lotsofcalculationsp.1326

Lots of calculations: Relativity Demystified (David McMahon, McGraw-Hill, 2006)

Susan Larsen Wednesday, June 27, 2012

http://physicssusan.mono.net/9035/General%20Relativity Page 1

Contents Chapter 2: Vectors, One Forms, and the metric ................................................................................................ 4

P.36: The signature of a metric. ................................................................................................................ 4

P. 46, Quiz 2-5: Spherical coordinates and vector transformation ........................................................... 4

Chapter 4: Tensor Calculus ................................................................................................................................ 5

P. 74, Example 4-4: Find the Christoffel symbols of the 2-sphere with radius ...................................... 5 P. 75, Example 4-5, quiz 4-7: Find the Christoffel symbols of the Kahn-Penrose metric .......................... 5

P. 78, Example 4-7: Alternative solution: Show that ............................................................ 6 New - P.80: One-forms: why ........................................................................................................ 6 P. 83, Example 4-9: Find the geodesic equations for cylindrical coordinates ........................................... 6

P. 84, Example 4-10: Use the geodesic equations to find the Christoffel symbols for the Rindler metric.

................................................................................................................................................................... 8

P. 86: The Riemann tensor eq. (4.42), (4.43) and (4.44) ........................................................................... 8

P. 87: Independent elements in the Riemann, Ricci and Weyl tensor .................................................... 10

P.87, Example 4-11: Compute the components of the Riemann tensor for the unit 2-sphere .............. 11

P.89, Example 4-12: Show that the Ricci scalar for the unit 2-sphere ......................................... 11 New - p.90: Proof: if a space is conformally flat, i.e. the Weyl tensor vanishes ......... 12 P.91 Quiz 4-2, Quiz 4-3: Calculate the Christoffel symbols for the spherical metric............................... 16

P.91 Quiz 4-4: The Riemann tensor of the spherical metric .................................................................... 16

P.91 Quiz 4-6: A Lie derivative in the spherical metric ............................................................................ 17

P.92, Quiz 4-8: The Ricci scalar of the Penrose Kahn metric ................................................................... 18

P.92, Quiz 4-9: The Christoffel symbols of a metric example .................................................................. 18

P.92, Quiz 4-10: The Ricci scalar of a metric example ............................................................................. 19

Chapter 5: Cartans Structure Equations ......................................................................................................... 19

P. 106, Example 5-1: Ricci rotation coefficients for the Tolman-Bondi- de Sitter metric ....................... 19

P.113: The curvature two forms and the Riemann tensor ...................................................................... 22

P.113, Example 5-2: Find the Ricci scalar using Cartans structure equations of the 2-sphere .............. 23

P.116, Example 5-3: Find the components of the Riemann tensor of the Robertson-Walker metric using

Cartans structure equations ................................................................................................................... 23

P. 120, Quiz 5-1, and 5-2: Ricci rotation coefficients of the spherical polar metric ................................ 26

P. 120, Quiz 5-3: Transformation of the Ricci rotation coefficients into the Christoffel symbols of the spherical polar metric ...................................................................................................... 27




P. 120, Quiz 5-4: Ricci rotation coefficients of the Rindler metric .......................................................... 27

P. 121, Quiz 5-7: The Einstein tensor for the Tolman-Bondi- de Sitter metric ........................................ 28

Chapter 6: The Einstein Field Equations .......................................................................................................... 32

P.138: The vacuum Einstein equations.................................................................................................... 32

P.138: The vacuum Einstein equations with a cosmological constant .................................................... 32

New - P.138: General remarks on the Einstein equations with a cosmological constant ....................... 33

P.139, example 6-2: Find the components of the curvature tensor for the metric in 2+1 dimensions

using Cartans structure equations .......................................................................................................... 34

P.139, Example 6-2: Find the components of the curvature tensor for the metric in 2+1 dimensions

using Cartans structure equations alternative solution ...................................................................... 36

P.147, Example 6-3: Find the components of the Einstein tensor in the coordinate basis for the metric

in 2+1 dimensions. ................................................................................................................................... 37

P.150, Example 6-4: The Einstein equations of the metric in 2+1 dimensions. ...................................... 39

P. 152, Quiz 6-1: Using the contracted Bianchi identities, prove that: .............................. 40 P.153-54, Quiz 6-5, Quiz 6-6, Quiz 6-7 and Quiz 6-8: Ricci rotation coefficients, Ricci scalar and Einstein

equations for a general 4-dimensional metric. ....................................................................................... 40

Chapter 7: The Energy-Momentum Tensor ..................................................................................................... 45

P.160: Perfect Fluids Alternative derivation ......................................................................................... 45

P.161, Example 7-2: The Einstein tensor and Friedmann-equations for the Robertson Walker metric . 46

P.161, Example 7-2: The Einstein tensor for the Robertson Walker metric Alternative version. ........ 48

P. 165, Quiz 7-3: Manipulating the Friedmann equations. ..................................................................... 49

Chapter 8: Killing Vectors ................................................................................................................................ 49

P.168, Example 8-1: Show that if the Lie derivative of the metric tensor with respect to vector X

vanishes ( ), the vector X satisfies the Killing equation. - Alternative version .................... 49 P.177, equation (8.7): Prove that ........................................................................... 50 P.178: Constructing a Conserved Current with Killing Vectors Alternative version: ............................ 50

P.179, Quiz 8-3: Given a Killing vector the Ricci scalar satisfies : ...................................... 50 Chapter 9: Null Tetrads and the Petrov Classification ..................................................................................... 50

P.186, Example 9-3, and 9-4: Construct a null tetrad for the flat space Minkowski metric ................... 50

P.195, Example 9-5: The Brinkmann metric ............................................................................................ 52

Chapter 10: The Schwarzschild Solution ......................................................................................................... 62

P.204: The Riemann and Ricci tensor of the general Schwarzschild metric ............................................ 62

P.215: The Riemann tensor of the Schwarzschild metric ........................................................................ 65

P.216: Calculation of the scalar in the Schwarzschild metric ......................................... 65




P.216: Geodesics in the Schwarzschild Spacetime .................................................................................. 66

P.218: The meaning of the integration constant: The choice of ...................................................... 67 P.229: Time Delay .................................................................................................................................... 68

P. 230, Quiz 10-1: Use the geodesic equations to find the Christoffel symbols for the general

Schwarzschild metric. .............................................................................................................................. 69

P.231, Quiz 10-2: The Ricci tensor for the general time dependent Schwarzschild metric. ................... 71

P.231-32, Quiz 10-3, and 10-4: The Ricci rotation coefficients and Ricci tensor for the Schwarzschild

metric with nonzero cosmological constant. .......................................................................................... 75

P.232, Quiz 10-5: The Petrov type of the Schwarzschild spacetime ....................................................... 77

Chapter 11: Black Holes ................................................................................................................................... 82

P.238: The Path of a Radially Infalling Particle ........................................................................................ 82

P. 242: The Schwarzschild metric in Kruskal Coordinates. ...................................................................... 84

P.246: The inverse metric of the Kerr Spinning Black Hole ..................................................................... 86

Chapter 12: Cosmology ................................................................................................................................... 88

P. 262: Spaces of Positive, Negative, and Zero Curvature ....................................................................... 88

P.277, Quiz 12-1: The general Schwarzschild metric in vacuum with a cosmological constant: The Ricci

scalar ........................................................................................................................................................ 89

P.277, Quiz 12-2, Quiz 12-3: The general Schwarzschild metric in vacuum with a cosmological constant:

Integration constants .............................................................................................................................. 90

P.277, Quiz 12-4: The general Schwarzschild metric in vacuum with a cosmological constant: The

spatial part of the line element. .............................................................................................................. 91

P.278, Quiz 12-6: Parameters in an flat universe with positive cosmological constant: Starting with use a change of variables .................................................................. 91 Chapter 13: Gravitational Waves .................................................................................................................... 92

P.286: Gauge transformation - The Einstein Gauge ................................................................................ 92

P.288 l.13: The Riemann tensor of a plane wave .................................................................................... 94

P. 290 l.12: The line element of a plane wave in the Einstein gauge ...................................................... 98

P. 291: The line element of a plane wave ............................................................................................... 99

P.298: The Rosen line element ................................................................................................................ 99

P.304: Colliding gravity waves - coordinate transformation ................................................................. 103

P.304: The delta and heavy-side functions: prove that ................................. 104 P.305: Example 13-1: Impulsive gravitational wave Region III .............................................................. 105

P.313: Example 13-2: Two interacting waves ........................................................................................ 110

P.318, Example 13-3: The Nariai spacetime .......................................................................................... 115




P.322: Quiz 13-1: Collision of a gravitational wave with an electromagnetic wave The non-zero spin

coefficients ............................................................................................................................................ 127

P.322,Quiz 13-2: The Aichelburg-Sexl Solution The passing of a black hole ...................................... 133

Final Exam ...................................................................................................................................................... 133

P.323 FE-1: ! ...................................................................................................................................... 133 P.324 FE-4:" ........................................................................................................................... 134 P.324 FE-6: and in a diagonal metric ................................................................................ 134 P.324 FE-7: Calculate the Christoffel symbols for a metric example .................................................... 134

P.325 FE-8: Calculate the Riemann tensor of metric example .............................................................. 135

P.325 FE-8: Calculate the Riemann tensor of metric example Alternative version ........................... 135

P.325 FE-9: Calculate the Christoffel symbols for a metric example .................................................... 136

P.325 FE-10: Calculate the Ricci rotation coefficients for a metric example ........................................ 137

P.325, FE-12: The non-zero Weyl scalars of the Reissner-Nordstrm spacetime ................................. 138

New - P.326 FE-13: The deflection of a light ray in a Schwarzschild metric with two different masses146

Bibliografi ....................................................................................................................................................... 147

Chapter 2: Vectors, One Forms, and the metric

P.36: The signature of a metric.

Here we are going to investigate what happens to various quantities when a metric,#$%,changes signature. Christoffel symbols %'$ 12 #$* +,#*%,-' ,#*',-% . ,#%',-* / No change Riemann tensor 0 %'*$ ,' %*$ . ,* %'$ %*1 1'$ . %'1 1*$ No change Ricci tensor 0$% 0 $'%' No change Ricci scalar 0 #$%0$% Changes sign Einstein tensor 2$% 0$% . 12 #$%0 No change Energy tensor 8425$% 2$% No change Cosmological constant 2$% 0$% . 12 #$%0 #$% Changes sign

P. 46, Quiz 2-5: Spherical coordinates and vector transformation

The line element 789 7:9 :97;9 :9 sin9 ; 7?9 The metric tensor: #$% @ 1 :9 :9 sin9 ;A

Given B$ +:, 1rsin ; , 1cos9 ;/ calculate B$ .




