above equation with respect to ,r we get .0),,(),( 2 =ztKrtBrt θθ We will discuss

here three different possibilities:

(a) ,0),( ≠rtBrt ,0),,(2 =ztK θθ (b) ,0),( =rtBrt ,0),,(2 ≠ztK θθ

(c) ,0),( =rtBrt .0),,(2 =ztK θθ

250


Case II (a):

In this case we have 0),( ≠rtBrt and .0),,(2 =ztK θθ Equation ⇒= 0),,(2 ztK θθ

),,(),,( 42 ztEztK =θ where ),(4 ztE is a function of integration. Substituting back

the above values in equation ,0),,(2),,(),( 32 =+ ztKztKrtBt θθ θθθ we get

⇒= 0),,(3 ztK θθθ ),,(),(),,( 653 ztEztEztK +=θθ where ),(5 ztE and ),(6 ztE are

functions of integration. Substituting all the above information in equation (5.2.42)

we get

( )

),,(),(),,(),(

,)(),(21

,),()(),(21

323652

11

410

θθθ

η

η

tEtzEXztEztEX

tFdrrtX

ztEdretFdrdrrteX Att

A

+=+=

+=

++=

∫

∫ ∫∫ −−

(5.2.43)


differentiating with respect to ,z we get )(),( 22 θθ FtE = and

),()(),( 434 tFtzFztE += where ),(2 θF )(3 tF and )(4 tF are functions of

integration. Substituting back the above values in (5.2.5) and solving, we get

0),(3 =θtEt and .0)(3 =tF Substituting back the above values in equation (5.2.5) we

get .0),,(2 =ztK z θ Which is a contradiction in this case. Hence this case is not

possible.

Case II (b):

In this case we have ,0),( =rtC tr ,0),,(2 ≠ztK z θ 0),( =rtBrt and .0),,(2 ≠ztK θθ

To solve the equation 0),,(2),,(),( 32 =+ ztKztKrtBt θθ θθθ we suppose 0),( =rtBt

which in turn implies ⇒= 0),,(3 ztK θθθ ),,(),(),,( 543 ztEztEztK += θθ where

251

),(4 ztE and ),(5 ztE are functions of integration. Refreshing the system of equations

(5.2.42) we get

( )

).,(),(),,(),(

,)(),(21

,),,()(),(21

323542

11

210

θθθ

η

θη

tEtzEXztEztEX

tFdrrtX

ztKdretFdrdrrteX Att

A

+=+=

+=

++=

∫

∫∫∫ −−

(5.2.44)



),,(),(),(21),,( 87622 θθθθ tEtEztEzztK ++= where ),,(6 θtE ),(7 θtE and

),(8 θtE are functions of integration. Substituting back the above value in (5.2.5) we

get 0),,(2 =ztK z θ which gives contradiction. Hence this case is also not possible.

Case II (c):

In this case we have ,0),( =rtC tr ,0),,(2 ≠ztK z θ 0),( =rtBrt and .0),,(2 =ztK θθ

Equation ⇒= 0),,(2 ztK θθ ),(),,( 42 ztEztK =θ where ),(4 ztE is a function of

integration. Substituting the above value in equation 0),,(2),,(),( 32 =+ ztKztKrtBt θθ θθθ

we get ),,(),(),,(0),,( 6533 ztEztEztKztK +=⇒= θθθθθ where ),(5 ztE and

),(6 ztE are functions of integration. Substituting all the above information in


( )

).,(),(),,(),(

,)(),(21

,),()(),(21

323652

11

410

θθθ

η

η

tEtzEXztEztEX

tFdrrtX

ztEdretFdrdrrteX Att

A

+=+=

+=

++=

∫

∫∫∫ −−

(5.2.45)


differentiating twice with respect to z we get

252

),()()(21),( 22212024 tFtFztFzztE ++= where ),(20 tF )(21 tF and )(22 tF are

functions of integration. Substituting back the above value in (5.2.5) and solving we

get ⇒== 0)()( 2120 tFtF ⇒= 0),(4 ztEz 0),,(2 =ztK z θ which gives contradiction.

Hence this case is also not possible.

Case III:

In this case we have 0),( =rtCrt and .0),,(2 =ztK z θ Equation ⇒= 0),,(2 ztK z θ

),,(),,( 22 θθ tEztK = where ),(2 θtE is a function of integration. Substituting back

the above value in (5.2.13) we get ),,(),(),,(0),,( 4344 θθθθ tEtzEztKztK zz +=⇒=

where ),(3 θtE and ),(4 θtE are functions of integration. Substituting all the above


( )

),,(),(,),,(),,(

,),(),(21

,),(),(),(21

433312

11

210

θθθθ

θη

θθη

θ tEtzEXztKdreztEX

tEdrrtX

tEdretEdrdrrteX

B

Att

A

+=+−=

+=

++=

∫∫

∫ ∫∫

−

−−

(5.2.46)


differentiating with respect to θ we get ⇒= 0),(1 θθ tE ),(),( 11 tFtE =θ where

)(1 tF is a function of integration. Substituting back the above value in equation

(5.2.9) and differentiating the resulting equation with respect to ,r we get

.0),(),( 2 =θθ tErtBrt To solve this equation we need to discuss three different

possibilities.

(a) ,0),( ≠rtBrt ,0),(2 =θθ tE (b) ,0),( =rtBrt ,0),(2 ≠θθ tE

(c) ,0),( =rtBrt .0),(2 =θθ tE


253

Case III (a):

In this case we have 0),( ≠rtBrt and .0),(2 =θθ tE Equation ⇒= 0),(2 θθ tE

),(),( 22 tFtE =θ where )(2 tF is a function of integration. Substituting back the

above value in (5.2.9) we get ),,(),(),,(0),,( 6533 ztEztEztKztK +=⇒= θθθθθ

where ),(5 ztE and ),(6 ztE are functions of integration. Substituting all the above


( )

),,(),(),,(),(

,)(),(21

,)()(),(21

433652

11

210

θθθ

η

η

tEtzEXztEztEX

tFdrrtX

tFdretFdrdrrteX Att

A

+=+=

+=

++=

∫

∫ ∫∫ −−

(5.2.47)


respect to ,θ we get ⇒= 0),(5 ztEt ),(),( 35 zFztE = where )(3 zF is a function of

integration. Substituting back the above value in (5.2.4) we get

),(),(0),( 466 zFztEztEt =⇒= where )(4 zF is a function of integration.

Substituting back all the above information in (5.2.47) we get

( )

).,(),(),()(

,)(),(21

,)()(),(21

433432

11

210

θθθ

η

η

tEtzEXzFzFX

tFdrrtX

tFdretFdrdrrteX Att

A

+=+=

+=

++=

∫

∫ ∫∫ −−

(5.2.48)


differentiating with respect to z we get ).(),(0),( 533 θθθ FtEtEt =⇒= Substituting

back the above value in (5.2.5) we get ),(),(0),( 644 θθθ FtEtEt =⇒= where

)(5 θF and )(6 θF are functions of integration. Substituting back all the above


254

( )

).()(),()(

),(),(21

,)()(),(21

653432

11

210

θθθ

η

η

FzFXzFzFX

tFdrrtX

tFdretFdrdrrteX Att

A

+=+=

+=

++=

∫

∫ ∫∫ −−

(5.2.49)


( ) ( ) .0)()()()( 4365 =+++− zFzFFFze zzBC θθθ θθ Differentiating the above equation

first with respect to r then with respect to ,t we get

( )( ) .0)()(),(),( 65 =+− θθ θθ FFzrtBrtC rtrt In this case we know that 0),( =rtCrt and

0),( ≠rtBrt which in turn gives ( ) ⇒=+ 0)()( 65 θθ θθ FFz 15 )( cF =θ and

.,,)( 2126 ℜ∈= cccF θ Substituting back these values in the above equation we get

33 )( czF = and .,,)( 434

4 ℜ∈= ccczF Refreshing our system of equations, we get

( ).,),(),(

21

,)()(),(21

213

43211

210

czcXccXtFdrrtX

tFdretFdrdrrteX Att

A

+=+=+=

++=

∫

∫ ∫∫ −−

θη

η (5.2.50)

Now subtracting equation (5.2.9) from equation (5.2.11) and solving after

differentiating with respect to ,r we get .0),( =rtη Which means that in this case no

proper conformal vector field exist.

Case III (b):

In this case we have 0),( =rtB tr and .0),(2 ≠θθ tE Substituting the above

information in equation (5.2.9) with 0),( =rtB t and solving we get

⇒= 0),,(3 ztK θθθ ),,(),(),,( 653 ztEztEztK += θθ where ),(5 ztE and ),(6 ztE are

functions of integration. Refreshing the system of equations (5.2.46) we get

255

( )

).,(),(),,(),(

,)(),(21

,),()(),(21

433652

11

210

θθθ

η

θη

tEtzEXztEztEX

tFdrrtX

tEdretFdrdrrteX Att

A

+=+=

+=

++=

∫

∫∫∫ −−

(5.2.51)


differentiating with respect to θ we get )(),( 25 zFztE = and

),()(),( 432 tFtFtE += θθ where ),(2 zF )(3 tF and )(4 tF are functions of

integration. Substituting back the above value in equation (5.2.4) and solve after


),()(),( 656 zFtFztE += where )(5 tF and )(6 zF are functions of integration.

Substituting back all the above information in (5.2.4) and solving we get

⇒= 0)(3 tF .0),(2 =θθ tE Which gives contradiction and hence this case is not

possible.

Case III (c):

In this case we have 0),( =rtB tr and .0),(2 =θθ tE Equation


Substituting the above information in (5.2.9) and solve after differentiating with

respect to ,θ we get ⇒= 0),,(3 ztK θθθ ),,(),(),,( 653 ztEztEztK += θθ where

),(5 ztE and ),(6 ztE are functions of integration. Substituting all the above


( )

).,(),(),,(),(

,)(),(21

,)()(),(21

433652

11

210

θθθ

η

η

tEtzEXztEztEX

tFdrrtX

tFdretFdrdrrteX Att

A

+=+=

+=

++=

∫

∫∫∫ −−

(5.2.52)

256


differentiating with respect to ,θ we get )(),(0),( 355 zFztEztEt =⇒= and

⇒= 0),(6 ztEt ),(),( 46 zFztE = where )(3 zF and )(4 zF are functions of

integration. Also considering equation (5.2.5) and using equation (5.2.52) then solve

after differentiating with respect to ,z we get )(),(0),( 533 θθθ FtEtEt =⇒= and

⇒= 0),(4 θtEt ),(),( 64 θθ FtE = where )(5 θF and )(6 θF are functions of

integration. Substituting all the above information in equation (5.2.52) we get

( )

).()(),()(

,)(),(21

,)()(),(21

653432

11

210

θθθ

η

η

FzFXzFzFX

tFdrrtX

tFdretFdrdrrteX Att

A

+=+=

+=

++=

∫

∫∫∫ −−

(5.2.53)


differentiating with respect to z and ,θ we get

ℜ∈+=⇒= 212155 ,,)(0)( ccccFF θθθθθ and .,,)(0)( 4343

33 ℜ∈+=⇒= ccczcFF θθθθ

Substituting back the above information in (5.2.10) and solving we get

ℜ∈+=⇒= 656544 ,,)(0)( ccczczFzFzz and ℜ∈+=⇒= 8787

66 ,,)(0)( ccccFF θθθθθ

which on back substitution give us .07531 ==== cccc Substituting all the above


( ).,,)(),(

21

,)()(),(21

823

64211

210

czcXccXtFdrrtX

tFdretFdrdrrteX Att

A

+=+=+=

++=

∫

∫∫∫ −−

θη

η (5.2.54)

Subtracting equation (5.2.9) from equation (5.2.11) we get

.0)(2)()( 4210 =−+−+− ccXBCXBC rrtt In order to solve this equation we

suppose ,0)( =− tt BC which in turn gives ( ).2)(),(21)( 24

1 cctFdrrtBC rr −=⎥⎦⎤

⎢⎣⎡ +− ∫η

Differentiating the above equation with respect to ,t we get

257

.0)(),(21)( 1 =⎥⎦

⎤⎢⎣⎡ +− ∫ tFdrrtBC ttrr η Solution of this equation involves three

different possibilities:

(l) ,0)( ≠− rr BC ,0)(),(21 1 =⎥⎦


(m) ,0)( =− rr BC ,0)(),(21 1 ≠⎥⎦


(n) ,0)( =− rr BC .0)(),(21 1 =⎥⎦



Case III (c) (l):

In this case we have 0)( ≠− rr BC and .0)(),(21 1 =⎥⎦


⇒=⎥⎦⎤

⎢⎣⎡ +∫ 0)(),(21 1 tFdrrt ttη )(rηη = and .,)( 99

1 ℜ∈= cctF Refreshing the


.,,),(21),( 82

364

29

120 czcXccXcdrrtXtFX +=+=+== ∫ θη (5.2.55)


respect to t we get

.0)(2)(21),()(),()(),( 2

922 =+⎟

⎠⎞

⎜⎝⎛ +++ ∫ tFcdrrrtAtFrtAtFrtA ttrttttt η (5.2.56)