B$ #$%B$ (2.17) BH #HHBH 1 : : BI #IIBI :9 + 1rsin ; / :sin ; BJ #JJBJ :9 sin9 ; + 1cos9 ;/ :9 tan9 ;

Which means the answer to the quiz 2-5 is (c)

Chapter 4: Tensor Calculus

P. 74, Example 4-4: Find the Christoffel symbols of the 2-sphere with radius The line element: 789 M97;9 M9 sin9 ; 7?9 The metric tensor and its inverse: #$% NM9 M9 sin9 ;O #$% P

1M9 1M9 sin9 ;Q

We have: $%' 12 ,'#$% ,%#$' . ,$#%' (4.15) %'$ #$**%' (4.16)

IJJ . 12 ,IM9 sin9 ; .M9 sin ; cos ; R JJI #IIIJJ . sin ; cos ; JIJ JJI 12 ,IM9 sin9 ; M9 sin ; cos ; R IJJ JIJ #JJJIJ cot ; P. 75, Example 4-5, quiz 4-7: Find the Christoffel symbols of the Kahn-Penrose metric

The line element: 789 27S7T . 1 . S97-9 . 1 S97U9 The metric tensor: #$% P 11 .1 . S9 .1 S9Q

and its inverse: #$% VWXWY 11 .11 . S9 .11 S9ZW

[W\

$%' 12 ,'#$% ,%#$' . ,$#%' (4.15) %'$ #$**%' (4.16) ]^^ . 12 ,].1 . S9 .1 . S R ^^_ #_]]^^ .1 . S ^ ]^ ^ ^] 12 ,].1 . S9 1 . S R ]^^ ^]^ #^^^ ]^ .11 . S ]`` . 12 ,].1 S9 1 S R ``_ #_]]`` 1 S




` ]` ` `] 12 ,].1 S9 .1 S R ]`` `]` #``` ]` 11 S Which means the answer to quiz 4-7 is (b)

P. 78, Example 4-7: Alternative solution: Show that p.69: a'#$% ,'#$% . #*% $'* . #$* '%*

If #*% #%*: ,'#$% . #%* $'* . #$* '%* ,'#$% . %$' . $'%

since $'% $%' 4.14: ,'#$% . %$' $%' and %$' $%' ,'#$% p. 73: ,'#$% . ,'#$% 0 Q.E.D.

New - P.80: One-forms: why For general forms, let g be a h-form and i be a j-form, we have

g k i .1lmi k g For one-forms this means

g k i .i k g R g k g .g k g 0 This also holds for 7n opo^q 7-$ because 7n is a one-form as well.

7n$ k 7n% ,n,-$ 7-$ k ,n,-% 7-%

,9n,-$,-% 7-$ k 7-% (i)

. ,9n,-$,-% 7-% k 7-$ (ii) R 7n$ k 7n$ ,9n,-$,-% 7-$ k 7-$ (i) and 7n$ k 7n$ . ,9n,-$,-% 7-$ k 7-$ (ii)

Now, because the partial derivatives commutate, this can only be true if 7-$ k 7-$ 0 and 79 0 (4.25)

P. 83, Example 4-9: Find the geodesic equations for cylindrical coordinates

The line element: 789 7:9 :97?9 7r9 The metric tensor and its inverse: #$% @ 1 :9 1 A #$% P

1 1:9 1 Q

$%' 12 ,'#$% ,%#$' . ,$#%' (4.15) %'$ #$**%' (4.16)

HJJ . 12 ,H:9 .: R JJH #HHHJJ .: JHJ JJH 12 ,H:9 : R HJJ JHJ #JJJHJ 1:




The geodesics equation: 79-$789 %'$ 7-%78 7-'78 0 (4.33)

-$ :: 79:789 %'H 7-%78 7-'78 0 R 79:789 . : +7?78 /9 0 -$ ?: 79?789 %'J 7-%78 7-'78 0 R 79?789 HJJ 7:78 7?78 JHJ 7?78 7:78 0 R 79?789 2: 7:78 7?78 0 -$ r: 79r789 %'s 7-%78 7-'78 0 R 79r789 0

The Christoffel symbols from the geodesic equations

We have t 12 #$%-u $-u % 12 :u9 12 :9v?u w9 12 ru9 (4.35) Now we need

,t,-$ 778 + ,t,-u $/ (4.36) -$ :: ,t,: 778 +,t,:u / R :?u 9 778 :u :y z 0 :y . :?u 9 -$ ?: ,t,? 778 {,t,?u | 0 778 v:9?u w 2::u?u :9?y z 0 ?y 1: :u?u 1: ?u :u -$ r: ,t,r 778 +,t,ru / R 0 778 ru ry

Collecting the results 0 :y . :?u 9 0 ?y 1: :u?u 1: ?u :u 0 ry




We can now find the Christoffel symbols from the geodesic equation : JJH .: HJJ 1: JHJ 1: P. 84, Example 4-10: Use the geodesic equations to find the Christoffel symbols for the Rindler

metric.

The line element: 789 }97~9 . 7}9 The metric tensor: #$% N}9 .1O

We have t 12 #$%-u $-u % 12 }9~u9 . 12 }u9 (4.35) Now we need

,t,-$ 778 + ,t,-u $/ (4.36) -$ }: ,t,} 778 {,t,}u | R }~u9 778 v.}uw .}y z 0 }y }~u9 -$ ~: ,t,~ 778 +,t,~u / 0 778 }9~u 2}}u~u }9~y z 0 ~y 1} }u~u 1} ~u}u

Collecting the results 0 }y }~u9 0 2}}u~u }9~y

We can now find the Christoffel symbols from the geodesic equation: } 1} 1} P. 86: The Riemann tensor eq. (4.42), (4.43) and (4.44)

Prove (4.42)

First we need a'#$% ,'#$% . $'* #*% . '%* #$* ,'#$% . %$' . $%' 0 (4.18) 0$%'* #$10 %'*1 #$1v,' %*1 . ,* %'1 p%* p'1 . p%' p*1 w (4.41) #$1,' %*1 . #$1,* %'1 #$1p%* p'1 . #$1p%' p*1 ,'#$1 %*1 . ,'#$1 %*1 . ,*#$1 %'1 ,*#$1 %'1 p%*#$1 p'1 . p%'#$1 p*1 ,'$%* . 1$' $1' %*1 . ,*$%' 1$* $1* %'1 p%*$p' . p%'$p* ,'$%* . ,*$%' . 1$' $1' %*1 1$* $1* %'1 1%*$1' . %'$1*




,'$%* . ,*$%' . 1$' %*1 1$* %'1 (4.42)

Prove (4.43) 0$%'* ,'$%* . ,*$%' . 1$' %*1 1$* %'1 (4.42) 12 ,' +,#$%,-* ,#$*,-% . ,#%*,-$ / . 12 ,* +,#$%,-' ,#$',-% . ,#%',-$ / . 1$' %*1 1$* %'1 12 { ,9#$%,-',-* ,9#$*,-',-% . ,9#%*,-',-$| . 12 { ,9#$%,-*,-' ,9#$',-*,-% . ,9#%',-*,-$| . 1$' %*1 1$* %'1 12 { ,9#$*,-',-% ,9#%',-*,-$ . ,9#$',-*,-% . ,9#%*,-',-$| . 1$' %*1 1$* %'1 (4.43)

Prove that (4.44):0$%'* 0'*$% .0$%*' .0%$'* and 0$%'* 0$'*% 0$*%' 0 0$%'* 12 { ,9#$*,-',-% ,9#%',-*,-$ . ,9#$',-*,-% . ,9#%*,-',-$| . 1$' %*1 1$* %'1 (4.43) 0'*$% 12 { ,9#'%,-$,-* ,9#*$,-%,-' . ,9#'$,-%,-* . ,9#*%,-$,-'| . 1'$ *%1 1'% *$1 12 { ,9#'%,-$,-* ,9#*$,-%,-' . ,9#'$,-%,-* . ,9#*%,-$,-'| . 1'$ *%1 #1p '%p #1pp*$ 12 { ,9#'%,-$,-* ,9#*$,-%,-' . ,9#'$,-%,-* . ,9#*%,-$,-'| . 1'$ *%1 v#1p#1pw '%p p*$ 1 12 { ,9#'%,-$,-* ,9#*$,-%,-' . ,9#'$,-%,-* . ,9#*%,-$,-'| . 1'$ *%1 '%p p*$ 12 { ,9#'%,-$,-* ,9#*$,-%,-' . ,9#'$,-%,-* . ,9#*%,-$,-'| . 1'$ *%1 '%1 1*$ 0$%'* Q.E.D. 0$%*' 12 { ,9#$',-*,-% ,9#%*,-',-$ . ,9#$*,-',-% . ,9#%',-*,-$| . 1$* %'1 1$' %*1 .0$%'* Q.E.D. 0%$'* 12 { ,9#%*,-',-$ ,9#$',-*,-% . ,9#%',-*,-$ . ,9#$*,-',-%| . 1%' $*1 1%* $'1 12 { ,9#%*,-',-$ ,9#$',-*,-% . ,9#%',-*,-$ . ,9#$*,-',-%| . #1p %'p #1pp$* #1p %*p #1pp$' 12 { ,9#%*,-',-$ ,9#$',-*,-% . ,9#%',-*,-$ . ,9#$*,-',-%| . #1p#1p %'p p$* #1p#1p %*p p$' 12 { ,9#%*,-',-$ ,9#$',-*,-% . ,9#%',-*,-$ . ,9#$*,-',-%| . %'p p$* %*p p$' 12 { ,9#%*,-',-$ ,9#$',-*,-% . ,9#%',-*,-$ . ,9#$*,-',-%| . %'1 1$* %*1 1$' .0$%'* Q.E.D. 0$'*% 12 { ,9#$%,-*,-' ,9#'*,-%,-$ . ,9#$*,-%,-' . ,9#'%,-*,-$| . 1$* '%1 1$% '*1 0$*%' 12 { ,9#$',-%,-* ,9#*%,-',-$ . ,9#$%,-',-* . ,9#*',-%,-$| . 1$% *'1 1$' *%1

1 #$%#%' $' if the metric is diagonal




0$'*% 0$*%' 12 { ,9#$%,-*,-' ,9#'*,-%,-$ . ,9#$*,-%,-' . ,9#'%,-*,-$| . 1$* '%1 1$% '*1 12 { ,9#$',-%,-* ,9#*%,-',-$ . ,9#$%,-',-* . ,9#*',-%,-$| . 1$% *'1 1$' *%1 12 { ,9#$',-%,-* ,9#*%,-',-$ . ,9#$*,-%,-' . ,9#'%,-*,-$| . 1$* '%1 1$' *%1 .0$%'* Q.E.D.