In order to solve (5.2.56) we suppose .0),( =rtAt Hence equation (5.2.56) gives

⇒= 0)(2 tFtt .,,)( 111011102 ℜ∈+= ccctctF Substituting back all the above

information in equation (5.2.2) we get ⇒=+⎥⎦⎤

⎢⎣⎡ +∫ )(2)(21

109 rccdrrAr ηη

258

( ) .)(

21

2)(ln)(1

10

2

9 ∫∫

∫+

−+=cdrr

drccdrrrAη

η Now differentiating equation (5.2.9)

with respect to t and using the fact that ,0),( =rtBrt we get

⇒=++ 0),()(),( 101110 rtBcctcrtB ttt ( ) ,)(),( 11110

1 −+= ctcrKrtBt where )(1 rK is

the function of integration. Differentiating the above equation with respect to ,r we

get .,)(0)(),( 121211 ℜ∈=⇒== ccrKrKrtB rrt Substituting this value in the above

equation and integrating with respect to ,t we get ( ) ),(ln),( 10

12

1110 rfctcrtB cc

++=

where )(rf is the function of integration. In this case 0)( =− tt BC therefore

( ) ),(ln),( 10

12

1110 rgctcrtC cc++= where )(rg is a function of integration. Substituting

back the above values in equation (5.2.9) then solving we get

( ) ,)()(ln)( 13

2

9 ∫∫ −+= drrVccdrrrf η ( ) ,)()(ln)( 14

2

9 ∫∫ −+= drrVccdrrrg η where

,)(

21

1)(9∫ +

=cdrr

rVη

41213 2ccc += and .2 12214 ccc += Finally, substituting all

the above information in equation (5.2.55) the conformal vector fields become [42]

.,,)(21, 82

364

29

11110

0 czcXccXcdrrXctcX +=+=+=+= ∫ θη (5.2.57)

The line element for non static cylindrically symmetric space-time in this case is

given as

,2),(2),(22)(2 dzededrdteds rtCrtBrA +++−= θ (5.2.58)

where

( ) ,)(

21

2)(ln)(1

10

2

9 ∫∫

∫+

−+=cdrr

drccdrrrAη

η

( ) ( ) ,)()(lnln),( 13

2

91110 10

12

∫∫ −+++= drrVccdrrctcrtB cc

η

259

( ) ( ) ,)()(lnln),( 14

2

91110 10

12

∫∫ −+++= drrVccdrrctcrtC cc

η ,)(

21

1)(9∫ +

=cdrr

rVη

41213 2ccc += and .2 12214 ccc += Subtracting Killing vector fields from (5.2.57) the

proper conformal vector fields for the above space-times (5.2.58) takes the form

( ).,),(, 241110 zccrUctcX θ+= (5.2.59)

Another possibility to solve the equation 0),()(),( 101110 =++ rtBcctcrtB ttt is to take

.0),( =rtBt Equation ).(0),( rBBrtBt =⇒= Substituting the above value in

equation (5.2.9) and solving we get ∫−= ,)(2)(ln)( 42 drrVcrUrB where

)(1)(rU

rV = and .)(21)( 9∫ += cdrrrU η Similarly considering equation (5.2.11) and

using equation (5.2.57) we get

).(2)(21),()(),( 291110 rccdrrrtCctcrtC rt ηη =+⎟

⎠⎞

⎜⎝⎛ +++ ∫ Solving this equation by

considering ,0),( =rtCt we get ∫−= ,)(2)(ln)( 22 drrVcrUrC where

)(1)(rU

rV =

and .)(21)( 9∫ += cdrrrU η In this case the space-time becomes static cylindrically

symmetric space-time and is given as


where

∫−= ,)(2)(ln)( 102 drrVcrUrA ,)(2)(ln)( 4

2 ∫−= drrVcrUrB ,)(2)(ln)( 22 ∫−= drrVcrUrC

,)(

1)(rU

rV = ,)(21)( 9∫ += cdrrrU η ,010 ≠c ,02 ≠c ,04 ≠c ,410 cc ≠ 21 cc ≠ and

.42 cc ≠ Conformal vector fields for the above space-time are given in equation

(5.2.57). Proper conformal vector fields for the above space-times (5.2.60) takes the

form

260

( ).,),(, 241110 zccrUctcX θ+= (5.2.61)

Case III (c) (m):

In this case we have 0)( =− rr BC and .0)(),(21 1 ≠⎥⎦

⎤⎢⎣⎡ +∫ tFdrrt ttη To solve

equations (5.2.2), (5.2.8) and (5.2.11) we need to take .0)( =rη Which implies that in

this case no proper conformal vector fields exist.

Case III (c) (n):

In this case we have 0)( =− rr BC and .0)(),(21 1 =⎥⎦


⇒=− 0)( rr BC ),(),( rtBrtC rr = and equation ⇒=⎥⎦⎤

⎢⎣⎡ +∫ 0)(),(21 1 tFdrrt ttη

)(rηη = and .,)( 111 ℜ∈= cctF Substituting the above information in equation

(5.2.54) we get

).()(),()(

,)(21),(

653432

1120

θθθ

η

FzFXzFzFX

cdrrXtFX

+=+=

+== ∫ (5.2.62)

Now subtracting equation (5.2.11) from equation (5.2.9) and using the fact that in this

case ),(),( rtBrtC rr = and ),(),( rtBrtC tt = we get ⇒=− 0)()( 35 zFF θ

.,)()( 2235 ℜ∈== cczFF θ Substituting back the above values in equation (5.2.9)

and solving we get ℜ∈+=⇒= 545466 ,,)(0)( ccccFF θθθθθ and

.,,)(0)( 646444 ℜ∈+−=⇒= ccczczFzFzz Substituting all the above information in


5423

6422

1120

,

)(21),(

cczcXcczcX

cdrrXtFX

++=+−=

+== ∫θθ

η (5.2.63)

261


respect to ,t we get

.0)(2)(21)()( 2

122 =+⎥⎦

⎤⎢⎣⎡ +++ ∫ tFcdrrAtFAtFA ttrttttt η (5.2.64)

In order to solve this equation we suppose ⇒= 0),( rtAt ).(rAA = Substituting the

above value in equation (5.2.64) we get .,,)(0)( 878722 ℜ∈+=⇒= ccctctFtFtt

Substituting back the above value in equation (5.2.64) and solving we get

( ) ∫∫ −+= ,)(2)(ln)( 7

2

1 drrVccdrrrA η where .)(

21

1)(1∫ +

=cdrr

rVη

Now

substituting the above information in equation (5.2.63), we get

5423

6422

11

870

,

)(21,

cczcXcczcX

cdrrXctcX

++=+−=

+=+= ∫θθ

η (5.2.65)


respect to ,t we get ⇒=++ 0)( 787 cBctcB ttt ( ) ,)(),( 187

1 −+= ctcrKrtBt where

)(1 rK is a function of integration. Differentiating the above equation with respect to

,r we get ℜ∈=⇒== 9911 ,)(0)(),( ccrKrKrtB rrt therefore

( ) ),(ln),( 21110 7

9

rKctcrtB cc++= where )(2 rK is a function of integration. Substituting

back the above value in equation (5.2.9) and solving we get

( ) ∫∫ −+= ,)()(ln)( 10

2

12 drrVccdrrrK η where

∫ +=

1)(21

1)(cdrr

rVη

and

.2 2910 ccc += The line element for non static cylindrically symmetric space-time in

this case after a suitable rescaling of z is given as

),( 22),(22)(2 dzdedrdteds rtBrA +++−= θ (5.2.66)

where

262

∫−= ,)(2)(ln)( 72 drrVcrUrA ,)()(ln)ln(),( 10

287

7

9

∫−++= drrVcrUctcrtB cc

,)(

1)(rU

rV = ∫ += 1)(21)( cdrrrU η and .2 2910 ccc += In this case the proper

conformal vector fields can be obtained by subtracting Killing vector fields from

(5.2.65) as

( ).,),(, 2287 zccrUctcX θ+= (5.2.67)

Another possible solution for the equation 0)( 787 =++ cBctcB ttt is to take

0),( =rtBt which in turn gives ,)(2)(ln)( 22 ∫−= drrVcrUrB where

.)(

21

1)(1∫ +

=cdrr

rVη

Conformal vector fields are given as

5423

6422

11

870

,

)(21,

cczcXcczcX

cdrrXctcX

++=+−=

+=+= ∫θθ

η (5.2.68)

In this case the space-time becomes static plane symmetric space-time and is given as

),( 22)(22)(2 dzdedrdteds rBrA +++−= θ (5.2.69)

where

∫−= ,)(2)(ln)( 72 drrVcrUrA ,)(2)(ln)( 2

2 ∫−= drrVcrUrB ,)(

1)(rU

rV =

.)(21)( 1∫ += cdrrrU η Subtracting Killing vector fields from (5.2.68) we get the

proper conformal vector field as

( ).,),(, 227 zccrUtcX θ= (5.2.70)

263

5.3. Proper Conformal Vector Fields in Non

Conformally Flat Kantowski-Sachs and

Bianchi Type III Space-Times

Consider Kantowski-Sachs and Bianchi type III space-times in usual coordinates

),,,( φθrt (labeled by ),,,,( 3210 xxxx respectively) with the line element [77]


where A and B are no where zero functions of t only. For θθ sinh)( =f the space-



independent Killing vector fields [79], which are

,r∂∂ ,

φ∂∂ ,sin

)()(cos

φφ

θθ

θφ

∂∂′

−∂∂

ff ,cos

)()(sin

φφ

θθ

θφ

∂∂′

+∂∂

ff (5.3.2)

where prime denotes the derivative with respect to .θ A vector field X is said to be a

conformal vector field if it satisfies equation (1.2.4). One can write (1.2.4) explicitly

using (5.3.1) as

,0,0 η=X (5.3.3)

,01,0

0,12 =− XXA (5.3.4)

,02,0

0,22 =− XXB (5.3.5)

,0)( 3,0

0,322 =− XXfB θ (5.3.6)

,1,10 η=+ XX

AAt (5.3.7)

,02,12

1,22 =+ XAXB (5.3.8)

264

,0)( 1,322

3,12 =+ XfBXA θ (5.3.9)

,2,20 η=+ XX

BBt (5.3.10)

,0)( 2,32

3,2 =+ XfX θ (5.3.11)

.)()(

3,320 η

θθ

=+′

+ XXffX

BBt (5.3.12)

Here it follows that [40] ).(tηη = Integrating (5.3.3), (5.3.4), (5.3.5) and (5.3.6) we

get

).,,(1),,()(

1

),,,(1),,(

),,,(1),,(),,,()(

42

12

3

32

12

22

1110

φθφθθ

φθφθ

φθφθφθη

φ

θ

rKdtB

rKf

X

rKdtB

rKX

rKdtA

rKXrKdttX r

+=

+=

+=+=

∫

∫

∫∫

(5.3.13)

where ),,,(1 φθrK ),,,(2 φθrK ),,(3 φθrK and ),,(4 φθrK are functions of

integration which are to be determined. Considering equation (5.3.11) and using

equation (5.3.13) then solve after differentiating with respect to t we get

),,(),()(),,( 211 θφφθφθ rFdrFfrK += ∫ where ),(1 φrF and ),(2 θrF are functions

of integration. Substituting the above value in equation (5.3.13) we get

( )( )

).,,(1),()(

1

),,,(1),(),()(

),,,(1),(),()(

),,(),()()(

42

13

32

212

22

211

210

φθφθ

φθθφφθ

φθθφφθ

θφφθη

θ

rKdtB

rFf

X

rKdtB

rFdrFfX

rKdtA

rFdrFfX

rFdrFfdttX

rr

+=

++′=

++=

++=

∫

∫∫

∫∫

∫∫

(5.3.14)

265

Considering equation (5.3.10) and using equation (5.3.14) then differentiating first

with respect to φ then with respect to ,t we get .0),(1 122

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛+

−φrF

BBBBB ttt In

order to solve this equation we will discuss three different possibilities:

(I) ,0122

2

≠⎟⎟⎠

⎞⎜⎜⎝

⎛+

−BB

BBB ttt ,0),(1 =φrF (II) ,0122

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛+

−BB

BBB ttt ,0),(1 ≠φrF

(III) ,0122

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛+

−BB

BBB ttt .0),(1 =φrF


Case I:

In this case we have 0122

2

≠⎟⎟⎠

⎞⎜⎜⎝

⎛+

−BB

BBB ttt and .0),(1 =φrF Substituting back the

above value in (5.3.10) and solve after differentiating with respect to ,φ we get

),,(),(),,(0),,( 4333 φθφθφθθφ rFrFrKrK +=⇒= where ),(3 θrF and ),(4 φrF are

functions of integration. Substituting all the above information in (5.3.14) we get

).,,(),,(),(1),(

),,,(1),(),,()(

43432

22

22

2120

φθφθθ

φθθθη

θ rKXrFrFdtB

rFX

rKdtA

rFXrFdttX r

=++=

+=+=

∫

∫∫ (5.3.15)


differentiating with respect to ,φ we get

),,(),(),,(0),,( 6522 φθθφθφθφ FrFrKrKr +=⇒= where ),(5 θrF and ),(6 φθF

are functions of integration. Considering equation (5.3.11) and using equation

(5.3.15) then solve after differentiating with respect to θ and we get

),,(),()()(),,( 874 φφ

θθφθ rFrF

ffrK +′

−= where ),(7 φrF and ),(8 φrF are functions

of integration. Now substituting all the above information in equation (5.3.15) we get

266

).,(),()()(),,(),(1),(

),,(),(1),(),,()(

873432

22

652

2120

φφθθφθθ

φθθθθη

θ rFrFffXrFrFdt

BrFX

FrFdtA

rFXrFdttX r

+′

−=++=

++=+=

∫

∫∫ (5.3.16)

Considering equation (5.3.10) and using equation (5.3.16) then differentiating first

with respect to r then twice with respect to ,t we get ,0),(22 =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛•••

θrFBBB r

where dot denotes differentiation with respect to .t In this case ,02 ≠⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛•••

BBB

therefore ),(),(0),( 122 θθθ DrFrFr =⇒= where )(1 θD is the function of

integration. Substituting back the above value in (5.3.10) and solve after

differentiating with respect to ,r we get ),()(),( 323 θθ DrDrF += where )(2 rD

and )(3 θD are functions of integration. Substituting all the above information in

(5.3.16), we get

).,(),()()(

),,()()(1)(

),,(),(),()(

873

4322

12

65110

φφθθ

φθθ

φθθθη

θ

rFrFffX

rFDrDdtB

DX

FrFXDdttX

+′

−=

+++=

+=+=

∫

∫ (5.3.17)


differentiating with respect to r we get ),()(),( 545 θθθ DrDrF += where )(4 θD

and )(5 θD are functions of integration. Substituting all the above information in


267

).,(),()()(

),,()()(1)(

),,()()(),()(

873

4322

12

654110

φφθθ

φθθ

φθθθθη

θ

rFrFffX

rFDrDdtB

DX

FDrDXDdttX

+′

−=

+++=

++=+=

∫

∫ (5.3.18)

Now subtracting equation (5.3.10) from equation (5.3.12) and using equation (5.3.18)

in the resulting equation then solve after differentiating with respect to ,t we get

,)()( 211 cfcD +′= θθ ., 21 ℜ∈cc Refreshing the above equation (5.3.18) we get

).,(),()()(

),,()()(1)(

),,()()(,)()(

873

43221

2

654121

0

φφθθ

φθθ

φθθθθη

rFrFffX

rFDrDdtB

fcX

FDrDXcfcdttX

+′

−=

+++=

++=+′+=

∫

∫ (5.3.19)


resulting equation with respect to ,t we get ( ) .0),()()( 6542

2

=++⎟⎟⎠

⎞⎜⎜⎝

⎛•

φθθθ θθθ FDrDBA

The solution of this equation can be obtained by considering three different

possibilities:

(a) ,02

2

≠⎟⎟⎠

⎞⎜⎜⎝

⎛•

BA ( ) ,0),()()( 654 =++ φθθθ θθθ FDrD

(b) ,02

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛•

BA ( ) ,0),()()( 654 ≠++ φθθθ θθθ FDrD

(c) ,02

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛•

BA ( ) .0),()()( 654 =++ φθθθ θθθ FDrD


268

Case I (a):


2

≠⎟⎟⎠

⎞⎜⎜⎝

⎛•

BA and ( ) .0),()()( 654 =++ φθθθ θθθ FDrD Differentiating

the above equation with respect to ,r we get .,)(0)( 3344 ℜ∈=⇒= ccDD θθθ

Substituting back this value in the above equation we get

),()(),( 656 φθφθ DDF +−= where )(6 φD is a function of integration. Also

substituting all the above information in equation (5.3.11) we get

).,(),( 47 φφ φ rFrF −= Refreshing the system of equations (5.3.19) we get

).,(),()()(

),,()()(1)(

),(,)()(

843

43221

2

63

121

0

φφθθ

φθθ

φθη

φ rFrFffX

rFDrDdtB

fcX

DrcXcfcdttX

+′

=

+++=

+=+′+=

∫

∫ (5.3.20)


differentiating first with respect to θ then with respect to t twice we get

,)( 543 ccD += θθ ℜ∈54 ,cc and .01 =c Substituting all the above values in

(5.3.20) we get

).,(),()()(),,()(

),(,)(

843454

22

63

12

0

φφθθφθ

φη

φ rFrFffXrFccrDX

DrcXcdttX

+′

=+++=

+=+= ∫ (5.3.21)


),()(),( 724 φφ DrDrF +−= where )(7 φD is the function of integration. Now

considering equation (5.3.12) and using equation (5.3.21) then solve after

differentiating the resulting equation first with respect to φ then with respect to ,θ

we get ,sincos)( 8767 φφφ cccD ++= .,, 876 ℜ∈ccc Substituting back the above

values in (5.3.8) and solving we get ),()(),( 988 rDrDrF += φφ where )(8 rD and

269

)(9 rD are functions of integration. Now substituting all the above information in

(5.3.21), we get

( ) ).()(sincos)()(

,sincos

),(,)(

9878

3

98742

63

12

0

rDrDccffX

ccccX

DrcXcdttX

++−′

=

+++=

+=+= ∫

φφφθθ

φφθ

φη

(5.3.22)


differentiating with respect to θ and φ we get ,)( 108 crD = 11

9 )( crD = and

,)( 126 cD =φ .,, 121110 ℜ∈ccc Substituting all the above information in equation

(5.3.22) we get

( ) .sincos)()(

,sincos

,,)(

1110783

98742

1231

20

ccccffX

ccccX

crcXcdttX

++−′

=

+++=

+=+= ∫

φφφθθ

φφθ

η

(5.3.23)

Subtracting equation (5.3.10) from equation (5.3.12) and using equation (5.3.23) in

the resulting equation then differentiating twice with respect to ,θ we get

.01094 === ccc Substituting all the above information in equation (5.3.23) we get

( ) .sincos)()(,sincos

,,)(

11783

872

1231

20

cccffXccX

crcXcdttX

+−′

=+=

+=+= ∫φφ

θθφφ

η (5.3.23)*

Considering equations (5.3.10) and (5.3.7) respectively and using equation (5.3.23)*

we get )()( tUtB = and ,)()( )(3 dttU

c

etUtA ∫=

−

where ( ).)()( 2cdtttU += ∫η Conformal

vector fields are given in equation (5.3.23)*. The line element for Kantowski-Sachs

and Bianchi type III space-times in this case is given in equation (5.3.1) with

270

dttU

c

etUtA ∫=

−)(

3

)()( and ),()( tUtB = where ( ).)()( 2cdtttU += ∫η Proper conformal

vector fields after subtracting Killing vector fields from (5.3.23)* is given as

( ).0,0,),( 3rctUX = (5.3.24)

Case I (b):

In this case we have ( ) 0),()()( 654 ≠++ φθθθ θθθ FDrD and .02

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛•

BA Equation

⇒=⎟⎟⎠

⎞⎜⎜⎝

⎛•

02

2

BA ),()( 22 tBtA α= where .0\ℜ∈α Substituting the above value in

(5.3.8) and solve after differentiating with respect to θ then with respect to ,r we get

434 )( ccD += θθ ., 43 ℜ∈cc Substituting back the above values in (5.3.8) and

solving we get )()()(),( 8766 θφφθφθ DDDF ++= and

),()()(),( 111094 rDDDrrF ++= φφφ where ),(6 φD ),(7 φD ),(8 θD ),(9 φD

)(10 φD and )(11 rD are all functions of integration. Refreshing the system of equations

(5.3.19) we get

( )

).,(),()()(

),()()()(1)(

),()()(

,)()(

873

10931221

2

761343

1

210

φφθθ

φφθθ

φφθθθ

θη

rFrFffX

DDrDrDdtB

fcX

DDDccrX

cfcdttX

+′

−=

++++=

++++=

+′+=

∫

∫

(5.3.25)


),()(),( 1497 φφφ φ DDrrF +−= where )(14 φD is a function of integration. Also

substituting the above value and equation (5.3.25) in equation (5.3.10) and solve after

differentiating with respect to θ and then twice with respect to t we get .01 =c

271

Substituting back the above value we get .,,)( 65653 ℜ∈+= ccccD θθ Substituting all

the above information in equation (5.3.25) we get

( )

( ) ).,()()()()(

),()()(

),()()(

,)(

81493

10965

122

761343

1

20

φφφθθ

φφθ

φφθθθ

η

φ rFDDrffX

DDrccrDX

DDDccrX

cdttX

++−′

−=

++++=

++++=

+= ∫

(5.3.26)


differentiating with respect to ,φ we get .,)()( 771014 ℜ∈+−= ccDD φφ Also

considering equation (5.3.12) and using equation (5.3.26) then solve after

differentiating with respect to ,φ r and θ we get

.,,,sincos)( 109810989 ℜ∈++= ccccccD φφφ Substituting back the above value in

(5.3.12) and solving we get ( ) .,,sinhcosh)( 1211121110 ℜ∈−+= ccccD φφφ

Substituting all the above information in equation (5.3.26) we get

( )( )

( ) ( )[ ] ),,(sinhcoshsincos)()(

,sinhcosh)sincos()(

),()()(

,)(

81312910

3

111091285122

761343

1

20

φφφφφθθ

φφφφθ

φφθθθ

η

rFccccrffX

cccrcrccrDX

DDDccrX

cdttX

++−−−−′

−=

−++++++=

++++=

+= ∫

(5.3.27)

where .11713 ccc −= Considering equation (5.3.9) and using equation (5.3.27) then

solve after differentiating twice with respect to θ we get .0109 == cc Substituting

back the above value in (5.3.27) and solving the remaining equations we reach to the

contradiction that ( ) .0),()()( 654 =++ φθθθ θθθ FDrD Hence this case is not possible.

272

Case I (c):

In this case we have ( ) 0),()()( 654 =++ φθθθ θθθ FDrD and .02

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛•

BA Equation

⇒=⎟⎟⎠

⎞⎜⎜⎝

⎛•

02

2

BA ),()( 22 tBtA α= where .0\ℜ∈α Differentiating

( ) 0),()()( 654 =++ φθθθ θθθ FDrD with respect to ,r we get

.,)(0)( 3344 ℜ∈=⇒= ccDD θθθ Substituting back this value in the above equation

we get ),()(),( 656 φθφθ DDF +−= where )(6 φD is a function of integration. Also

considering equation (5.3.11) and using equation (5.3.19) we get

).,(),( 47 φφ φ rFrF −= Now substituting all the above values in equation (5.3.19) we

get

).,(),()()(

),,()()(1)(

),(,)()(

843

43221

2

63

121

0

φφθθ

φθθ

φθη

φ rFrFffX

rFDrDdtB

fcX

DrcXcfcdttX

+′

=

+++=

+=+′+=

∫

∫ (5.3.28)


differentiating with respect to θ and with respect to t twice we obtain

,)( 543 ccD += θθ ℜ∈54 ,cc and .01 =c Substituting all the above information in

(5.3.28) we get

).,(),()()(),,()(

),(,)(

843454

22

63

12

0

φφθθφθ

φη

φ rFrFffXrFccrDX

DrcXcdttX

+′

=+++=

+=+= ∫ (5.3.29)


),()(),( 724 φφ DrDrF +−= where )(7 φD is the function of integration. Now

considering equation (5.3.12) and using equation (5.3.29) then solving after

273

differentiating the resulting equation first with respect to φ then with respect to ,θ

we get ,sincos)( 8767 φφφ cccD ++= .,, 876 ℜ∈ccc Substituting back the above

value in equation (5.3.8) we get ),()(),( 988 rDrDrF += φφ where )(8 rD and

)(9 rD are functions of integration. Now substituting all the above values in (5.3.29),

we get

( ) ).()(sincos)()(

,sincos

),(,)(

9878

3

98742

63

12

0

rDrDccffX

ccccX

DrcXcdttX

++−′

=

+++=

+=+= ∫

φφφθθ

φφθ

φη

(5.3.30)


differentiating with respect to θ and φ we get ,)( 108 crD = 11

9 )( crD = and

,)( 126 cD =φ .,, 121110 ℜ∈ccc Substituting all the above information in (5.3.30) we

get

( ) .sincos)()(

,sincos

,,)(

1110783

98742

1231

20

ccccffX

ccccX

crcXcdttX

++−′

=

+++=

+=+= ∫

φφφθθ

φφθ

η

(5.3.31)

Subtracting equation (5.3.10) from (5.3.12) and substituting equation (5.3.31) in the

resulting equation then solve after differentiating twice with respect to ,θ we get

.01094 === ccc Refreshing the system of equations (5.3.31) we get the conformal

vector fields in this case as


,,)(

11783

872

1231

20

cccffXccX

crcXcdttX

+−′

=+=

+=+= ∫φφ

θθφφ

η (5.3.31)*

274

Considering equations (5.3.10) and (5.3.7) respectively and using equation (5.3.31)*

we get )()( tUtB = and ,)()( )(3 dttU

c

etUtA ∫=

−

where ( ).)()( 2cdtttU += ∫η In this case

we have ).()( 22 tBtA α= Substituting back the values of A and B in the above

equation we conclude that 03 =c and .1=α The solution of equations (5.3.3) to

(5.3.12) is given as [40]


,),(

11783

872

1210

cccffXccX

cXtUX

+−′

=+=

==

φφθθφφ

(5.3.32)

The line element for Kantowski-Sachs and Bianchi type III space-times in this case

becomes

),)()(( 2222222 φθθ dfddrtAdtds +++−= (5.3.33)

with )()( tUtA = where ( ).)()( 2cdtttU += ∫η Proper conformal vector fields after


( ).0,0,0),(tUX = (5.3.34)

Case II:

In this case we have 02 =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛•••