Also notice 0$$$$ ,'$%* . ,*$%' . 1$' %*1 1$* %'1 ,$$$$ . ,$$$$ . 1$$ $$1 1$$ $$1 (4.42) 0 0%$$$ ,$%$$ . ,$%$$ . 1%$ $$1 1%$ $$1 0 0$$%$ 12 { ,9#$*,-',-% ,9#%',-*,-$ . ,9#$',-*,-% . ,9#%*,-',-$| . 1$' %*1 1$* %'1 (4.43) 12 { ,9#$$,-%,-$ ,9#$%,-$,-$ . ,9#$%,-$,-$ . ,9#$$,-%,-$| . 1$% $$1 1$$ $%1 .#p1p$%#p1p$$ p$$ $%p .#p1#p1p$%p$$ p$$ $%p 0 0$$%% 12 { ,9#$%,-%,-$ ,9#$%,-%,-$ . ,9#$%,-%,-$ . ,9#$%,-%,-$| . 1$% $%1 1$% $%1 0 0$$%' 12 { ,9#$',-%,-$ ,9#$%,-',-$ . ,9#$%,-',-$ . ,9#$',-%,-$| . 1$% $'1 1$' $%1 0 0$%'' 12 { ,9#$',-',-% ,9#%',-',-$ . ,9#$',-',-% . ,9#%',-',-$| . 1$' %'1 1$' %'1 0 P. 87: Independent elements in the Riemann, Ricci and Weyl tensor

In dimensions, there are 1$ 99 . 1/12 independent elements in the Riemann tensor. In the Ricci tensor there are '' 1/2 independent elements, and in the Weyl tensor there are 1` 10 independent elements if 4, if 4 there are none. Summarized: 1$ 99 . 112 '' 12 1` 2 1 099 3 0 09 099

0

3 6 099 09 099 0 09 099

6 0 099 09 0 09 0

0

4 20 0$%'* 10 0$% 0'$'% 10 $%'*2 2 The Weyl tensor possesses the same symmetries as the Riemann tensor: $%'* .$%*' .%$'* '*$% and $%'* $*%' $'*% 0. It possesses an additional symmetry: '$'% 0. It follows that the Weyl tensor is trace-

free, in other words, it vanishes for any pair of contracted indices. One can think of the Weyl tensor as that part of the

curvature tensor for which all contractions vanish (d'Inverno, 1992, p. 88)




099 09 09 099 099 09 0 0 09 09 0 0 093 09 0 099 099 09 099 09 0

0 099 09 09 0 0 0 09 0 0

99 99 9 9 9 94 9 9 9

P.87, Example 4-11: Compute the components of the Riemann tensor for the unit 2-sphere

The number of independent elements in the Riemann tensor in a metric of dimension 2 is vw9 1 so we can choose to calculate 0 JIJI

The Riemann tensor 0 %'*$ ,' %*$ . ,* %'$ %*1 1'$ . %'1 1*$ (4.41) We choose M ;, and 7 ?

0 JIJI ,I JJI . ,J JII JJ1 1II . JI1 1JI ,I JJI JJ1 1II . JI1 1JI Sum over : ,I JJI JJI III . JII IJI JJJ JII . JIJ JJI ,I JJI . JIJ JJI ,I. sin ; cos ; . cot ; . sin ; cos ; . cos9 ; sin9 ; cos9 ; sin9 ; P.89, Example 4-12: Show that the Ricci scalar for the unit 2-sphere

The line element: 789 7;9 sin9 ; 7?9 The metric tensor and its inverse: #$% 1 sin9 ; #$% @ 1 1sin9 ;A

3 Because: 0$%'* 0$'*% 0$*%' 0

4 Because: $%'* $'*% $*%' 0




The Ricci scalar: 0 #$%0$% (4.47)

Sum over a: 0 #I%0I% #J%0J% Sum over b: #II0II #IJ0IJ #JI0JI #JJ0JJ #IJ #JI 0 #II0II #JJ0JJ #II0 I'I' #JJ0 J'J' 0$% 0 $'%' (4.46) Sum over c #II0 IIII #II0 IJIJ #JJ0 JIJI #JJ0 JJJJ 0 IIII 0 JJJJ 0 #II0 IJIJ #JJ0 JIJI #II#JJ0JIJI #JJ#II0IJIJ 0JIJI 0IJIJ (4.44) 2#II#JJ0IJIJ 2#JJ0 JIJI 0 JIJI sin9 ; ex 4-11 2 1sin9 ; sin9 ; 2

Remark that 0 2#JJ0 JIJI is a general solution for a 2-dimensional diagonal metric if we write: 0 2#990 99 (S1) New - p.90: Proof: if a space is conformally flat, i.e. the Weyl tensor vanishes The Weyl scalar $%'* 0$%'* 12 #$*0'% #%'0*$ . #$'0*% . #%*0'$ 16 #$'#*% . #$*#'%0 (4.49) 0$%'* 12 #$*01'1% #%'01*1$ . #$'01*1% . #%*01'1$ 16 #$'#*% . #$*#'%#1p01p 0$%'* 12 v#$*#1p0p'1% #%'#1p0p*1$ . #$'#1p0p*1% . #%*#1p0p'1$w 16 #$'#*% . #$*#'%#1p#01p 0$%'* 12 n-9v$*1p0p'1% %'1p0p*1$ . $'1p0p*1% . %*1p0p'1$w 16 n-$'*% . $*'%1p01p 0$%'* 12 n-9$*01'1% %'01*1$ . $'01*1% . %*01'1$ 16 n-$'*% . $*'%011




12 { ,9#$*,-',-% ,9#%',-*,-$ . ,9#$',-*,-% . ,9#%*,-',-$| . 1$' %*1 1$* %'1 12 n-9 $* {12 { ,9#1%,-1,-' ,9#'1,-%,-1 . ,9#11,-%,-' . ,9#'%,-1,-1| . p11 '%p p1% '1p | %' {12 { ,9#1$,-1,-* ,9#*1,-$,-1 . ,9#11,-$,-* . ,9#*$,-1,-1| . p11 *$p p1$ *1p |. $' {12 { ,9#1%,-1,-* ,9#*1,-%,-1 . ,9#11,-%,-* . ,9#*%,-1,-1| . p11 *%p p1% *1p |. %* {12 { ,9#1$,-1,-' ,9#'1,-$,-1 . ,9#11,-$,-' . ,9#'$,-1,-1| . p11 '$p p1$ '1p | 16 n-$'*% . $*'% {12 { ,9#1,-,-1 ,9#1,-1,- . ,9#,-1,-1 . ,9#11,-,-|. p 11p p1 1p |

5 12 $*9,',% %'9,*,$ . $'9,*,% . %*9,',$n- . 1$' %*1 1$* %'1 12 n-9 {$* +12 1%9,1,' '19,%,1 . 119,%,' . '%9,1,1n- . p11 '%p p1% '1p / %' +12 1$9,1,* *19,$,1 . 119,$,* . *$9,1,1n- . p11 *$p p1$ *1p /. $' +12 1%9,1,* *19,%,1 . 119,%,* . *%9,1,1n- . p11 *%p p1% *1p /. %* +12 1$9,1,' '19,$,1 . 119,$,' . '$9,1,1n- . p11 '$p p1$ '1p /| 16 n-$'*% . $*'% +12 19,,1 19,1, . 9,1,1 . 119,,n-. p 11p p1 1p / .1$' %*1 1$* %'1 12 n-9 $*v.p11 '%p p1% '1p w %'v.p11 *$p p1$ *1p w. $'v.p11 *%p p1% *1p w . %*v.p11 '$p p1$ '1p w 16 n-$'*% . $*'%v.p 11p p1 1p w #1pv.1$'p%* 1$*p%'w 12 n-9 $*#pv.p11'% p1%'1w %'#pv.p11*$ p1$*1w. $'#pv.p11*% p1%*1w . %*#pv.p11'$ p1$'1w 16 n-#p$'*% . $*'%v.p11 p11w

5 $*9,',% %'9,*,$ . $'9,*,% . %*9,',$n- N1 1 . 1 . 1 0 n M 7 -0 n M : : : 7 0




n-1pv.1$'p%* 1$*p%'w 12 n- $*pv.p11'% p1%'1w %'pv.p11*$ p1$*1w. $'pv.p11*% p1%*1w . %*pv.p11'$ p1$'1w 16 n-p$'*% . $*'%v.p11 p11w n-1p . {12 ,'#1$ ,$#1' . ,1#$'| {12 v,*#p% ,%#p* . ,p#%*w|

{12 ,*#$1 ,$#1* . ,1#$*| {12 v,'#p% ,%#p' . ,p#%'w| 12 n- $*p . {12 v,1#p1 ,1#p1 . ,p#11w| {12 ,%#' ,'#%. ,#'%| {12 v,%#p1 ,1#p% . ,p#1%w| {12 ,1#' ,'#1 . ,#'1| %'p . {12 v,1#p1 ,1#p1 . ,p#11w| {12 ,$#* ,*#$ . ,#*$| {12 v,$#p1 ,1#p$ . ,p#1$w| {12 ,1#* ,*#1 . ,#*1|. $'p . {12 v,1#p1 ,1#p1 . ,p#11w| {12 ,%#* ,*#% . ,#*%| {12 v,%#p1 ,1#p% . ,p#1%w| {12 ,1#* ,*#1 . ,#*1|. %*p . {12 v,1#p1 ,1#p1 . ,p#11w| {12 ,$#' ,'#$ . ,#'$| {12 v,$#p1 ,1#p$ . ,p#1$w| {12 ,1#' ,'#1 . ,#'1| 16 n-p$'*%. $*'% . {12 v,#p ,#p . ,p#w| {12 v,1#1 ,1#1 . ,#11w| {12 v,1#p ,#p1 . ,p#1w| {12 v,#1 ,1# . ,#1w|




n-1p . {12 1$,' 1',$ . $',1| {12 vp%,* p*,% . %*,pw| {12 1$,* 1*,$ . $*,1| {12 vp%,' p',% . %',pw| n- 12 n- $*p . {12 vp1,1 p1,1 . 11,p w| {12 ',% %,'. '%,| {12 vp1,% p%,1 . 1%,pw| {12 v',1 p1,' . '1,w| %'p . {12 vp1,1 p1,1 . 11,pw| {12 *,$ $,* . *$,| {12 vp1,$ p$,1 . 1$,pw| {12 *,1 1,* . *1,|. $'p . {12 vp1,1 p1,1 . 11,pw| {12 *,% %,* . *%,| {12 vp1,% p%,1 . 1%,pw| {12 *,1 1,* . *1,|. %*p . {12 vp1,1 p1,1 . 11,pw| {12 ',$ $,' . '$,| {12 vp1,$ p$,1 . 1$,pw| {12 ',1 1,' . '1,| n- 16 n-p$'*%. $*'% . {12 vp, p, . ,pw| {12 v1,1 1,1 . 11,w| {12 vp,1 p1, . 1,pw| {12 v1, ,1 . 1,w| n-

6n-1p {. +12 1$,'/ +12 p%,*/ +12 1$,*/ +12 p%,'/| n- 12 n- $*p {. +12 p1,1/ +12 ',%/ +12 p1,%/ +12 ',1/| %'p {. +12 p1,1/ +12 *,$/ +12 p1,$/ +12 *,1/|. $'p {. +12 p1,1/ +12 *,%/ +12 p1,%/ +12 *,1/|. %*p {. +12 p1,1/ +12 ',$/ +12 p1,$/ +12 ',1/| n- 16 n-p$'*% . $*'% {. +12 p,/ +12 1,1/ +12 p,1/ +12 1,/| n-

6 1',$ . $',1n- N1 . 1 0 n M -0 n : : M 0




14 n-1pv.1$p%,',* 1$p%,*,'wn- 18 n- $*pv.p1',1,% p1',%,1w %'pv.p1*,1,$ p1*,$,1w . $'pv.p1*,1,% p1*,%,1w. %*pv.p1',1,$ p1',$,1w n- 124 n-p$'*% . $*'%v.p1,,1 p1,1,wn- 0 P.91 Quiz 4-2, Quiz 4-3: Calculate the Christoffel symbols for the spherical metric

The line element: 789 7:9 :97;9 :9 sin9 ; 7?9 The metric tensor and its inverse: #$% @ 1 :9 :9 sin9 ;A #$% VWX

WY 1 1:9 1:9 sin9 ;ZW[W\

$%' 12 ,'#$% ,%#$' . ,$#%' (4.15) %'$ #$**%' (4.16)

HII . 12 ,H:9 .: R IIH #HHHII .: IHI IIH 12 ,H:9 : R HII IHI #IIIHI 1: HJJ . 12 ,H:9 sin9 ; .: sin9 ; R JJH #HHHJJ .: sin9 ; JHJ JJH 12 ,H:9 sin9 ; : sin9 ; R HJJ JHJ #JJJHJ 1: IJJ . 12 ,I:9 sin9 ; .:9 sin ; cos ; R JJI #IIIJJ . sin ; cos ; JIJ JJI 12 ,I:9 sin9 ; :9 sin ; cos ; R IJJ JIJ #JJJIJ cot ;

So the answer to quiz 4-2 is (a) .:9 sin ; cos ; And the answer to quiz 4-3 is (c) cot ;

P.91 Quiz 4-4: The Riemann tensor of the spherical metric

The number of independent elements in the Riemann tensor in a metric of dimension 3 is vw9 6 and we have to calculate: 0 IHIH ; 0 JHJH ; 0 JIJI ; 0 IHJH ; 0 HIJI ; 0 HJIJ

The Riemann tensor 0 %'*$ ,' %*$ . ,* %'$ %*1 1'$ . %'1 1*$ (4.41)

M, :, , 7 ;: 0 IHIH ,H IIH . ,I IHH II1 1HH . IH1 1IH :, ;, ?: ,H IIH . IHI IIH .1 . +1:/ .: 0 M, :, , 7 ?: 0 JHJH ,H JJH . ,J JHH JJ1 1HH . JH1 1JH




:, ;, ?: ,H JJH . JHJ JJH . sin9 ; . +1:/ .: sin9 ; 0 M, ;, , 7 ?: 0 JIJI ,I JJI . ,J JII JJ1 1II . JI1 1JI :, ;, ?: ,I JJI JJH HII . JIJ JJI . cos9 ; sin9 ; .: sin9 ; +1:/ . cot ;. sin ; cos ; 0 M, :, ;, 7 ?: 0 IHJH ,H IJH . ,J IHH IJ1 1HH . IH1 1JH :, ;, ?: 0 M, ;, :, 7 ?: 0 HIJI ,I HJI . ,J HII HJ1 1II . HI1 1JI :, ;, ?: 0 M, ?, :, 7 ;: 0 HJIJ ,J HIJ . ,I HJJ HI1 1JJ . HJ1 1IJ :, ;, ?: +1:/ cot ; . +1:/ cot ; 0 0HIIJ .0IHIJ .#II0 HIJI 0

We see that all the elements of the Riemann tensor equals 0, and the answer to quiz 4-4 is (d) P.91 Quiz 4-6: A Lie derivative in the spherical metric

Let $ :, sin ; , sin ; cos ? and T$ :, :9 cos ; , sin ? Calculate the Lie derivative S _ T%a% . %a%T (4.27) The covariant derivative: a%$ ,$,-% %'$ ' (4.6)

S$ T%a%$ . %a%T$ T% {,$,-% %'$ '| . % {,T$,-% %'$ T'| T% ,$,-% T% %'$ ' . % ,T$,-% . % %'$ T' T% ,$,-% . % ,T$,-% T% %'$ ' . % %'$ T' M :: SH T% ,H,-% . % ,TH,-% T% %'H ' . % %'H T' :, ;, ?: TI IIH I . I IIH TI TJ JJH J . J JJH TJ 0 M ;: SI T% ,I,-% . % ,TI,-% T% %'I ' . % %'I T' :, ;, ?: TI ,I,; . H ,TI,: . I ,TI,; TH H'I ' . H H'I T' TI I'I ' .I I'I T' TJ J'I ' . J J'I T' :, ;, ?: TI ,I,; . H ,TI,: . I ,TI,; TH HII I . H HII TI TI IHI H .I IHI TH TJ JJI J . J JI TJ TI ,I,; . H ,TI,: . I ,TI,; :9 cos ;cos ; . :2: cos ; . sin ;.:9 sin ; :91 . 2 cos ; M ?: SJ T% ,J,-% . % ,TJ,-% T% %'J ' . % %'J T' :, ;, ?: TI ,J,-I TJ ,J,-J . J ,TJ,-J TH H'J ' . H H'J T' TI I'J ' .I I'J T' TJ J'J ' . J J'J T'




:, ;, ?: TI ,J,-I TJ ,J,-J . J ,TJ,-J TH HJJ J . H HJJ TJ TI IJJ J .I IJJ TJ TJ JHJ H . J JHJ TH TJ JIJ I . J JIJ TI

TI ,J,-I TJ ,J,-J . J ,TJ,-J :9 cos ;cos ; cos ? sin ? . sin ; sin ? . sin ; cos ? cos ? :9 cos9 ; cos ? . sin ;

So we can conclude:

S SHSISJ 0:91 . 2 cos ;:9 cos9 ; cos ? . sin ;

And the answer to quiz 4-6 is (a)

P.92, Quiz 4-8: The Ricci scalar of the Penrose Kahn metric

The Ricci scalar: 0 #$%0$% (4.47) The Ricci tensor 0$% 0 $'%' (4.46)

Sum over M S, T, -, U: 0 #]%0]% #_%0_% #^%0^% #`%0`% Sum over S, T, -, U: #]_0]_ #_]0_] #^^0^^ #``0`` #]_0 ]'_' #_]0 _']' #^^0 ^'^' #``0 `'`' Sum over S, T, -, U: #]_0 ]^_^ #]_0 ]`_` #_]0 _^]^ #_]0 _`]` #^^0 ^]^] #^^0 ^_^_ #^^0 ^`^` #``0 `]`] #``0 `_`_ #``0 `^`^ #]_ #_]: #]_#^^0^]^_ #]_#``0`]`_ #_]#^^0^_^] #_]#``0`_`] #^^#]_0_^]^ #^^#_]0]^_^ #^^#``0`^`^ #``#]_0_`]` #``#_]0]`_` #``#^^0^`^` 0$%'* 0'*$% .0%$'* .0$%*': 4#]_#^^0^]^_ 4#]_#``0`]`_ 2#^^#``0`^`^ 4#]_0 ]^_^ 4#]_0 ]`_` 2#^^0 ^`^`

Remark we can rewrite this into to a general expression for a non-diagonal metric of the type:

#$% VXY #9#9 # #Z[

\

We write 0 4#90 9 4#90 9 2#0 (S2) Now we need to calculate the three elements in the Riemann tensor: 0 ]^_^ ; 0 ]`_` ; 0 ^`^`

0 ]^_^ ,^ ]_^ . ,_ ]^^ ]_1 1^^ . ]^1 1_^ 0 0 ]`_` ,` ]_` . ,_ ]`` ]_1 1`` . ]`1 1_` 0 0 ^`^` ,` ^^` . ,^ ^`` ^^1 1`` . ^`1 1^` 0

So we can conclude 0 0 Which means the answer to quiz 4-8 is (b)

P.92, Quiz 4-9: The Christoffel symbols of a metric example

The line element: 789 U9 sin - 7-9 -9 tan U 7U9 The metric tensor and its inverse: #$% NU9 sin - -9 tan UO #$% VX

Y 1U9 sin - 1-9 tan UZ[\




$%' 12 ,'#$% ,%#$' . ,$#%' (4.15) %'$ #$**%' (4.16) ` ^^ 12 v.,`#^^w .U sin - R ^^` #``` ^^ . U sin --9 tan U ^ `^ ^ ^` 12 U sin - R `^^ ^`^ #^^^ `^ 1U ^ ^^ 12 U9 cos - R ^^^ #^^^ ^^ 12 cot - ^ `` .- tan U R ``^ #^^^ `` . - tan UU9 sin - ` ^` ` `^ - tan U R ^`` `^` #``` ^` 1- ` `` 12 -91 tan9 U R ``` #``` `` 1 tan9 U2 tan U