BBB and .0),(1 ≠φrF Considering equation

(5.3.10) and using equation (5.3.14) then solve after differentiating with respect to ,φ

we get ),,(),(),,(0),,( 4333 φθφθφθθφ rFrFrKrK +=⇒= where ),(3 θrF and

),(4 φrF are functions of integration. Refreshing the system of equations (5.3.14) we

get

275

( )( )

).,,(1),()(

1

),,(),(1),(),()(

),,,(1),(),()(

),,(),()()(

42

13

432

212

22

211

210

φθφθ

φθθφφθ

φθθφφθ

θφφθη

θ

rKdtB

rFf

X

rFrFdtB

rFdrFfX

rKdtA

rFdrFfX

rFdrFfdttX

rr

+=

+++′=

++=

++=

∫

∫∫

∫∫

∫∫

(5.3.35)

Considering equation (5.375) and using equation (5.3.35) then differentiating first

with respect to φ then with respect to t twice we get ⇒=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛•••

02

AAA

,2∫=•

dtAA

A β where .0\ℜ∈β Substituting back the above value in equation

(5.3.7) and solving we get rDrDrF βφβφφ sin)(cos)(),( 211 += and

),,(),(),,( 652 θφθφθ rFFrK += where ),(1 φD ),(2 φD ),(5 φθF and ),(6 θrF are

functions of integration. Refreshing the system of equation (5.3.35) we get

[ ]( )[ ]( )

( )( )

( ) ).,,(1)(sin)(cos)(

1),,(),(

1),()(sin)(cos)(

),,(),(

1),()(sin)(cos)(

),,()(sin)(cos)()(

42

213

43

22212

65

22121

2210

φθφβφβθ

φθ

θφφβφφβθ

θφθ

θφφβφφββθ

θφφβφφβθη

θ

rKdtB

rDrDf

X

rFrF

dtB

rFdDrdDrfX

rFF

dtA

rFdDrdDrfX

rFdDrdDrfdttX

r

++=

++

++′=

++

+−=

+++=

∫

∫∫∫

∫∫∫

∫∫∫

(5.3.36)


differentiating first with respect to φ then with respect to ,t we get

,0)()( 21 == φφ DD which in turn implies .0),(1 =φrF Which gives a contradiction.

Hence this case is not possible.

276

Case III:

In this case we have 02 =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛•••

BBB and .0),(1 =φrF Considering equation

(5.3.10) and using equation (5.3.14) then solve after differentiating with respect to ,φ

we get ),,(),(),,( 433 φθφθ rFrFrK += where ),(3 θrF and ),(4 φrF are functions


( )

( ) ).,,(),,(),(1),(

),,,(1),(

),,()(

43432

22

22

21

20

φθφθθ

φθθ

θη

θ rKXrFrFdtB

rFX

rKdtA

rFX

rFdttX

r

=++=

+=

+=

∫

∫

∫ (5.3.37)

Considering equation (5.3.7), (5.3.10), (5.3.11) and using equations (5.3.37), then

solving in a similar fashion, we get ),,(),(),,( 652 φθθφθ FrFrK +=

),,(),()()(

),,( 874 φφθθ

φθ rFrFff

rK +′

−= ,cos)(sin)()(),( 3212 θβθβθ rDrDrDrF ++=

)()(),( 543 rDrDrF += θθ and ),,(),( 47 φφ φ rFrF −= where ),,(5 θrF ),,(6 φθF

),,(7 φrF ),,(8 θrF ),(1 rD ),(2 rD ),(3 rD )(4 rD and )(5 rD are functions of

integration. Refreshing the system of equation (5.3.37) we get

( )

( )

).,(),()()(

),,()()(

1sin)(cos)(

),,(),(

1cos)(sin)()(

,cos)(sin)()()(

873

454

2322

65

23211

3210

φφθθ

φθ

θβθββ

φθ

θβθβ

θβθβη

rFrFffX

rFrDrD

dtB

rDrDX

rFrF

dtA

rDrDrDX

rDrDrDdttX

rrr

+′

−=

+++

−=

++

++=

+++=

∫

∫

∫

(5.3.38)

277

Subtracting equation (5.3.10) from equation (5.3.12) and substituting equation

(5.3.38) in the resulting equation then solve after differentiating with respect to t and

θ we get .cos)(sin)()(),( 8764 φφφ rDrDrDrF ++= Substituting back the above

value and solving we get ),()(),( 1098 rDrDrF += φφ where ),(6 rD ),(7 rD ),(8 rD

)(9 rD and )(10 rD are functions of integration. Substituting back all the above

information in equation [(5.3.12)-(5.3.10)] and solving we get

.0)()()()()( 119432 ===== rDrDrDrDrD Refreshing equation (5.3.38) we get

( )

( ) ).(sin)(cos)()()(

,cos)(sin)(

),,(),(1)(),()(

10873

872

652

1110

rDrDrDffX

rDrDX

rFrFdtA

rDXrDdttX r

+−′

−=

+=

++=+= ∫∫

φφθθ

φφ

φθη

(5.3.39)


differentiating the resulting equation with respect to ,φ we get ,)( 17 crD =

.,,)( 2128 ℜ∈= cccrD Substituting back the above values and solving we get

),(),( 125 rDrF =θ where )(12 rD is a function of integration. Considering equation

(5.3.10) and using equation (5.3.39) then solve after differentiating with respect to r

and ,t we get ℜ∈= 331 ,)( ccrD and ),()(),( 14136 φφ DrDrF += where )(13 rD and

)(14 φD are functions of integration. Substituting all the above information in


( ) ).(sincos)()(,cossin

),()(,)(

1021

321

2

141513

0

rDccffXccX

DrDXcdttX

+−′

−=+=

+=+= ∫φφ

θθφφ

φη (5.3.40)

Considering equation (5.3.7), (5.3.9) respectively and using equation (5.3.40) then

solving we get ,)( 410 crD = 5

14 )( cD =φ and .,,,,)( 76547615 ℜ∈+= cccccrcrD

Refreshing equation (5.3.40) we get

278

( ) ,sincos)()(,cossin

,,)(

4213

212

861

30

cccffXccX

crcXcdttX

+−′

−=+=

+=+= ∫φφ

θθφφ

η (5.3.41)

where ).(,,,,, 758864321 ccccccccc +=ℜ∈ Considering equation (5.3.7) and using

equation (5.3.41) then solving we get ( ) .)()( 3

6

)(3

dtcdtt

c

ecdtttA∫ ∫+=

+−

∫η

η Also

substituting the above information in equation (5.3.10) and solving we get

.)()( 2cdtttB += ∫η The solution of equations (5.3.3) to (5.3.12) is given as [40]

( ) ,sincos)()(,cossin

,),(

4213

212

8610

cccffXccX

crcXtUX

+−′

−=+=

+==

φφθθφφ

(5.3.42)

where ).(,,,,, 758864321 ccccccccc +=ℜ∈ ( ).)()( 2cdtttU += ∫η and the line

element for Kantowski-Sachs and Bianchi type III space-times is given as in (5.3.1)

with the metric functions given as above. Proper conformal vector fields after


( ).0,0,),( 6rctUX = (5.3.43)

5.4. Summary of the Chapter In this chapter we explored proper conformal vector fields for non conformally flat

non static cylindrically symmetric, Kantowski-Sachs and Bianchi type III space-times

in general relativity theory. Following results are obtained from the above study:

(1) In non static cylindrically symmetric space-times: Different possibilities for the

existence of proper conformal vector fields have been found for non conformally flat

non static cylindrically symmetric space-times by using direct integration technique.

It has been shown that very special classes of these space-times admit proper

conformal vector fields. It is important to note that the classification of non static

cylindrically symmetric space-times also covered non static plane symmetric, static

279

cylindrically symmetric and static plane symmetric space-times. It turns out that the

above space-times admit 3, 4 or 5 conformal vector fields. In all the cases when the

above space-times admit conformal vector fields the number of proper conformal

vector field is one. When the above space-time admits three conformal vector fields,

results for proper conformal vector fields are given in equations (5.2.23),

(5.2.26),(5.2.59) and (5.2.61). When the above space-time admits four conformal

vector fields, results for proper conformal vector fields are given in equations

(5.2.32), (5.2.35) and (5.2.67). When the above space-time admits five conformal

vector fields, result for proper conformal vector fields is given in equation (5.2.70).

(2) In Kantowski-Sachs and Bianchi type III space-times: Proper conformal vector

fields are investigated in non conformally flat Kantowski-Sachs and Bianchi type III

space-times by using direct integration technique. It has been shown that special

classes of the above space-times admit proper conformal vector fields. It turns out

that the above space-times admit only five conformal vector fields. In all the cases

when the above space-times admit conformal vector fields the number of proper

conformal vector field is one. The results for proper conformal vector fields are given

in equations (5.3.24), (5.3.34) and (5.3.43).

280

Chapter 6

Self Similar Vector Fields in Kantowski-

Sachs, Bianchi Type III, Static Plane

Symmetric, Static Spherically Symmetric

and Static Cylindrically Symmetric Space-

Times

6.1. Introduction

This chapter is devoted to investigate self similar vector fields in Kantowski-Sachs,

Bianchi type III, static plane symmetric, static spherically symmetric and static

cylindrically symmetric space-times by using algebraic and direct integration method.

In this chapter we will discuss tilted and non tilted self similar vector fields admitted

by the above mentioned space-times. This chapter is organized as follows: In section

(6.2) self similar vector fields of Kantowski-Sachs and Bianchi type III space-times

are investigated. In the next section (6.3) self similar vector fields in static plane

symmetric space-times have been explored. In section (6.4) self similar vector fields

of static spherically symmetric space-times are explored. In section (6.5) self similar

vector fields of static cylindrically symmetric space-time are investigated. For the

above space-times we will study self similar vector fields of first, second, zeroth and

infinite kinds in tilted and non tilted cases. Last section (6.6) of the chapter is

dedicated to a detailed summary of the work.

281

6.2. Self Similar Vector Fields in Kantowski-Sachs

and Bianchi Type III Space-Times

Consider Kantowski-Sachs and Bianchi type III space-times in the spherical

coordinate system ),,,( φθrt (labeled by ),,,,( 3210 xxxx respectively) with the line

element [77]


where A and B are no where zero function of t only. For θθ sinh)( =f the space-



independent Killing vector fields [77, 79] given as

,r∂∂ ,

φ∂∂ ,sincos

φφ

θφ

∂∂′

−∂∂

ff ,cossin

φφ

θφ

∂∂′

+∂∂

ff (6.2.2)

where prime denotes the derivative with respect to .θ Here we shall take the four-

velocity vector as space like vector field u and define 1)( aa tAu δ= so that

.1=aauu The line element (6.2.1) becomes

).)()()(( 222222 φθθ dftBdtBdtduds ++−+= (6.2.3)

Clearly from equation (1.2.5) we get ⇒=αaa Xu ,

1 ,1 βα += uX where ., ℜ∈βα

Writing equation (1.2.6) explicitly using (6.2.3) we get

,0,0 δ=X (6.2.4)

,0,)(, 02

20 =− XtBX (6.2.5)

,0,)()(, 032

30 =− XftBX θ (6.2.6)

282

,2

2,20 δ=+

•

XXB

B (6.2.7)

,0)( 3,2

2,32 =+ XXf θ (6.2.8)

,)()(

23,

320 δθθ

=+′

+•

XXffX

BB (6.2.9)

where dot represents differentiation with respect to .t Solving equations (6.2.4),

(6.2.5) and (6.2.6) we get

,),(1),()(

1

,),(1),(),,(

312

3

21210

∫

∫

+=

+=+=

φθφθθ

φθφθφθδ

φ

θ

PdtB

Pf

X

PdtB

PXPtX (6.2.10)

where ),,(1 φθP ),(2 φθP and ),(3 φθP are functions of integration which are to be

determined. Considering equation (6.2.8) and using equation (6.2.10) we get

.0),(),()(1),(21),()()(2 23211 =+++

′− ∫∫ φθφθθφθφθ

θθ

φθθφφ PPfdtB

PdtB

Pff Differentiating

the above equation with respect to ,t and solving we get

),()()(),( 211 θφθφθ KKfP += where )(1 φK and )(2 θK are functions of


.),(1)()(

1

,),(1))()()((

),()()(

313

2212

210

∫

∫

+=

++′=

++=

φθφθ

φθθφθ

θφθδ

φ

θ

PdtB

Kf

X

PdtB

KKfX

KKftX

(6.2.11)

Subtracting equation (6.2.7) from equation (6.2.9), we get

.0)()(

3,32

2,2 =+

′+− XX

ffXθθ Now considering this equation and using equation

(6.2.11) we get

283

.0)()()()()(

)(1)(

)()]([)()( 2211

21 =

′+−+

′+ θ

θθθφ

θφ

θθφθ θθθφφ K

ffKK

fK

ffKf Solving the

above equation we get φφφ cossin)( 211 ccK += and ,)( 3

2 cK =θ .,, 321 ℜ∈ccc

Substituting the above information in (6.2.11), we get

( )

.),(1)sincos()(

1

,),(1)cossin()(

,)cossin()(

321

3

221

2

3210

∫

∫

+−=

++′=

+++=

φθφφθ

φθφφθ

φφθδ

PdtB

ccf

X

PdtB

ccfX

cccftX

(6.2.12)


respect to φ we get .0)sincos(12 21 =−⎟

⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛∫

•

φφ ccdtBB

B Here we have to discuss

three different possibilities:

(I) 012

≠⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛∫

•

dtBB

B and ,0)sincos( 21 =− φφ cc

(II) 012

=⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛∫

•

dtBB

B and ,0)sincos( 21 ≠− φφ cc

(III) 012

=⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛∫

•

dtBB

B and .0)sincos( 21 =− φφ cc

Case I:


≠⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛∫

•

dtBB

B and .0)sincos( 21 =− φφ cc Equation

⇒=− 0)sincos( 21 φφ cc .021 == cc Refreshing the system of equations (6.2.12)

we get

).,(),,(, 33223

0 φθφθδ PXPXctX ==+= (6.2.13)

284


.),()(2

23 δφθδ θ =++

•

PctB

B Solving this equation we get ),(),( 32 φφθ KP = where

)(3 φK is the function of integration and .)()( 23cttB += δ Substituting all the above


),,(),(, 33323

0 φθφδ PXKXctX ==+= (6.2.14)


⇒=+ 0),()()( 323 φθθφ θφ PfK ),()()()(),( 433 φφ

θθφθ φ KK

ffP +′

= where )(4 φK is a

function of integration. Now considering equation (6.2.9) and using equation (6.2.14),

we get ⇒=++′

0)())()(()()( 433 φφφ

θθ

φφφ KKKff φφφ cossin)( 54

3 ccK += and

,)( 62 cK =φ .,, 654 ℜ∈ccc Substituting all the above information in equation (6.2.14)

we get

.)sincos()()(,cossin, 654

354

23

0 cccffXccXctX +−′

=+=+= φφθθφφδ (6.2.15)

The line element for Kantowski-Sachs and Bianchi type III space-times in this case is

given as

).)(()( 22223

222 φθθδ dfdctdudtds ++++−= (6.2.16)

Self similar vector fields in this case are given as

.)sincos()()(,cossin

,,

6543

542

13

0

cccffXccX

uXctX

+−′

=+=

+=+=

φφθθφφ

βαδ (6.2.17)

In the following we will discuss the self similar vector fields of different kinds.