Which means that the answer to quiz 4-9 is (c)

P.92, Quiz 4-10: The Ricci scalar of a metric example

From example 4-12 we know that for a 2-dimensional diagonal metric: 0 2#990 99 which means we only have to calculate 0 `^`^

0 `^`^ ,^ ``^ . ,` `^^ ``1 1^^ . `^1 1`^ -, U: ,^ ``^ . ,` `^^ ``^ ^^^ . `^^ ^`^ ``` `^^ . `^` ``^

,^ +. - tan UU9 sin -/ . ,` +1U/ +. - tan UU9 sin -/ +12 cot -/ . +1U/9 {1 tan9 U2 tan U | +1U/ . +1-/ +. - tan UU9 sin -/

. tan UU9 + 1sin - . - cos -sin9 - / +1U/9 . +12 -U9 tan U cos -sin9 - / . +1U/9 {1 tan9 U2U tan U | + tan UU9 sin -/

+-tan U cos -U9 sin9 - / . +12 -U9 tan U cos -sin9 - / {1 tan9 U2U tan U |

{- cos - tan9 U U sin9 - U sin9 - tan9 U2U9 sin9 - tan U | R 0 2#``0 `^`^ 2 + 1-9 tan U/ {- cos - tan9 U U sin9 - U sin9 - tan9 U2U9 sin9 - tan U | {- cos - tan9 U U sin9 - U sin9 - tan9 U-9U9 sin9 - tan9 U |

And the answer to quiz 4-10 is (b)

Chapter 5: Cartans Structure Equations

P. 106, Example 5-1: Ricci rotation coefficients for the Tolman-Bondi- de Sitter metric

The line element: 789 79 . 9,H7:9 . 09, :7;9 . 09, : sin9 ; 7?9

The Basis one forms




7 P 1 .1 .1 .1Q

H ,H7: 7: ,HH I 0, :7; 7; 10, : I J 0, : sin ; 7? 7? 10, : sin ; J

Cartans First Structure equation and the calculation of the Ricci rotation coefficients %'$ :

7$ . %$ k % (5.9) %$ %'$ ' (5.10)

7 0 7H 7v,H7:w . u,H7 k 7: . u k H 7I 70, :7; 0u 7 k 7; 07: k 7; 0u0 k I 00 ,HH k I 7J 70, : sin ; 7? 0u sin ; 7 k 7? 0 sin ; 7: k 7? 0, : cos ; 7; k 7? 0u0 k J 00 ,HH k J cot ;0 I k J

Summarizing the curvature one forms in a matrix:

%$ VWWWXWWWY 0 . uH 0u0 I 0u0 J. uH 0 00 ,HI 00 ,HJ0u0 I . 00 ,HI 0 cot ;0 J0u0 J . 00 ,HJ . cot ;0 J 0 ZWW

W[WWW\

Where M refers to column and to row.

Now we can read off the Ricci rotation coefficients HH . u HH . u II 0u0 JJ 0u0 II 0u0 IIH . 00 ,H HII 00 ,H HJJ 00 ,H J J 0u0 J JH . 00 ,H J JI . cot ;0 IJJ cot ;0 Which means the answer to quiz 5-6 is (b)

Transformation of the Ricci rotation coefficients %'$ into the Christoffel symbols $




We have the transformation $ *$ 1p* %1 'p (5.14)

HH *H 1p* 1 Hp HH . u 1 . u HH * 1p* H1 Hp HH v HH w9 1v. uwv,Hw9 . u 9,H II *I 1p* 1 Ip II 0u0 1 0u0 II * 1p* I1 Ip II II 9 1 0u0 09 00u HII *I 1p* H1 Ip HII HH 00 ,H,H 00 IIH *H 1p* I1 Ip HH IIH II 9 ,H {. 00 ,H| 09 .009,H JJ *J 1p* 1 Jp JJ 0u0 1 0u0 JJ * 1p* J1 Jp J J JJ 9 1 0u0 09 sin9 ; 00u sin9 ; HJJ *J 1p* H1 Jp HJJ HH 00 ,H,H 00 JJH *H 1p* J1 Jp HH J JH JJ 9 ,H {. 00 ,H| 09 sin9 ; .009,H sin9 ; IJJ *J 1p* I1 Jp IJJ II cot ;0 0 cot ; JJI *I 1p* J1 Jp II J JI JJ 9 10 +. cot ;0 / 09 sin9 ; . cos ; sin ; However by this method we do not obtain HHH .

To check we calculate the Christoffel symbols directly from the metric

The line element: 789 79 . 9,H7:9 . 09, :7;9 . 09, : sin9 ; 7?9 The metric tensor #$% P1 0 0 00 .9,H 0 00 0 .09, : 00 0 0 .09, : sin9 ;Q

and its inverse: #$% VWXWY1 0 0 00 .9,H 0 00 0 . 109, : 00 0 0 . 109, : sin9 ;ZW

[W\

$%' 12 ,'#$% ,%#$' . ,$#%' (4.15) %'$ #$**%' (4.16)

HHH 12 ,Hv.9,Hw 9,H




R HHH #HHHHH .9,H9,H . HH . 12 ,v.9,Hw . u9,H R HH #HH 1v. u9,Hw . u9,H HH HH 12 ,v.9,Hw u9,H R HH HH #HHHH .9,H u 9,H . u II . 12 ,v.09, :w 0u 0 R II #II 1 0u 0 0u 0 II II 12 ,v.09, :w .0u 0 R II II #IIII . 109 v.0u 0w 0u0 HII . 12 ,Hv.09, :w 00 R IIH #HHHII .9,H00 IHI IIH 12 ,Hv.09, :w .00 R HII IHI #IIIHI . 109 .00 00 JJ . 12 ,.09, : sin9 ; 0u 0 sin9 ; R JJ #JJ 1 0u 0 sin9 ; 0u 0 sin9 ; JJ JJ 12 ,.09, : sin9 ; .0u 0 sin9 ; R JJ JJ #JJJJ +. 109 sin9 ;/ v.0u 0 sin9 ;w 0u0 HJJ . 12 ,H.09, : sin9 ; 00 sin9 ; R JJH #HHHJJ .9,H00 sin9 ; JHJ JJH 12 ,H.09, : sin9 ; .00 sin9 ; R HJJ JHJ #JJJHJ +. 109 sin9 ;/ .00 sin9 ; 00 IJJ . 12 ,I.09, : sin9 ; 09 sin ; cos ; R JJI #IIIJJ . 109 09 sin ; cos ; . sin ; cos ; JIJ JJI 12 ,I.09, : sin9 ; .09 sin ; cos ; R IJJ JIJ #JJJIJ . 109 sin9 ; .09 sin ; cos ; cot ; P.113: The curvature two forms and the Riemann tensor

To find the Riemann tensor from the curvature two forms, it can sometimes be more convenient to use the

following expression %$ 12 0 %'*$ ' k * (5.28) 0 %'*$ ' k * no summations




P.113, Example 5-2: Find the Ricci scalar using Cartans structure equations of the 2-sphere

The line element: 789 7;9 sin9 ; 7?9

The Basis one forms

I 7; 7; I 1 1 J sin ; 7? 7? 1sin ; J

Cartans First Structure equation and the calculation of the curvature one forms:

7$ . %$ k % (5.9) 7I 0 7J 7sin ; 7? cos ; 7; k 7? cot ; I k J R IJ cot ; J

Curvature two forms:

%$ 7 %$ '$ k % ' 12 0 %'*$ ' k * (5.27), (5.28)

IJ : 7 IJ 7cot ; J 7cos ; 7? . sin ; 7; k 7? .I k J 'J k I ' IJ k I I JJ k I J 0 R IJ .I k J

From which we can identify the single independent element of the Riemann tensor 0 J IJI 1 And the Ricci scalar 0 2J J 0 J IJI 2 (S1)

P.116, Example 5-3: Find the components of the Riemann tensor of the Robertson-Walker met-

ric using Cartans structure equations

The metric: 789 .79 M91 . :9 7:9 M9:97;9 M9:9 sin9 ; 7?9

The Basis one forms

7




H M1 . :9 7: 7: 1 . :9M H I M:7; 7; 1M: I P

.1 1 1 1 Q J M: sin ; 7? 7? 1M: sin ; J

Cartans First Structure equation and the calculation of the curvature one forms:

7$ . %$ k % 7 0 7H 7 { M1 . :9 7:| Mu1 . :9 7 k 7: . MuM H k 7I 7M:7; Mu :7 k 7; M7: k 7; MuM k I 1 . :9M: H k I 7J 7M: sin ; 7? Mu: sin ; 7 k 7? M sin ; 7: k 7? M: cos ; 7; k 7? Mu: sin ; k 1M: sin ; J M sin ; 1 . :9M H k 1M: sin ; J M: cos ; 1M: I k 1M: sin ; J MuM k J 1 . :9M: H k J cot ;M: I k J

The curvature one-forms summerized in a matrix

%$ VWWWXWWWY 0 MuM H MuM I MuM JMuM H 0 1 . :9M: I 1 . :9M: JMuM I . 1 . :9M: I 0 cot ;M: JMuM J . 1 . :9M: J . cot ;M: J 0 ZWW

W[WWW\

Where M refers to the column and the row

Curvature two forms:

%$ 7 %$ '$ k % ' 12 0 %'*$ ' k * (5.27), (5.28)




H : 7 H 7 +MuM H/ 7 +MuM M1 . :9 7:/ 7 + Mu1 . :9 7:/ My1 . :9 7 k 7: My1 . :9 k 1 . :9M H MyM k H 'H k ' H k HH k H IH k I JH k J 0 R H MyM k H I : 7 I 7 +MuM I/ 7 +MuM M:7;/ 7Mu :7; My:7 k 7; Mu7: k 7; My: k 1M: I Mu 1 . :9M H k 1M: I MyM k I Mu 1 . :9M9: H k I 'I k ' I k HI k H II k I JI k J HI k H Mu1 . :9M9: I k H R I MyM k I J : 7 J 7($u$ J 7 $u$ M: sin ; 7? 7Mu: sin ; 7? My: sin ; 7 k 7? Mu sin ; 7: k 7? Mu: cos ; 7; k 7? MyM k J Mu 1 . :9M9: H k J Mu cot ;M9: I k J 'J k ' J k HJ k H IJ k I JJ k J HJ k H IJ k I Mu1 . :9M9: J k H Mu cot ;M9: J k I R J MyM k J HI : 7 HI 7 {1 . :9M: I| 7 {1 . :9M: M:7;| 7 1 . :97; .:1 . :9 7: k 7; .M9 H k I 'I k H ' HI k HH I k H II k HI JI k HJ I k H +MuM/9 I k H R HI .Mu 9 M9 H k I HJ : 7 HJ 7 {1 . :9M: J | 7 1 . :9 sin ; 7? .:1 . :9 sin ; 7: k 7? 1 . :9 cos ; 7; k 7? .:M9: H k J 1 . :9 cot ;M9:9 I k J 'J k H ' HJ k HH J k H IJ k HI JJ k HJ J k H IJ k HI . Mu 9M9 H k J . 1 . :9 cot ;M9:9 I k J R HJ .Mu 9 M9 H k J