(a) First consider that ,δα = then the line element for Kantowski-Sachs and Bianchi

type III space-times become

285

).)(()( 22223

222 φθθα dfdctdudtds ++++−= (6.2.18)

Proper self similar vector field after subtracting Killing vector fields from (6.2.17)

can be written as ).0,0,,( 3 uctX αα += In this case the proper self similar vector

field is tilted and of first kind.

(b) Now taking 0=α and ,0≠δ the line element for Kantowski-Sachs and Bianchi

type III space-times become

).)(()( 22223


Proper self similar vector field after subtracting Killing vector fields become

).0,0,0,( 3ctX += δ In this case the proper self similar vector field is of zeroth kind

orthogonal to the space like vector .au

(c) Now taking 0≠α 0≠δ and ,δα ≠ the line element for Kantowski-Sachs and

Bianchi type III space-times become

).)(()( 22223


Proper self similar vector field after subtracting Killing vector fields become

).0,0,,( 3 uctX αδ += In this case the proper self similar vector field is of second

kind tilted to the space like vector .au

(d) Taking 0≠α and ,0=δ the line element for Kantowski-Sachs and Bianchi type

III space-times become

),)(( 222222 φθθγ dfddrdtds +++−= (6.2.23)

where .0\ℜ∈γ Proper self similar vector field in this case is ).0,0,,0( uX α= In

this case the proper self similar vector field is of infinite kind parallel to the space like

vector .au

286

Case II:


=⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛∫

•

dtBB

B and .0)sincos( 21 ≠− φφ cc Substituting

back the above information in (6.2.7) and solving we get ),(),( 32 φφθ KP =

,0321 ==== cccδ where )(3 φK is a function of integration. In this case 01 ≠c and

.02 ≠c Hence this case is not possible.

Case III:


=⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛∫

•

dtBB

B and .0)sincos( 21 =− φφ cc Equation

⇒=− 0)sincos( 21 φφ cc .021 == cc It is important to note that in this case

equation 012

=⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛∫

•

dtBB

B has no real solution. We therefore leave this case.

6.3. Self Similar Vector Fields in Static Plane


Consider the static plane symmetric space-times in the usual coordinate system

),,,( zyxt (labeled by ),,,,( 3210 xxxx respectively) with the line element [78]

),( 22)(22)(2 dzdyedxdteds xBxA +++−= (6.3.1)

where A and B are arbitrary functions of x only. The above space-times (6.3.1)

admit at least four independent Killing vector fields [19] which are ,t∂∂ ,

y∂∂

z∂∂

and .z

yy

z∂∂

−∂∂ To find self similar vector fields for the above space-time we shall

287

take the four-velocity vector as time like vector and define 02a

A

a eu δ= where

.1−=aauu The line element (6.3.1) will now become [48]

).( 22)(222 dzdyedxduds xB +++−= (6.3.2)

Clearly from equation (1.2.5), we get

,0 βα += uX (6.3.3)

where ., ℜ∈βα Writing equation (1.2.6) explicitly using (6.3.2) we get

,1,1 δ=X (6.3.4)

,02,1

1,2)( =+ XXe xB (6.3.5)

,03,1

1,3)( =+ XXe xB (6.3.6)

,2,2)( 221 δ=+′ XXxB (6.3.7)

,03,2

2,3 =+ XX (6.3.8)

.2,2)( 331 δ=+′ XXxB (6.3.9)

where ‘dash’ represents differentiation with respect to .x Solving equations (6.3.4),

(6.3.5) and (6.3.6) we get

),,(),(

),,(),(),,(3)(13

2)(1211

zyKdxezyKX

zyKdxezyKXzyKxXxB

z

xBy

+−=

+−=+=

∫∫

−

−δ (6.3.10)

where ),(and),(),,( 321 zyKzyKzyK are functions of integration which are to be

determined. Considering equation (6.3.7) and using equation (6.3.10) then

differentiating the resulting equation with respect to y and ,x respectively we get

.0),(2),()( )(11 =−′′ − xByyyy ezyKzyKxB There exist the following two possibilities:

(I) ,0),(1 =zyK y (II) .0),(1 ≠zyK y


288

Case I:

In this case we have ⇒= 0),(1 zyK y ),(),( 11 zFzyK = where )(1 zF is a function of

integration. Substituting back the above value in equation (6.3.7), we get

.2),(2)()()( 21 δδ =+′+′ zyKzFxBxBx y Differentiating this equation with respect to

,y we get ⇒= 0),(2 zyK yy ),()(),( 322 zFzyFzyK += where )(2 zF and )(3 zF are


.2)(2)()()( 21 δδ =+′+′ zFzFxBxBx Now differentiating this equation with respect

to z and ,x respectively we get .0)()( 1 =′′ zFxB z Here we have to discuss the

following three possibilities.

(a) 0)( ≠′′ xB and ,0)(1 =zFz (b) 0)( =′′ xB and ,0)(1 ≠zFz

(c) 0)( =′′ xB and .0)(1 =zFz


Case I(a):

In this case we have 0)( ≠′′ xB and .0)(1 =zFz Equation ⇒= 0)(1 zFz

.,)( 111 ℜ∈= ddzF Substituting back in equation (6.3.7) and differentiating with

respect to ,z we get .,)(0)( 2222 ℜ∈=⇒= ddzFzFz Substituting the above values

in equation (6.3.7) and solving we get ( ) .ln)()(2

1

2

δδ

δd

dxxB−

+= Substituting all the


).,(),(, 3332

21

1 zyKXzFydXdxX =+=+= δ (6.3.11)


.0)(),( 33 =+ zFzyK zy Differentiating this equation with respect to ,y we get

),()(),(0),( 5433 zFzyFzyKzyK yy +=⇒= where )(4 zF and )(5 zF are functions

289

of integration. Equation ⇒=+ 0)(),( 33 zFzyK zy .0)()( 34 =+ zFzF z Substituting

back the above information in equation (6.3.11) we get

).()(),(, 53332

21

1 zFzyFXzFydXdxX z +−=+=+= δ (6.3.12)


.2))()((2))(( 531 δδ =+−++′ zFzyFdxxB zzz Differentiating this equation with respect

to ,y we get ⇒= 0)(3 zFzz .,,)( 43433 ℜ∈+= dddzdzF Substituting back in the

above equation and differentiating with respect to ,z we get

.,,)(0)( 656555 ℜ∈+=⇒= dddzdzFzFzz Substituting back the above values in

equation (6.3.9) we get .25 dd = Substituting all the above information in equation

(6.3.12) we get

.,, 6233

4322

11 dzdydXdzdydXdxX ++−=++=+= δ (6.3.13)

Finally the line element for static plane symmetric space-times becomes [48]

( ) ).( 22)(2

1222 2

dzdydxdxdudsd

++++−=−δ

δδ (6.3.14)

Self similar vector fields for the above space-times after subtracting Killing vector

fields are given as

.0,, 321

10 ==+== XXdxXuX δα (6.3.15)


(i) First consider that ,δα = then the proper self similar vector fields for static plane

symmetric space-times can be written as

).0,0,,( 1dxuX += αα (6.3.16)

The line element in this case takes the form

( ) ).( 22)(2

1222 2

dzdydxdxdudsd

++++−=−δ

δδ (6.3.16)*

In this case the proper self similar vector field is tilted and of first kind.

290

(ii) Now taking 0=α and ,0≠δ then the proper self similar vector fields for static

plane symmetric space-times can be written as

).0,0,,0( 1dxX += δ (6.3.17)


( ) ).( 22)(2

1222 2

dzdydxdxdudsd

++++−=−δ

δδ (6.3.17)*

In this case the proper self similar vector field is of zeroth kind orthogonal to the time

like vector .au

(ii) Now taking 0≠α 0≠δ and ,δα ≠ then the proper self similar vector fields for

static plane symmetric space-times can be written as

).0,0,,( 1dxuX += δα (6.3.18)


( ) ).( 22)(2

1222 2

dzdydxdxdudsd

++++−=−δ

δδ (6.3.18)*

In this case the proper self similar vector field is of second kind tilted to the time like

vector .au

(iv) Taking 0≠α and ,0=δ then the proper self similar vector fields for static plane

symmetric space-times can be written as

)0,0,,( 1duX α= (6.3.19)

with line element

( ) ,22)(222 dzdyedxduds cax +++−= + (6.3.20)

where ,21

2

dd

a −= 01 ≠d and .,,, 21 ℜ∈cadd The proper self similar vector field is

tilted to the time like vector au and represents the self-similarity of infinite kind.

291

Case I(b):

In this case we have 0)(1 ≠zFz and .0)( =′′ xB Equation

.,)(0)( 11 ℜ∈=′⇒=′′ γγxBxB Substituting back in equation (6.3.7), we get

.2)(2)( 2111 δγγδ =++ zFzFx Differentiating this equation with respect to ,x we

get ⇒= 01γ .,)(0)( 22 ℜ∈=⇒=′ γγxBxB The line element for static plane

symmetric space-times after a suitable rescaling of y and z becomes

.22222 dzdydxduds +++−= (6.3.21)

The above space-time represents Minkowski space-time and we will leave this case.

Case I(c):

Ina this case we have 0)(1 =zFz and .0)( =′′ xB Equation

ℜ∈=⇒= 1111 ,)(0)( ddzFzFz and equation .,)(0)( 11 ℜ∈=′⇒=′′ γγxBxB

Substituting back the above values in equation (6.3.7) and differentiating with respect

to ,x we get ⇒= 01γ .,)(0)( 22 ℜ∈=⇒=′ γγxBxB Once again we reach to

Minkowski space-time.

Case II:

In this case we have .0),(1 ≠zyK y Therefore equation

⇒=−′′ − 0),(2),()( )(11 xByyyy ezyKzyKxB ⇒ℜ∈==′′ qq

zyKzyK

exBy

yyyxB ,),(),(

)(21

1

1)(

⇒=− 0),(),( 11 zyqKzyK yyyy ,cos)(sin)()(),( 3211 yqzFyqzFzFzyK ++=

where ),(1 zF )(2 zF and )(3 zF are functions of integration. Substituting back this

value in the above equation we get ⇒== 0)()( 32 zFzF .0),(1 =zyK y Which is a

contradiction. Hence this case is not possible.