IJ : 7 IJ 7 +cot ;M: J / 7 +cot ;M: M: sin ; 7?/ 7cos ; 7? . sin ; 7; k 7? . 1M9:9 I k J 'J k I ' J k I HJ k IH IJ k II JJ k IJ J k I HJ k IH +MuM/9 J k I . 1 . :9M:9 J k I R IJ Mu 9 M9 J k I

Summarized in a matrix

%$ VWWXWWY0

MyM k H MyM k I MyM k J 0 .Mu 9 M9 H k I .Mu 9 M9 H k J 0 Mu 9 M9 J k I 0 ZWW[WW\

Now we can read off the elements in the Riemann tensor in the non-coordinate basis 0 HH . MyM 0 II . MyM 0 J J . MyM 0 HIHI Mu 9 M9 0 HJ HJ Mu 9 M9

IJ IJ Mu 9 M9 P. 120, Quiz 5-1, and 5-2: Ricci rotation coefficients of the spherical polar metric

The line element: 789 7:9 :97;9 :9 sin9 ; 7?9

The Basis one forms H 7: 7: H @ 1 1 1 A I

:7; 7; 1: I J : sin ; 7? 7? 1: sin ; J


We have: 7$ . %$ k % (5.9) %$ %'$ ' (5.10)

7H 0 7I 7:7; 7: k 7; 1: H k I




7J 7: sin ; 7? sin ; 7: k 7? : cos ; 7; k 7? 1: H k J cot ;: I k J


%$ VWXWY 0 1: I 1: J. 1: I 0 cot ;: J. 1: J . cot ;: J 0 ZW

[W\


Now we can read off the Ricci rotation coefficients IIH . 1: HII 1: HJJ 1: J JH . 1: J JI . cot ;: IJJ cot ;: So the answer to quiz 5-1 is (a) and to 5-2 is (c).

P. 120, Quiz 5-3: Transformation of the Ricci rotation coefficients into the Christoffel sym-bols of the spherical polar metric

We have the transformation $ *$ 1p* %1 'p (5.14)

HII *I 1p* H1 Ip HII HH 1: 1 1: IIH *H 1p* I1 Ip HH IIH II 9 1 +. 1:/ :9 .: HJJ *J 1p* H1 Jp HJJ HH 1: 1 1: JJH *H 1p* J1 Jp HH J JH JJ 9 1 +. 1:/ :9 sin9 ; .: sin9 ; IJJ *J 1p* I1 Jp IJJ II cot ;: : cot ; JJI *I 1p* J1 Jp II J JI JJ 9 1: +. cot ;: / :9 sin9 ; . sin ; cos ; So the answer to quiz 5-3 is JJH .: sin9 ;

P. 120, Quiz 5-4: Ricci rotation coefficients of the Rindler metric

The line element: 789 S97T9 . 7S9

The Basis one forms ] 7S 7S ] 1 .1 _ S7T 7T 1S _





We have 7$ . %$ k % (5.9) %$ %'$ ' (5.10)

7] 0 7_ 7S7T 7S k 7T 1S ] k _ R ]_ 1S _ R ]__ 1S We calculate __] ]]]__ .]]_]_ .]]__ ]__ 1S

So the answer to quiz 5-4 is ]__ __] ] P. 121, Quiz 5-7: The Einstein tensor for the Tolman-Bondi- de Sitter metric

The curvature two forms:

%$ 7 %$ '$ k % ' 12 0 %'*$ ' k * (5.27), (5.28)

H : 7 H 7v. uHw 7v. u,H7:w . y,H v uw9,H7 k 7: . y v u w9 k H 'H k ' H k HH k H IH k I JH k J 0 R H . y v u w9 k H I : 7 I 7 {0u0 I| 7v0u , :7;w 0y 7 k 7; 0u 7: k 7; 0y0 k I v0u w0 ,HH k I 'I k ' I k HI k H II k I JI k J HI k H ,HI k v. uHw R I 0y0 k I v0u w 0 u0 ,HH k I J 7 J 7(u J 7v0u , : sin ; 7?w 0y sin ; 7 k 7? v0u w sin ; 7: k 7? 0u cos ; 7; k 7? 0y0 k J v0u w0 ,HH k J 0u09 cot ; I k J 'J k ' J k HJ k H IJ k I JJ k J HJ k H IJ k I 00 ,HJ k v. uHw cot ;0 J k 0u0 I R J 0y0 k J v0u w 0 u0 ,HH k J




HI : 7 HI 7 {00 ,HI | 7v0,H7;w v0u w,H 0 u ,H 7 k 7; 0,H 0,H7: k 7; v0u w 0 u ,H0 k I 0 0 9,H0 H k I 'I k H ' I k H HI k H H II k H I JI k H J I k H u I k(. uH R HI v0u w 0 u ,H0 k I 0 0 9,H0 0u u0 H k I HJ : 7 HJ 7 {00 ,HJ | 7v0,H sin ; 7?w v0u w,H sin ; 0,H sin ; 7 k 7? 0,H sin ; 0,H sin ;7: k 7? 0,H cos ; 7; k 7? v0u w 0 ,H0 k J 0 0 9,H0 H k J 009 ,H cot ; I k J 'J k H ' J k H HJ k H H IJ k H I JJ k H J J k H IJ k H I 0u0 J k v. uHw cot ;0 J k 00 ,HI R HJ v0u w 0 ,H0 k J 0 0 2,:0 0u u0 H k J IJ : 7 IJ 7 +cot ;0 J / 7cos ; 7? . sin ; 7; k 7? . 109 I k J 'J k I ' J k I HJ k I H IJ k I I JJ k I J J k I HJ k I H 0u0 J k 0u0 I 00 ,HJ k {. 00 ,HI |

R IJ 109 v0u w909 . 0909 9,H J k I

Summarized in a matrix:

%$ VWWWXWWWY0 . y v uw9 k H 0y0 k I v0u w 0 u0 H k I 0y0 k J v0u w 0 u0 H k J 0 v0u w 0 u 0 k I 0 0 90 0u u0 H k I v0u w 0 0 k J 0 0 90 0u u0 H k J

0 109 v0u w909 . 0909 9,H J k I 0 ZWWW[WWW\

Now we can find the independent elements of the Riemann tensor in the non-coordinate basis:

R HH y . v u w9 0 II . 0y0 0 J J . 0y0 0 IHI .v0u w 0 u ,H0 0 J HJ .v0u w 0 u ,H0 0 HIHI . 0 0 9,H0 0u u0 0 HJ HJ . 0 0 9,H0 0u u0 0 IJ IJ 109 v0u w909 . 0909 9,H Where A,B,C,D,E will be used later, to make the calculations easier




The Ricci tensor:

0$% 0 $'%' (4.46) 0 0 '' 0 0 HH 0 II 0 JJ 0 HH 0 II 0 J J y . v u w9 . 2 0y0 2 0H 0 H'' 0 H 0 HHH 0 HII 0 HJ J 0 HII 0 HJ J . v0u w 0 u ,H0 . v0u w 0 u ,H0 .2 v0u w 0 u ,H0 2 0I 0 I'' 0 I 0 IHH 0 III 0 IJ J 0 0J 0 J '' 0 J 0 J HH 0 J II 0 J J J 0 0HH 0 H'H' 0 HH 0 HHHH 0 HIHI 0 HJ HJ .R HH 0 HIHI 0 HJ HJ . y . v u w9 . 2 0 0 9,H0 0u u0 . 2 0II 0 I'I' 0 II 0 IHIH 0 IIII 0 IJ IJ .0 II 0 HIHI 0 IJ IJ 0y0 . 0 0 9,H0 0u u0 109 v0u w909 . 0909 9,H . 0J J 0 J 'J' 0 J J 0 J HJH 0 J IJI 0 J J JJ .0 J J 0 HJ HJ 0 IJ IJ 0y0 . 0 0 9,H0 0u u0 109 v0u w909 . 0909 9,H . 0II


0$% VWWWWXWWWWY y . v u w9 . 2 0y0 .2 v0u w 0 u 0 0 0 . y . v u w9 . 2 0 0 90 0u u0 0 0

0 0 0y0 . 0 0 90 0u u0 109 v0u w909 . 0909 9 00 0 0 0y0 . 0 0 90 0u u0 109 v0u w909 . 0909 9ZWW

WW[WWWW\

Where M refers to column and to row

The Ricci scalar:

0 $% 0$% (4.47) 0 0 HH0HH II0II J J 0J J 0 . 0HH . 0II . 0J J 2 . . 2 . 2. 2 4 . 4 . 2 20 HH 40 II . 40 HIHI . 20 IJ IJ 2 y . v u w9 . 4 0y0 4 0 0 9,H0 0u u0 . 2 109 v0u w909 . 0909 9,H




The Einstein tensor:

2$% 0$% . 12 $% 0 (4.48) 2 0 . 12 0 0 . 12 0 2 . 12 2 4 . 4 . 2 2 20 HIHI 0 IJ IJ .2 0 0 9,H0 0u u0 109 v0u w909 . 0909 9,H 109 1 . 200u u v0u w9 . 200 200 099,H 2H 0H . 12 H0 0H 0 HII 0 HJ J .2 v0u w 0 u ,H0 2I 0I . 12 I0 0 2HH 0HH . 12 HH0 0HH 12 0 . 2 12 2 4 . 4 . 2 2 . 20 II . 0 IJ IJ .2 0y0 . 109 v0u w909 . 0909 9,H 109 099,H . 200y . 1 . v0u w9 2II 0II . 12 II0 0II 12 0 . 12 2 4 . 4 . 2 . R HH 0 II . 0 HIHI y . v u w9 . 0y0 0 0 9,H0 0u u0 y . v u w9 10 0 09,H 0u u . 0y 2J J 2II y . v u w9 10 0 09,H 0u u . 0y


2$% VWWWXWWWY 109 1 . 200u u v0u w9 . 200 200 099 .2 v0u w 0 u 0 0 0 109 099 . 200y . 1 . v0u w9 0 00 0 y . v u w9 10 0 09 0u u . 0y 00 0 0 y . v u w9 10 0 09 0u u .


The Einstein tensor in the coordinate basis:

The transformation: 2$% $' %* 2'* (6.34)

2 ' * 2'* v w92 109 1 . 200u u v0u w9 . 200 200 099 2H H' * 2'* HH 2H .2 v0u w0 0 u0




2HH H' H* 2'* v HH w92HH 109 09 . 200y 1 v0u w9 9 2II I' I* 2'* II 9 2II 09 + y . v u w9 10 0 09 0u u . 0y / 2JJ J ' J* 2'* J J 9 2J J 09 sin9 ; + y . v u w9 10 0 09 0u u . 0y / Which means the answer to quiz 5-7 is (d), almost!


2$% VWWWXWWWY 109 1 . 200u u v0u w9 . 200 200 099 .2 v0u w0 0 u0 0 0

109 09 . 200y 1 v0u w9 9 0 00 0 09 + y . v uw9 10 0 09 0u u . 0y / 00 0 0 09 sin9 ; + y . v u w9 10 0 09 0u u . 0y /ZWWW[WWW\

Where M refers to column and to row Chapter 6: The Einstein Field Equations

P.138: The vacuum Einstein equations

Prove that the Einstein field equations 2$% 5$% reduces to the vacuum Einstein equations 0$% 0 if we set 5$% 0

The Einstein tensor

2$% 0$% . 12 #$%0 (4.48) Now setting 2$% 5$% 0 and calculating 2$% 0$% . 12 #$%0 0 z 0$% 12 #$%0 Multiplying by #$% R #$%0$% 12 #$%#$%0 using the definition

0 #$%0$% (4.47) and that in 4 dimensions #$%#$% 4 R 0 12 40 20 Now this can only be true if 0$% 0 Q.E.D. P.138: The vacuum Einstein equations with a cosmological constant

Prove that the Einstein field equations 2$% 5$% reduces to 0$% #$% and 0 4 for metrics with positive signature and 0$% .#$% and 0 .4 for metrics with negative signature in vacuum with a cosmological constant

7.

The Einstein equation in vacuum with a cosmological constant and positive signature is

7 A excellent qualitative explanation of the cosmological constant, you can find in (Greene, 2004, s. 273-279)




0 0$% . 12 #$%0 #$% (6.6) R 0 #$%0$% . 12 #$%#$%0 #$%#$%

0 . 12 40 4 R 0 4 Q.E.D. Next we rewrite the Einstein equation

0 0$% . 12 #$%0 #$% 0$% . 12 #$%4 #$% 0$% . #$% R 0$% #$% Q.E.D. In the non coordinate basis

0$% $% In the case of metrics with negative signature the Einstein equation in vacuum with a cosmological constant

0 0$% . 12 #$%0 . #$% (6.6) and we can see that

0 .4 Q.E.D. 0$% .#$% Q.E.D. In the non coordinate basis

0$% .$% New - P.138: General remarks on the Einstein equations with a cosmological constant8

If we demand that the gravitational field equations are

(1) generally covariant

(2) be of second differential order in #$% (3) involve the energy-momentum 5$% linearly it can be shown that the only equation which meets these requirements is

0$% 0#$% . #$% 5$% where , , and are constants. The demand that 5$% satisfies the conservation equation a%5$% 0

leads to . 12 Proof:

if a%5$% 0 R a%v0$% 0#$% . #$%w 0 R a%0$% a%v0#$%w . a%#$% 0 R 9 a%0$% a%0#$% 0va%#$%w 0 R a%0$% a%0#$% 0 Next we use the Bianchi identity:

a$0*1%' a%0*1'$ a'0*1$% 0 R #*%a$0*1%' a%0*1'$ a'0*1$% 0 8 (d'Inverno, 1992, p. 172)

9 a%#$% 0




R a$#*%0*1%' a%#*%0*1'$ a'#*%0*1$% 0 R a$0%1%' a%0%1'$ a'0%1$% 0 R a$01' a%0%1'$ . a'01$ 0 R #$1va$01' a%0%1'$ . a'01$w 0 R a$#$101' a%#$10%1'$ . a'#$101$ 0 R 10 a$0$' a%0%' . a'0 0 R 2 +a$0$' . 12 a'0/ 0 R 2#%' +a$0$' . 12 a'0/ 0 R 2 +a$#%'0$' . 12 a'0#%'/ 0 R 2 +a$0$% . 12 a$0#$%/ 0 Now if we compare with

a%0$% a%0#$% 0 we see that

. 12

P.139, example 6-2: Find the components of the curvature tensor for the metric in 2+1 dimen-

sions using Cartans structure equations

The line element: 789 .79 9%,H7:9 09, :7?9

The Basis one forms

7 @.1 1 1 A H %,H7: 7: %,HH

J 0, :7? 7? 10, : J

Cartans First Structure equation and the calculation of the curvature two-forms

7$ . %$ k % (5.9) %$ %'$ ' (5.10)

7 0 7H 7v%,H7:w u %,H7 k 7: u k H 7J 70, :7? 0u 7 k 7? 07: k 7? 0u0 k J 00 %,HH k J


10

#$10 1'$% #$101 $' % 0 $'$% 0 '%




%$ VWWXWWY 0 u H 0u0 Ju H 0 00 %,HJ0u0 J . 00 %,HJ 0 ZWW

[WW\



%$ 7 %$ '$ k % ' 12 0 %'*$ ' k * (5.27), (5.28)

H : 7 H 7vu Hw 7vu %,H7:w y %,H vu w9%,H 7 k 7: . y vu w9 H k 'H k ' H k HH k H JH k J 0 R H . y vu w9 H k J : 7 J 7 {0u0 J | 7v0u , :7?w 0y 7 k 7? v0u w7: k 7? . 0y0 J k . v0u w0 %,HJ k H 'J k ' J k HJ k H JJ k J HJ k H 00 %,HJ k u H R J . 0y0 J k . v0u w0 . 0u0 %,HJ k H HJ : 7 HJ 7 {00 %,HJ | 7v0%,H7?w v0u w%,H . 0u %,H 7 k 7? 0%,H 0%,H7: k 7? . v0u w . 0u %,H0 J k . 0 0 9%,H0 J k H 'J k H ' J k H HJ k H H JJ k H J J k H 0u u0 J k H R HJ . v0u w . 0u %,H0 J k . {0 0 9%,H0 . 0u u0 | J k H


%$ VWXWY 0 . y vu w9 k H . 0y0 J k . v0u w0 . 0u0 %,HJ k H

0 .v0u w . 0u %,H0 J k . {0 0 9%,H0 . 0u u0 | J k H0 ZW[W\





R HH . y vu w9 H k 0 J J . 0y0 0 J HJ (C) . v0u w 0u %,H0 0 HJ HJ (D) . {0 0 9%,H0 . 0u u0 | Where A, B, C and D will be used later to make the calculations easier

P.139, Example 6-2: Find the components of the curvature tensor for the metric in 2+1 dimen-

sions using Cartans structure equations alternative solution

The line element: 789 .79 9%,H7:9 09, :7?9

Now we can compare with the Tolman-Bondi de Sitter line element, where the primes should not be mis-

taken for the derivative 7/7:. 789 79 . 9,H7:9 . 09, :7;9 . 09, : sin9 ; 7?9 And chose: 7 7 ,H7: %,H7: 0, :7; 0 0, : sin ; 7? 0, :7? Comparing the two metrics we see: 7? 7?, ; 9 , 0, : 0, :, 7 7 Next we can use the former calculations of the Tolman-Bondi de Sitter metric to find the Riemann and

Einstein tensor for the 2+1 metric.

But first we need to find u 7, :7 v,Hw 77 vv,Hww %,H 7:7: 77 {%,H 7:7: | . 7, :7 .u , : y 79, :79 77 v.u w .y , : 7, :7: v,Hw 77: vv,Hww v,Hw 7:7: 77: {%,H 7:7: | .v,Hw%,H, : 0u 70, :7 70, :7 0u , : 0y 790, :79 790, :79 0y , : 0 70, :7: 7:7: 70, :7: v,Hw%,H0, : 0u 790, :77: 77: {70, :7 | 7:7: 77: 0u , : v,Hw%,H0u , :

The Riemann tensor




Tolman Bondi de Sitter 2+1 R HH y . v u w9 R R HH . y vu w9 0 II . 0y0 0 IHI . v0u w 0 u ,H0 0 HIHI . 0 0 9,H0 0u u0 0 J J . 0y0 R 0 J J . 0y0 0 J HJ . v0u w 0 u ,H0 R 0 J HJ . v0u w . 0u %,H0 0 HJ HJ . 0 0 9,H0 0u u0 R 0 HJ HJ . 0 . 0 9%,H0 . 0u u0 0 IJ IJ 109 v0u w909 . 0909 9,H Where A, B, C and D will be used later to make the calculations easier

P.147, Example 6-3: Find the components of the Einstein tensor in the coordinate basis for the

metric in 2+1 dimensions.

The Ricci tensor:

0$% 0 $'%' (4.46) 0 0 '' 0 0 HH 0 J J 0 HH 0 JJ . y vu w9 . 0y0 0H 0 H'' 0 H 0 HHH 0 HJ J 0 HJ J . v0u w . 0u %,H0 0J 0 J '' 0 J 0 J HH 0 J II 0 J J J 0 0HH 0 H'H' 0 HH 0 HHHH 0 HJ HJ .R HH 0 HJ HJ y vu w9 . 0 . 0 9%,H0 . 0u u0 . 0J J 0 J 'J' 0 J J 0 J HJH 0 J J JJ .0 J J 0 HJ HJ 0y0 . 0 . 0 9%,H0 . 0u u0 .





0$% VWWXWWY. y vu w9 .