292

6.4. Self Similar Vector Fields in Static Spherically


Consider static spherically symmetric space-times in the usual coordinate system

),,,( φθrt (labeled by ),,,,( 3210 xxxx respectively) with the line element [16, 78]

),sin( 22222)(2)(2 φθθ ddrdredteds rBrA +++−= (6.4.1)

where A and B are functions of r only. The above space-times (6.4.1) admit at least

four independent Killing vector fields [16] which are

,t∂∂ ,

φ∂∂ ,sincotcos

φφθ

θφ

∂∂

−∂∂ .coscotsin

φφθ

θφ

∂∂

+∂∂ (6.4.2)

We are taking the four-velocity vector as time like vector field u and define

,02)(

a

rA

a eu δ= so that .1−=aauu The line element (6.4.1) after a suitable rescaling

becomes

).sin( 22222)(22 φθθ ddrdreduds rB +++−= (6.4.3)

Clearly from equation (1.2.5) we get ,0 βα += uX ., ℜ∈βα Writing (1.2.6)

explicitly using (6.4.3) we get

,2)( 1,11 δ=+′ XXrB (6.4.4)

,0,, 21)(

122 =+ XeXr rB (6.4.5)

,0,sin, 1322

31)( =+ XrXe rB θ (6.4.6)

,21

2,21 δrrXX =+ (6.4.7)

,0sin 3,2

2,32 =+ XXθ (6.4.8)

293

,21cot 3,

321 δθ rXrXrX =++ (6.4.9)

where ‘dash’ denotes differentiation with respect to .r Solving equations (6.4.4),

(6.4.5) and (6.4.6) we get

,),(),(sin

1

,),(),(

),,(2

32

21

23

22

212

12221

∫

∫

∫

+=

+−=

+=−−

φθφθθ

φθφθ

φθδ

φ

θ

PdrrePX

PdrrePX

PedreeX

B

B

BBB

(6.4.10)

where ),,(1 φθP ),(2 φθP and ),(3 φθP are functions of integration which are to be


differentiating with respect to ,r we get ⇒=− 0),(cot),( 11 φθθφθ φθφ PP

),()(sin),( 211 θφθφθ KKP += where )(1 φK and )(2 θK are functions of


.),(),(sin

1

,),())()((cos

)),()((sin2

32

213

22

2212

212221

∫

∫

∫

+=

++−=

++=−−

φθφθθ

φθθφθ

θφθδ

φ

θ

PdrreKX

PdrreKKX

KKedreeX

B

B

BBB

(6.4.11)


respect to φ and ,r we get .0)( 2

221 =

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

′

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −

re

reK

BB

φφ In order to solve this equation

we need to discuss the following three possibilities:

294

(I) 0)(1 ≠φφK and ,02

22=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

′

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −

re

re

BB

(II) 0)(1 =φφK and ,02

22≠

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

′

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −

re

re

BB

(III) 0)(1 =φφK and .02

22=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

′

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −

re

re

BB

Case I:

In this case we have 0)(1 ≠φφK and .02

22=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

′

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −

re

re

BB

Equation

⇒=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

′

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −

02

22

re

re

BB

.0\,22

2)( ℜ∈

−= a

raae rB Substituting back the above

information in (6.4.7) and differentiating with respect to ,φ we get ⇒= 0),(2 φθθφP

),()(),( 432 φθφθ KKP += where )(3 θK and )(4 φK are functions of integration.

Now substituting the above values in (6.4.7) and differentiating with respect to θ and

,r we get ⇒=+ 0)()( 22 θθ θθθθ KK ,sincos)( 3212 θθθ cccK ++= .,, 321 ℜ∈ccc

Substituting back in equation (6.4.7) and solve after differentiating with respect to ,θ

we get ,)( 543 ccK += θθ ., 54 ℜ∈cc Finally, substituting all the above information

in equation (6.4.7) we get an equation

.2

1sin2 412

22122 δδ

=+−

+⎟⎠⎞

⎜⎝⎛− − cc

ara

rarra

r This equation will satisfy only if

.041 === ccδ Now substituting all the above information in (6.4.11), we get

295

( )

( )

).,(),(sin1

),(sincos)(cos1

,sincos)(sin

32

2213

4523

12

222

321

2

221

φθφθθ

φθθφθ

θθφθ

φ Pa

raKr

X

KcccKa

rar

X

ccKa

raX

+−

=

++−+−

=

++−

=

(6.4.12)

Considering equation (6.4.9) and using equation (6.4.12) and differentiating with

respect to ,r we get ⇒=+ 0)()( 11 φφφφ KK ℜ∈+= 76761 ,,sincos)( ccccK φφφ

and .03 =c Substituting back the above values in (6.4.9) and solving we get 05 =c

and ),()(cot),( 643 θφφθφθ KdKP +−= ∫ where )(6 θK is the function of


( )

( )

),()(cot)sincos(sin1

),(sin)sincos(cos1

,cos)sincos(sin

64672

223

42762

222

2762

221

θφφθφφθ

φθφφθ

θφφθ

KdKcca

rar

X

Kccca

rar

X

ccca

raX

+−−−

=

+−+−

=

++−

=

∫

(6.4.13)

where .,, 762 ℜ∈ccc Considering equation (6.4.8) and using equation (6.4.13) then

differentiating with respect to ,φ we get ⇒=+ 0)()( 44 φφφφ KK

,sincos)( 984 φφφ ccK += ., 98 ℜ∈cc Substituting back we get

.,)(0)( 101066 ℜ∈=⇒= ccKK θθθ Refreshing the system of equations (6.4.13) we

get

296

( )

( )

,)cossin(cot)sincos(sin1

,sincossin)sincos(cos1

,cos)sincos(sin

1098672

223

982762

222

2762

221

ccccca

rar

X

ccccca

rar

X

ccca

raX

+−−−−

=

++−+−

=

++−

=

φφθφφθ

φφθφφθ

θφφθ

(6.4.14)

where .,,,,, 1098762 ℜ∈cccccc Finally the line element for static spherically

symmetric space-times in this case takes the form

),sin( 2222222

222 φθθ ddrdr

raaduds ++−

+−= (6.4.15)

and the self similar vector fields for the above space-time (6.4.15) is given as [49]

( )

( )

,)cossin(cot)sincos(sin1

,sincossin)sincos(cos1

,cos)sincos(sin

,

1098672

223

982762

222

2762

221

0

ccccca

rar

X

ccccca

rar

X

ccca

raX

uX

+−−−−

=

++−+−

=

++−

=

+=

φφθφφθ

φφθφφθ

θφφθ

βα

(6.4.16)

where .,,,,, 1098762 ℜ∈cccccc Since in this case 0=δ so the above space-time

(6.4.15) admits only proper self similar vector field of infinite kind given as

),0,0,0,( uX α= which is parallel to the time like vector .au

Case II:

In this case we have 0)(1 =φφK and .02

22≠

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

′

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −

re

re

BB

Equation ⇒= 0)(1 φφK

.,)( 111 ℜ∈= ccK φ Substituting back these information in (6.4.7) and differentiating

with respect to ,φ we get ⇒= 0),(2 φθθφP ),()(),( 432 φθφθ KKP += where )(3 θK

297

and )(4 φK are functions of integration. Now substituting the above values in (6.4.7)

we get an equation

.0)()()(sin12

32

2222

2

22

12

2=+−+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛++

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛− ∫∫∫

−−−

θθθθδθθθ Kdr

reKKedr

reecdre

re

BB

BBB

B

In order to solve this equation we must take ⇒=∫−

122

drer

e BB

).0\( ℜ∈= ηηB

Substituting back this value in the above equation and solve after differentiating with

respect to r we get .,,sin)( 323212 ℜ∈++−= cccccK θθθ Also Substituting back

the above values in the above given equation we get

.,2

)( 4432

223 ℜ∈⎟⎟

⎠

⎞⎜⎜⎝

⎛++−=

−

cccceK θθθη

Substituting all these information in


( )

).,(),(2

,2

334432

22

2

22

3221

φθφθθ

θδ

ηη

η

PXKcccecr

eX

ccerX

=+⎟⎟⎠

⎞⎜⎜⎝

⎛++−=

++=

−

−

(6.4.17)

Considering equation (6.4.7) and using equation (6.4.17) we get .032 ==cc Also

considering equation (6.4.8) and using equation (6.4.17) with the above information

and solving we get ),()(cot),( 643 φφθφθ φ KKP += where )(6 φK is a function of


).()(cot),(,2

64344

221 φφθφδφ

η

KKXKceXrX +=+−==−

(6.4.18)


differentiating with respect to ,θ we get ,sincos)( 7654 φφφ cccK ++= 5

24 cec

η

=

298

and .,,,,)( 876586 ℜ∈= cccccK φ Now substituting all the above values in equation

(6.4.18) we get

( ) .sincoscot,sincos,2 867

376

21 cccXccXrX +−=+== φφθφφδ (6.4.19)

Thus the line element for static spherically symmetric space-time in this case

becomes

).sin( 2222222 φθθη ddrdreduds +++−= (6.4.20)

Self similar vector fields in this case becomes [49]

( ) .sincoscot,sincos

,2

,

8673

762

10

cccXccX

rXuX

+−=+=

=+=

φφθφφ

δβα (6.4.21)


(i) First consider that ,δα = then the proper self similar vector fields for static

spherically symmetric space-times after subtracting Killing vector fields from

(6.2.21) can be written as

).0,0,2

,( ruX αα= (6.4.22)



spherically symmetric space-times after subtracting Killing vector fields from


).0,0,2

,0( rX δ= (6.4.23)


like vector .au

299

(iii) Now taking 0≠α 0≠δ and ,δα ≠ then the proper self similar vector fields for

static spherically symmetric space-times after subtracting Killing vector fields from


).0,0,2

,( ruX δα= (6.4.24)


vector .au

(iv) Taking 0≠α and ,0=δ proper self similar vector fields for spherically

symmetric static space-times after subtracting Killing vector fields from (6.2.21) can

be written as

).0,0,0,( uX α= (6.4.25)

In this case the proper self similar vector field is of infinite kind parallel to the time

like vector .au

Case III:

In this case we have 0)(1 =φφK and .02

22=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

′

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −

re

re

BB

Equation

⇒=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

′

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −

02

22

re

re

BB

0\,22

2)( ℜ∈

−= a

raae rB and equation ⇒= 0)(1 φφK

.,)( 111 ℜ∈= ccK φ Substituting back the above information in (6.4.7) and

differentiating with respect to ,φ we get ⇒= 0),(2 φθθφP ),()(),( 432 φθφθ KKP +=

where )(3 θK and )(4 φK are functions of integration. Now substituting the above

values in (6.4.7) and differentiating with respect to θ and ,r we get

⇒=+ 0)()( 22 θθ θθθθ KK ,sincos)( 4322 θθθ cccK ++= .,, 432 ℜ∈ccc Substituting

300

back in equation (6.4.7) and solve after differentiating with respect to ,θ we get

,)( 653 ccK += θθ ., 65 ℜ∈cc Finally, substituting all the above information in

equation (6.4.7) we get an equation .2

1sin2 522

22122 δδ

=+−

+⎟⎠⎞

⎜⎝⎛− − cc

ara

rarra

r

This equation will satisfy only if .052 === ccδ Now substituting all the above

information in (6.4.11), we get

( )

( )

).,(

,)(sincoscos1

,sincossin

33

64

3412

222

4312

221

φθ

φθθθ

θθθ

PX

cKccca

rar

X

ccca

raX

=

++−+−

=

++−

=

(6.4.26)

Considering equation (6.4.9) and using equation (6.4.26) and differentiating with

respect to ,φ we get ⇒−= )(cot),( 43 φθφθ φφφ KP

),()()(cot),( 6543 θθφφφθφθ KKdKP ++−= ∫ where )(5 θK and )(6 θK are

functions of integration. Substituting back the above values in equation (6.4.9) we get

⇒=++−

+ 0sin)(cos)( 562

22

41 θθθ rKrca

racc .0)(,0, 5641 ==−= θKccc


).()(cot

),(sin1,cos

643

432

222

32

221

θφφθ

φθθ

KdKX

Kca

rar

Xca

raX

+−=

+−

−=−

=

∫ (6.4.27)


respect to ,φ we get ⇒=+ 0)()( 44 φφφφ KK ,sincos)( 984 φφφ ccK += ., 98 ℜ∈cc

Substituting back we get .,)(0)( 101066 ℜ∈=⇒= ccKK θθθ Refreshing the system

of equations (6.4.27) we get

301

.)sincos(cot

,sincossin1,cos

10893

9832

222

32

221

cccX

ccca

rar

Xca

raX

+−=

++−

−=−

=

θθθ

φφθθ (6.4.28)

where .,,, 10983 ℜ∈cccc Also considering equation (6.4.4) and using equation

(6.4.28) we get .0)1( 23 =− ac If we take 03 ≠c then we must take .12 =a Finally,

the line element for static spherically symmetric space-times in this case takes the

form

),sin(1

1 222222

22 φθθ ddrdrr

duds ++−

+−= (6.4.29)

and the self similar vector fields for the above space-time (6.4.29) is given as [49]

.)sincos(cot,sincossin11,cos1,

10893

98322

3210

cccXcccrr

X

crXuX

+−=++−−=

−=+=

θθθφφθ

θβα (6.4.30)

where .,,, 10983 ℜ∈cccc Since in this case 0=δ so the above space-time (6.4.29)

admits only proper self similar vector field of infinite kind given as

),0,0,0,( uX α= which is parallel to the time like vector .au

6.5. Self Similar Vector Fields in Static

Cylindrically Symmetric Space-Times




302

where ,A B and C are functions of r only. The space-time (6.5.1) admits minimum

three Killing vector fields [36] which are .,,φθ ∂∂

∂∂

∂∂t

It is important to note that the

above space-times (6.5.1) become static plane symmetric space-times when

).()( rCrB = We have already discussed self similar vector fields for static plane

symmetric space-times in section (6.3) so we will not discuss this case here in this

section. To find self similar vector fields for the above space-time we shall take the

four-velocity vector as time like vector field u and define 02a

A

a eu δ= so that

.1−=aauu The line element (6.5.1) will now become

,2)(2)(222 dzededrduds rCrB +++−= θ (6.5.2)

Clearly from equation (1.2.5), we get

,0 βα += uX (6.5.3)

where ., ℜ∈βα Writing equation (1.2.6) explicitly using (6.5.2) we get

,1,1 δ=X (6.5.4)

,02,1

1,2)( =+ XXe rB (6.5.5)

,03,1

1,3)( =+ XXe rC (6.5.6)

,2,2)( 221 δ=+• XXrB (6.5.7)

,03,2)(

2,3)( =+ XeXe rBrC (6.5.8)

,2,2)( 331 δ=+• XXrC (6.5.9)

where ‘dot’ represents differentiation with respect to .r Solving equations (6.5.4),

(6.5.5) and (6.5.6) we get

),,(),(

),,(),(),,(3)(13

2)(1211

zKdrezKX

zKdrezKXzKrXrC

z

rB

θθ

θθθδ θ

+−=

+−=+=

∫∫

−

−

(6.5.10)

303

where ),(and),(),,( 321 zKzKzK θθθ are functions of integration which are to be


differentiating the resulting equation with respect to θ and ,r respectively we get

.0),(2),()( )(11 =− −•• rBezKzKrB θθ θθθθ There exist the following two possibilities:

(I) ,0),(1 =zK θθ (II) .0),(1 ≠zK θθ


Case I:

In this case we have ⇒= 0),(1 zK θθ ),(),( 11 zFzK =θ where )(1 zF is a function of

integration. Substituting back the above value in equation (6.5.7), we get

.2),(2)()()( 21 δθδ θ =++ •• zKzFrBrrB Differentiating this equation with respect to

,θ we get ⇒= 0),(2 zK θθθ ),()(),( 322 zFzFzK += θθ where )(2 zF and )(3 zF

are functions of integration. Substituting back this value in the above equation we get

.2)(2)()()( 21 δδ =++ •• zFzFrBrrB Now differentiating this equation with respect

to z and ,r respectively we get .0)()( 1 =•• zFrB z Here we have to discuss the

following three possibilities.

(a) 0)( ≠•• rB and ,0)(1 =zFz (b) 0)( =•• rB and ,0)(1 ≠zFz

(c) 0)( =•• rB and .0)(1 =zFz


Case I(a):

In this case we have 0)( ≠•• rB and .0)(1 =zFz Equation ⇒= 0)(1 zFz

.,)( 111 ℜ∈= ddzF Substituting back in equation (6.5.7) and differentiating with

respect to ,z we get .,)(0)( 2222 ℜ∈=⇒= ddzFzFz Substituting the above values

in equation (6.5.7) and solving we get ( ) .ln)()(2

1

2

δδ

δd

drrB−



304

).,(),(, 3332

21

1 zKXzFdXdrX θθδ =+=+= (6.5.11)


.0)(),( 3)(3)( =+ zFezKe zrBrC θθ Differentiating this equation with respect to ,θ we get

),()(),(0),( 5433 zFzFzKzK +=⇒= θθθθθ where )(4 zF and )(5 zF are functions


).()(),(, 54332

21

1 zFzFXzFdXdrX +=+=+= θθδ (6.5.12)


.2))()((2))(( 541 δθδ =+++• zFzFdrrC zz Differentiating this equation with respect

to ,θ we get ⇒= 0)(4 zFz .,)( 334 ℜ∈= ddzF Substituting back in the above

equation and differentiating with respect to ,z we get

.,,)(0)( 545455 ℜ∈+=⇒= dddzdzFzFzz Substituting back the above values in

equation ,2))()((2))(( 541 δθδ =+++• zFzFdrrC zz we get ⇒=++• δδ 22))(( 41 ddrrC

( ) .ln)()(2

1

4

δδ

δd

drrC−

+= Substituting all the above information in equation (6.5.12)

we get

.),(, 54333

22

11 dzddXzFdXdrX ++=+=+= θθδ (6.5.13)


.0)()()( 3

)(2

13

)(2

1

42

=+++−−

ddrzFdrd

z

dδ

δδ

δ

δδ Solving this equation we get 03 =d

and .,)(0)( 6633 ℜ∈=⇒= ddzFzFz Refreshing the system of equations (6.5.13)

we get

,,,, 543

622

110 dzdXddXdrXuX +=+=+=+= θδβα (6.5.14)

where ).0(,,,,,,, 65421 ≠ℜ∈ δδβαddddd Finally the line element for static

cylindrically symmetric space-times becomes [80]

.)()( 2)1(2

12)1(2

1222

42

dzdrddrdrdudsddδδ δθδ

−−+++++−= (6.5.15)

305

Self similar vector fields for the above space-times after subtracting Killing vector

fields are given as

.0,, 321

10 ==+== XXdrXuX δα (6.5.16)


(i) First consider that ,0≠= δα then the proper self similar vector fields for static

cylindrically symmetric space-times can be written as

).0,0,,( 1druX += αα (6.5.17)


.)()( 2)1(2

12)1(2

1222

42

dzdrddrdrdudsddαα αθα

−−+++++−= (6.5.18)




).0,0,,0( 1drX += δ (6.5.19)


.)()( 2)1(2

12)1(2

1222

42


−−+++++−= (6.5.20)


like vector .au


static cylindrically symmetric space-times can be written as

).0,0,,( 1druX += δα (6.5.21)

The line element in this case takes the form [80]

.)()( 2)1(2

12)1(2

1222

42


−−+++++−= (6.5.22)

306


vector .au

(iv) Taking 0≠α and ,0=δ then the proper self similar vector fields for static


)0,0,,( 1duX α= (6.5.23)

with line element

,2)(2)(222 dzededrduds frcbar ++ +++−= θ (6.5.24)

where ,0,2 11

2 ≠−= ddda

1

42ddc −= and .,,,,,, 421 ℜ∈fcbaddd The proper self

similar vector field is tilted to the time like vector au and represents the self-

similarity of infinite kind.

Case I(b):

In this case we have 0)(1 ≠zFz and .0)( =•• rB Equation

.,,)(0)( 2121 ℜ∈+=⇒= ••• γγγγ rrBrB Substituting back in equation (6.5.7), we

get .2)(2)( 2111 δγγδ =++ zFzFr Differentiating this equation with respect to ,r

we get .01 =γ Substituting back in the above equation we get .)(2 δ=zF Refreshing

the system of equations (6.5.10), we get

).,()(),(),( 3)(133211 zKdrezFXzFXzFrX rCz θθδδ +−=+=+= ∫ − (6.5.25)


resulting equation with respect to ,θ we get ⇒= 0),(3 zK θθθ

),()(),( 543 zFzFzK += θθ where )(4 zF and )(5 zF are functions of integration.


).()()(

),(),(54)(13

3211

zFzFdrezFX

zFXzFrXrC

z ++−=

+=+=

∫ − θ

θδδ (6.5.26)

307

Now considering equation (6.5.9) and using equation (6.5.26) we get

.2))()()((2))()(( 54)(11 δθδ =++−++ ∫ −• zFzFdrezFzFrrC zzrC

zz Differentiating this

equation with respect to r and z respectively, we get

.0)(2)()( )(11 =− −•• rCzzzz ezFzFrC In this case we know that 0)(1 ≠zFz therefore we

have ⇒==•• qzFzFerC

z

zzzrC

)()()(

21

1

1)( ⇒=− 0)()( 11 zqFzF zzzz

.,,,cossin)( 3213211 ℜ∈++= dddzqdzqddzF Substituting back this value in

the above equation we get ⇒== 032 dd .0)(1 =zFz Which gives a contradiction.

Hence this case is not possible.

Case I(c):

In this case we have 0)(1 =zFz and .0)( =•• rB Equation

ℜ∈+=⇒= •••2121 ,,)(0)( γγγγ rrBrB and equation ⇒= 0)(1 zFz

.,)( 111 ℜ∈= ddzF Substituting back in equation (6.5.7), we get

.2)(2 2111 δγγδ =++ zFdr Differentiating this equation with respect to ,r we get

.01 =γ Substituting back in the above equation we get .)(2 δ=zF Refreshing the

system of equations (6.5.10), we get

).,(),(, 33321

1 zKXzFXdrX θθδδ =+=+= (6.5.27)


resulting equation with respect to ,θ we get ⇒= 0),(3 zK θθθ

),()(),( 543 zFzFzK += θθ where )(4 zF and )(5 zF are functions of integration.


).()(),(, 543321

1 zFzFXzFXdrX +=+=+= θθδδ (6.5.28)

Now considering equation (6.5.9) and using equation (6.5.28) we get

.2))()((2))(( 541 δθδ =+++• zFzFdrrC zz Differentiating this equation with respect

308

to θ we get .,)(0)( 3344 ℜ∈=⇒= ddzFzFz Substituting this value in the above

equation and differentiating the resulting equation with respect to ,z we get

.,,)(0)( 545455 ℜ∈+=⇒= dddzdzFzFzz Substituting back this value in (6.5.9)

we get ⇒=++• δδ 22))(( 41 ddrrC ( ) .ln)()(2

1

4

δδ

δd

drrC−



.),(, 543332

11 dzddXzFXdrX ++=+=+= θδθδ (6.5.29)


.0)( 3)(32 =+ dezFe rC

zγ Differentiating this equation with respect to ,r we get

.03 =d Substituting back we get .,)(0)( 6633 ℜ∈=⇒= ddzFzFz Finally, the

solution of equations (6.5.4) to (6.5.9) becomes

.,, 543

62

11 dzdXdXdrX +=+=+= δθδ (6.5.30)

The line element for static cylindrically symmetric space-times in this case after a

suitable rescaling of θ takes the form [80]

,)( 2)1(2

12222

4

dzdrddrdudsdδδθ

−++++−= (6.5.31)

where ).0(,,,, 5461 ≠ℜ∈ δδ dddd Proper self similar vector fields for the above

space-times after subtracting Killing vector fields are given as

.0,, 321

10 ==+== XXdrXuX δα (6.5.32)


(i) First consider that ,0≠= δα then the proper self similar vector fields for static


).0,0,,( 1druX += αα (6.5.33)


309

.)( 2)1(2

12222

4

dzdrddrdudsdααθ

−++++−= (6.5.34)




).0,0,,0( 1drX += δ (6.5.35)


.)( 2)1(2

12222

4

dzdrddrdudsdδδθ

−++++−= (6.5.36)


like vector .au


static cylindrically symmetric space-times can be written as

).0,0,,( 1druX += δα (6.5.37)


.)( 2)1(2

12222

4

dzdrddrdudsdδδθ

−++++−= (6.5.38)


vector .au

(iv) Taking 0≠α and ,0=δ then the proper self similar vector fields for static


)0,0,,( 1duX α= (6.5.39)

with line element

,2)(2222 dzeddrduds frc ++++−= θ (6.5.40)

310

where ,21

4

ddc −= ,01 ≠d and .,,, 41 ℜ∈fcdd The proper self similar vector field is

tilted to the time like vector au and represents the self-similarity of infinite kind.

Case II:

In this case we have .0),(1 ≠zK θθ Equation ⇒=− −•• 0),(2),()( )(11 rBezKzKrB θθ θθθθ

⇒−==•• qzKzK

erB rB

),(),(

)(21

1

1)(

θθ

θ

θθθ ⇒=+ 0),(),( 11 zqKzK θθ θθθθ

.cos)(sin)()(),( 3211 θθθ qzFqzFzFzK ++= Substituting back this value in the

above equation we get .0)()( 32 == zFzF Therefore, ⇒= )(),( 11 zFzK θ

.0),(1 =zK θθ Which is a contradiction. Hence this case is not possible.


In this chapter we explored self similar vector fields for Kantowski-Sachs, Bianchi

type III, static plane symmetric, static spherically symmetric and static cylindrically

symmetric space-times by using algebraic and direct integration techniques.

Following results are obtained from the above study:

(1) In Kantowski-Sachs and Bianchi type III space-times: Different possibilities for

the existence of proper self similar vector fields have been found for the above space-

times by using direct integration technique. It turns out that the above space-times

admit tilted and non tilted proper self similar vector fields. It turns out that the above

space-times admit tilted proper self similar vector field of first kind and second kind.

The above space-time also admit non tilted orthogonal proper self similar vector

fields of zeroth kind and non tilted parallel proper self similar vector fields of infinite

kind. This space-times admit the above kinds of self similar vector fields for a special

choice of the metric functions.

311

(2) In static plane symmetric space-times: Proper self similar vector fields in both

tilted and non tilted cases have been explored for the above space-times by using

direct integration technique. It turns out that the above space-times admit tilted and

non tilted proper self similar vector fields. It turns out that the above space-times

admit tilted proper self similar vector field of first kind, second kind and infinite kind.

The above space-time also admits non tilted orthogonal proper self similar vector

fields of zeroth kind. This space-time admits the above kinds of self similar vector

fields for a special choice of the metric functions.

(3) In static spherically symmetric space-times: Proper self similar vector fields of

tilted and non tilted kinds have been found for the above space-times by using direct

integration technique. It turns out that the above space-times admit tilted and non

tilted proper self similar vector fields. The above space-times admit tilted proper self

similar vector field of first kind and second kind. The above space-time also admit

non tilted orthogonal proper self similar vector fields of zeroth kind and non tilted

parallel proper self similar vector fields of infinite kind. This space-time admits the

above kinds of self similar vector fields for a special choice of the metric functions.

(4) In static cylindrically symmetric space-times: Proper self similar vector fields of

tilted and non tilted kinds have been found for static cylindrically symmetric space-

times. The above space-time admits tilted proper self similar vector fields of first,

second and infinite kind. This space-time also admits non tilted zeroth kind proper

self similar vector field orthogonal to the time like vector field .au The above space-

time admits the above kinds of self similar vector fields for a special choice of metric

functions.

312

References

1. A. Qadir, Relativity: An introduction to the special theory, World scientific

(1989).

2. T. Sauer, Field equations in teleparallel space-time: Einstein’s

Fernparallelismus approach towards unified field theory, Historia

Mathematica, 33 (2006) 399.

3. R. Aldrovendri and J. G. Pereira, An introduction to teleparallel gravity

(lecture notes) (2005).

4. G. S. Hall, Symmetries and curvature structure in general relativity, World

scientific (2004).

5. R. M. Wald, General Relativity, University of Chicago Press (1984).

6. G. S. Hall, Physical and geometrical classification in general relativity, CBPF-

MO-001/93, (1993).

7. R. Aldrovendri and J. G. Pereira, An introduction to gravitation theory

(lecture notes) (2001).

8. M. Tsamparlis and P. S. Apostolopoulos, Symmetries of Bianchi-I space-

times, Journal of Mathematical Physics, 41 (2000) 7573.

9. H. Maeda and T. Harada, H. Iguchi and N. Okuyama, A classification of

spherically symmetric kinematic self-similar perfect fluid solutions, Progress

of Theoretical Physics, 108 (2002) 819.

10. J. G. Pereira, T. Vargas and C. M. Zhang, Axial-vector torsion and the

teleparallel Kerr space-time, Classical and Quantum Gravity, 18 (2001) 833.

11. Y. N. Obukhov and T. Vargas, Gödel type solution in teleparallel gravity,

Physics letters A, 327 (2004) 365.

12. V. C. de Andrade, L. C. T. Guillen and J. G. Pereira, Teleparallel gravity: an

overview, arxiv: gr-qc/0011087v1 (2000).

13. M. Sharif and M. J. Amir, Teleparallel Killing vectors of the Einstein

universe, Modern Physics Letters A, 23 (2008) 963.

14. A. Z. Petrov, Einstein spaces, Pergamon, Oxford University Press, (1969).

313

15. T. Harada and H. Maeda, Stability criterion for self similar solutions with a

scalar field and those with a stiff fluid in general relativity, Classical and

Quantum Gravity, 21 (2004) 371.

16. A. H. Bokhari and A. Qadir, Symmetries of static spherically symmetric

space-times, Journal of Mathematical Physics, 28 (1987) 1019.

17. A. Qadir and M. Ziad, Classification of static cylindrically symmetric space-

times, Nuovo Cimento B, 110 (1995) 277.

18. A. Qadir and M. Ziad, The classification of spherically symmetric space-

times, Nuovo Cimento B, 110 (1995) 317.

19. T. Feroze, A. Qadir and M. Ziad, The classification of plane symmetric space-

times by isometries, Journal of Mathematical Physics, 42 (2001) 4947.

20. L. P. Eisenhart, Continuous groups of transformation, Princeton University

Press (1933).

21. G. S. Hall and J. D. Steele, Homothety groups in space-time, General

Relativity and Gravitation, 22 (1990) 457.

22. D. Ahmad and M. Ziad, Homothetic motions of spherically symmetric space-

times, Journal of Mathematical Physics, 38 (1997) 2547.

23. M. Ziad, The classification of static plane symmetric space-time, Nuovo

Cimento B, 114 (1999) 683.

24. A. Qadir, M. Sharif and M. Ziad, Homotheties of cylindrically symmetric

static manifolds and their global extension, Classical and Quantum Gravity,

17 (2000) 345.

25. G. Shabbir and K. B. Amur, Proper homothetic vector fields in Bianchi type I

space-time, Applied Sciences, 8 (2006) 153.

26. G. Shabbir and M. Ramzan, Classification of cylindrically symmetric static

space-times according to their proper homothetic vector fields, Applied

Sciences, 9 (2007) 148.

314

27. M. Z. Bashir and S. Ehsan, Classification of plane symmetric Loretnzian

manifolds according to their homotheties and metrics, Nuovo Cimento B, 123

(2008) 71.

28. R. F. Bilyalov, Conformal transformation groups in gravitational field, Soviet

Physics Doklady, 8 (1964) 878.

29. L. Defrise-Carter, Conformal groups and conformally equivalent isometry

groups, Communication in Mathematical Physics, 40 (1975) 273.

30. G. S. Hall, Conformal symmetries and fixed points in space-time, Journal of

Mathematical Physics, 31 (1990) 1198.

31. G. S. Hall and J. D. Steele, Conformal vector fields in general relativity,

Journal of Mathematical Physics, 32 (1991) 1847.

32. D. Kramer and J. Carot, Conformal symmetry of perfect fluids in general

relativity, Journal of Mathematical Physics, 32 (1991) 1857.

33. R. Maartens, S. D. Maharaj and B. O. J. Tupper, General solution and

classification of conformal motions in static spherical space-times, Classical

and Quantum Gravity, 12 (1995) 2577.

34. G. S. Hall, Conformal vector fields and conformal type collineations in space-

times, General Relativity and Gravitation, 32 (2000) 933.

35. G. Shabbir and S. Iqbal, A note on proper conformal vector fields in Bianchi

type I space-times, Applied Sciences, 10 (2008) 231.

36. G. Shabbir and S. Iqbal, A note on proper conformal vector fields in

cylindrically symmetric static space-times, University of Politehnica

Bucharest Scientific Bulletin Series A, Applied Mathematics and Physics, 70

(2008) 23-28.

37. G. Shabbir, M. Ramzan and A. Ali, Classification of non-static spherically

symmetric space-times according to their proper conformal vector fields,

University of Politehnica Bucharest Scientific Bulletin Series A, Applied

Mathematics and Physics, 71 (2009) 3.

315

38. G. Shabbir and A. Ali, A note on proper conformal symmetry in Bianchi types

VIII and IX space-times, Advanced Studies in Theoretical Physics, 3 (2009)

93.

39. G. Shabbir and A. Ali, Classification of spatially homogeneous rotating space-

times according to their conformal vector fields, Applied Mathematical

Sciences, 3 (2009) 869.

40. G. Shabbir and S. Khan, Classification of Kantowski-Sachs and Bianchi type

III space-times according to their proper conformal vector fields (submitted).

41. G. Shabbir and S. Khan, Proper conformal vector fields in Bianchi type III

space-times, Proceedings of the 3rd International Conference on Mathematical

Sciences, United Arab Emirates University, Al-Ain UAE (2008) 521.

42. G. Shabbir and S. Khan, Classification of non conformally flat non static

cylindrically symmetric space-times according to their proper conformal

vector fields (submitted).

43. A. A. Coley, Kinematic self similarity, Classical and Quantum Gravity, 14

(1997) 87.

44. H. Maeda and T. Harada, H. Iguchi and N. Okuyama, A classification of

spherically symmetric kinematic self-similar perfect fluid solutions, Progress

of Theoretical Physics, 108 (2002) 819.

45. M. Sharif and S. Aziz, On the physical properties of spherically symmetric

self similar solutions, International Journal of Modern Physics D 14 (2005)

73.

46. M. Sharif and S. Aziz, On physical properties of cylindrically symmetric self

similar solutions, International Journal of Modern Physics A, 20 (2005) 7579.

47. G. Shabbir and S. Khan, Self similar solutions of Bianchi type III space-times

using partial differential equations, Journal of the association of Arab

universities for basic and applied sciences, 7 (2009) 76.

48. G. Shabbir and S. Khan, A note on self similar vector fields in plane

symmetric static space-times, TWMS Journal of Pure and Applied

316

Mathematics, 1 (2010) 252.

49. G. Shabbir and S. Khan, A note on self similar vector fields in static

spherically symmetric space-times (submitted).

50. F. I. Mikhail, M. I. Wanas, A. Hindawi and E. I. Lashin, Energy momentum

complex in Moller’s tetrad theory of gravitation, International Journal of

Theoretical Physics, 32 (1993) 1627.

51. G. G. L. Nashed, General spherically symmetric non singular black hole

solutions in a teleparallel theory of gravitation, Physical Review D, 66 (2002)

064015.

52. T. Vargas, The energy of the universe in teleparallel gravity, General

Relativity and Gravitation, 36 (2004) 1255.

53. M. Salti and A. Havare, Energy-momentum in viscous Kasner type universe

in Bergmann-Thomson formulations, International Journal of Modern Physics

A, 20 (2005) 2169.

54. O. Aydogdu, M. Salti and M. Korunur, Energy in Reboucas-Tiomno-Korotki-

obukhov and Godel type space-times in Bergmann-Thomson’s formulations,

Acta Physica Slovaca, 55 (2005) 537.

55. N. Rosen, The energy of the universe, General Relativity and Gravitation, 26

(1994) 319.

56. F. W. Hehl and A. Macias, Metric affine gauge theory of gravity II: Exact

solutions, International Journal of Modern Physics D, 8 (1999) 399.

57. Y. N. Obukhov, E. J. Vlachynsky, W. Esser, R. Tresguerres and F. W. Hehl,

An exact solution of the metric affine guage theory with dilation, shear and

spin charges, Physics Letters A, 220 (1996) 01.

58. P. Baekler, M. Gurses, F. W. Hehl and J. D. McCrea, The exterior

gravitational field if a charged spinning source in the poincare guage theory:

A Kerr-Newmann metric with dynamic torsion, Physics Letters A, 128 (1988)

245.

317

59. E. J. Vlachynsky, R. Tresguerres, Y. N. Obukhov and F. W. Hehl, An axially

symmetric solution of metric affine gravity, Classical Quantum Gravity, 13

(1996) 3253.

60. J. K. Ho, De C. Chern and J. M. Nester, Some spherically symmetric exact

solutions of the metric affine gravity theory, Chinese Journal of Physics, 35

(1997) 640.

61. T. Kawai and N. Toma, A charged Kerr metric solution in new general

relativity, Progress of Theoretical Physics, 87 (1992) 583.

62. M. Sharif and M. J. Amir, Teleparallel version of the stationary axisymmetric

solutions and their energy contents, General Relativity and Gravitation, 39

(2007) 989.

63. M. Sharif and M. J. Amir, Teleparallel versions of Friedman and Lewis-

Papapetrou space-times, General Relativity and Gravitation, 38 (2006) 1735.

64. M. Sharif and B. Majeed, Teleparallel Killing vectors of spherically

symmetric space-times, Communication in Theoretical Physics, 52 (2009)

435.

65. G. Shabbir and S. Khan, A note on classification of Bianchi type I space-times

according to their teleparallel Killing vector fields, Modern Physics Letters A,

25 (2010) 55.

66. G. Shabbir and S. Khan, A note on Killing vector fields of Bianchi type II

space-times in teleparallel theory of gravitation, Modern Physics Letters A, 25

(2010) 1733.

67. G. Shabbir, A. Ali and S. Khan, A note on teleparallel Killing vector fields in

Bianchi type VIII and IX space-times in teleparallel theory of gravitation,

Chinese Physics B, 20 (2011) 070401.

68. G. Shabbir and S. Khan, Classification of Kantowski-Sachs and Bianchi type

III space-times according to their Killing vector fields in teleparallel theory of

gravitation, Communication in Theoretical Physics, 54 (2010) 469.

318

69. G. Shabbir and S. Khan, Classification of cylindrically symmetric static

space-times according to their Killing vector fields in teleparallel theory of

gravitation, Modern Physics Letters A, 25 (2010) 525.

70. G. Shabbir, S. Khan and A. Ali, A note on classification of spatially

homogeneous rotating space-times according to their teleparallel Killing

vector fields in teleparallel theory of gravitation, Communication in

Theoretical Physics, 55 (2011) 268.

71. G. Shabbir, S. Khan and M. J. Amir, A note on classification of cylindrically

symmetric non static space-times according to their Killing vector fields in

teleparallel theory of gravitation, Brazilian Journal of Physics, 41 (2011) 184.

72. G. Shabbir and S. Khan, Classification of Bianchi type I space-times

according to their proper teleparallel homothetic vector fields in the

teleparallel theory of gravitation, Modern Physics Letters A, 25 (2010) 2145.

73. G. Shabbir and S. Khan, Classification of teleparallel homothetic vector fields

in cylindrically symmetric static space-times in teleparallel theory of

gravitation, Communications in Theoretical Physics, 54 (2010) 675.

74. G. Shabbir and S. Khan, Teleparallel proper homothetic vector fields in non

static plane symmetric space-times in the teleparallel theory of gravitation

(submitted).

75. L. L. So and T. Vargas, The energy of Bianchi type I and II Universes in

teleparallel gravity, Chinese Journal of Physics, 43 (2005) 901.

76. U. Camci and E. Sahin, Matter collineation classification of Bianchi type II

space-time, General Relativity and Gravitation, 38 (2006) 1331.

77. R. Kantowski and R. K. Sachs, Some spatially homogeneous anisotropic

relativistic cosmological models, Journal of Mathematical Physics, 7 (1966)

443.

78. H. Stephani, D. Kramer, M. Maccallum, C. Hoenselaers and E. Herlt, Exact

Solutions of Einstein’s Field Equations (Second Edition), Cambridge

University Press (2003).

319

79. G. Shabbir and A. B. Mehmood, Proper curvature collineations in Kantowski-

Sachs and Bianchi type III space-times, Modern Physics Letters A, 22 (2007)

807.

80. G. Shabbir and S. Khan, Proper self similar vector fields in static cylindrically

symmetric space-times (submitted).

some important lie symmetries in both general relativity

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