0y0 . v0u w . 0u %0 0 y vu w9 . 0 . 0 9%0 . 0u u0 00 0 0y0 . 0 . 0 9%0 . 0u u0 ZWW

[WW\


The Ricci scalar:

0 $% 0$% (4.47) 0 0 HH0HH J J 0J J .0 0HH 0J J . . . .2 . 2 2 .20 HH . 20 J J 20 HJ HJ 2 y vu w9 2 0y0 . 2 0 . 0 9%,H0 . 0u u0


2$% 0$% . 12 $% 0 (4.48) 2 0 . 12 0 0 12 0 12 .2 . 2 2 0 HJ HJ . 0 . 0 9%0 . 0u u0 2H 0H . 12 H0 0H 0 HJ J . v0u w . 0u %0 2I 0I . 12 I0 0 2HH 0HH . 12 HH0 0HH . 12 0 . . 12 .2 . 2 2 0 J J . 0y0 2J J 0J J . 12 J J 0 0J J . 12 0 . . 12 .2 . 2 2 R HH . y vu w9


2$% VWXWY. 0 . 0 9%0 . 0u u0 . v0u w . 0u %0 0 . 0y0 00 0 . y vu w9ZW

[W\





The Einstein tensor in the coordinate basis:

The transformation: 2$% $' %* 2'* (6.34)

2 ' * 2'* v w92 . 0 . 0 9%0 . 0u u0 2H H' * 2'* HH 2H . v0u w . 0u 0 2HH H' H* 2'* v HH w92HH . 0y0 9% 2JJ J' J* 2'* JJ 9 2J J .09 y vu w9


2$% VWWXWWY. 0 . 0 9%0 . 0u u0 . v0u w

. 0u 0 0 . 0y0 9% 00 0 .09 y vu w9ZWW

[WW\

Where M refers to column and to row P.150, Example 6-4: The Einstein equations of the metric in 2+1 dimensions.

Given the Einstein equation ( if 2 1: 2$% $% 5$% (6.40) with .9 you get 2$% . 9$% 5$% and the stress-energy tensor: 5$% @ 0 00 0 00 0 0A

You can find the Einstein equations

VWXWY. 0 . 0 9%0 . 0u u0 . v0u w . 0u %0 0 . 0y0 00 0 . y vu w9ZW

[W\ . 9 .1 1 1 @

0 00 0 00 0 0A

2: . 0 . 0 9%0 . 0u u0 9 p.152 2H: . v0u w . 0u %0 0 z v0u w . 0u 0 p.152 2HH: . 0y0 . 9 0




z 0y 90 0 (6.41) 2J J : . y vu w9 . 9 0 (6.42) P. 152, Quiz 6-1: Using the contracted Bianchi identities, prove that:

Expressions needed:

Bianchi identity: 0 a0*1%' a%0*1'$ a'0*1$% (4.45) Proved page 78: 0 a'#$% 0 a'#$% z #*$#1%a'#$% 0 z a'#*$#1%#$% 0 (S3) z 0 a'#*1 Riemann tensor: 0$%'* .0$%*' (4.44) Ricci tensor: 0 $'%' 0$% (4.46) Ricci scalar: 0 #$%0$% 0 $$ (4.47) #$10 1'$% #$101 $' % 0 $'$% 0 '% (S4) The Einstein tensor: 2$% 0$% . 12 #$%0 (4.48) Kronecker delta # '$ #$%#%' '$ (2.15) a$2 '$ a$ +0 '$ . 12 # '$ 0/ a$0 '$ . 12 a$# '$ 0 use (2.15) a$0 '$ . 12 # '$ a$0 a$0 '$ . 12 '$a$0 a$0 '$ . 12 a'0 (S5)

The proof:

Multiply (4.45) by #*%: 0 #*%a0*1%' a%0*1'$ a'0*1$% z (use (S3)) 0 a#*%0*1%' a%#*%0*1'$ a'#*%0*1$% z 0 a0 1%'% a%0 1'$% a'0 1$%% use 4.44 and 4.46 0 a$01' a%0 1'$% . a'0 1%$% (use (4.46)) 0 a$01' a%0 1'$% . a'01$ Multiply by #$1 0 #$1a$01' a%0 1'$% . a'01$ 0 a$#$101' a%#$10 1'$% . a'#$101$ 0 a$0 '$ a%#$10 1'$% . a'0 $$ (use (4.47)) 0 a$0 '$ a%#$10 1'$% . a'0 (use (S4)) 0 a$0 '$ a%0 '% . a'0 0 2 a$0 '$ . 12 a'0 z (use (S5)) 0 a$2 '$ Multiply by #%' 0 #%'a$2 '$ a$#%'2 '$ a$2$%

Which means the answer to quiz 6-1 is (c) P.153-54, Quiz 6-5, Quiz 6-6, Quiz 6-7 and Quiz 6-8: Ricci rotation coefficients, Ricci scalar and

Einstein equations for a general 4-dimensional metric.

The line element: 789 .79 9(, :)7:9 9(, :)7?9 9(, :)7r9

The Basis one forms

7 P.1 1 1 1 Q H (, :)7: 7: 1(, :) H




J , :7? 7? 1, : J s , :7r 7r 1, : s


7$ . %$ k % (5.9) %$ %'$ ' (5.10)

7 0 7H 7, :7: u 7 k 7: u k H 7J 7, :7? u 7 k 7? 7: k 7? u k J H k J 7s 7, :7r u 7 k 7r 7: k 7r u k s H k s


%$ VWWWXWWWY 0 u H u J u su H 0 J su J . J 0 0u s . s 0 0 ZWW

W[WWW\


Now we can read off the Ricci rotation coefficients HH u HH u JJ u ss u J J u J JH . HJJ Hss ss u ssH . Which means the answer to quiz 6-5 is (a) and quiz 6-6 is (c)


%$ 7 %$ '$ k % ' 12 0 %'*$ ' k * (5.27), (5.28)




H : 7 H 7 {u H| 7 u , : 7: y 7 k 7: u 7: k 7: y k H 'H k ' H k HH k H JH k J sH k s 0 R H . y H k J : 7 J 7 {u J | 7vu , :7?w y 7 k 7? u 7: k 7? y k J u H k J 'J k ' J k HJ k H JJ k J sJ k s HJ k H J k u H R J y k J u H k J J k u H . y J k {u9 . u| J k H s : 7 s 7 {u s| 7vu , :7rw y 7 k 7r u 7: k 7r y k s u H k s 's k ' s k Hs k H Js k J ss k s Hs k H s k u H s y k s u H k s s k u H . y s k {u9 . u | s k H HJ : 7 HJ 7 + J / 7 + 7?/ u . u9 7 k 7? . 9 7: k 7? u . u 9 J k . J k H 'J k H' HJ k HH J k H JJ k HJ sJ k Hs J k H u u J k H R HJ u . u 9 J k . J k H u u J k H u . u 9 J k { . u u| J k H Hs : 7 Hs 7 + s/ 7 + 7r/ u . u9 7 k 7r . 9 7: k 7r u . u 9 s k . s k H 's k H' Hs k HH s k H Js k HJ ss k Hs s k H u u s k H R Hs u . u 9 s k . s k H u u s k H u . u 9 s k { . u u| s k H Js : 7 Js 0 R Js 's k J' s k J Hs k JH Js k JJ ss k Js s k J Hs k JH u s k u J s k {. J |




{u u . 9| s k J


%$ VWWXWWY0 . y H k . y J k {u9 . u| J k H . y s k {u9 . u | s k H

0 {u9 . u| J k { . u u| J k H {u9 . u | s k { . u u| s k H 0 {u u . 9| s k J 0 ZW

W[WW\


R HH . y 0 J J . y 0 ss . y 0 J HJ u9 . u 0 sHs u9 . u 0 HJ HJ . u u 0 HsHs 2 . u u 0 J sJs u u . 9 Where A,B,C,D,E,F,G,H will be used later, to make the calculations easier

The Ricci tensor:

0$% 0 $'%' (4.46) 0 0 '' 0 0 HH 0 J J 0 ss 0 HH 0 J J 0 ss . y . y . y 0H 0 H'' 0 H 0 HHH 0 HJ J 0 Hss 0 HJ J 0 Hss u9 . u u9 . u 0J 0 J '' 0 J 0 J HH 0 J J J 0 J ss 0 0s 0 s'' 0 s 0 sHH 0 sJ J 0 sss 0 0HH 0 H'H' 0 HH 0 HHHH 0 HJ HJ 0 HsHs .R HH 0 HJ HJ 0 HsHs y . u u . u u . 2 0J J 0 J 'J' 0 J J 0 J HJH 0 J J JJ 0 J sJs .0 J J 0 HJ HJ 0 J sJs y . u u u u . 9 . 0ss 0 s's' 0 ss 0 sHsH 0 sJ sJ 0 ssss .0 ss 0 HsHs 0 J sJs y . u u u u . 9 . 2





0$% VWWWXWWWY. y . y . y u9 . u u9 . u 0 0 y . u u . u u 0 00 0 y . u u u u . 9 00 0 0 y . u u u u . 9ZWW

W[WWW\


The Ricci scalar:

0 $% 0$% (4.47) 0 0 HH0HH II0II J J 0J J .0 0HH 0J J 0ss . . 2 . . 2 .2 . 2 . 2 2 22 2 .2R HH . 20 J J . 20 ss 20 HJ HJ 20 HsHs 20 J sJs 2 {y y y . u u . u u u u . 9| And the answer to quiz 6-8 is (a)


2$% 0$% . 12 $% 0 (4.48) 2 0 . 12 0 0 12 0 12 .2 . 2 . 2 2 22 2 2 0 HJ HJ 0 HsHs 0 J sJs . u u . u u u u . 9 2H 0H . 12 H0 0H u9 . u u9 . u 2J 0J . 12 J 0 0 2s 0s . 12 s0 0 2HH 0HH . 12 HH0 0HH . 12 0 . 2 . 12 .2 . 2 . 2 2 22 2 0 J J 0 ss 0 J sJs . y . y u u . 9 2J H 0J H . 12 J H0 0 2sH 0sH . 12 sH0 0 2J J 0J J . 12 J J 0 0J J . 12 0 . . 12 .2 . 2 . 2 2 22 2




2 R HH 0 ss 0 HsHs . y . y . u u 2sJ 0sJ . 12 sJ 0 0 2ss 0ss . 12 ss0 0ss . 12 0 . 2 . 12 .2 . 2 . 2 2 22 2 R HH 0 J J 0 HJ HJ . y . y . u u And the answer to quiz 6-7 is (a)


2$% VWWWXWWWY . u u . u u u u . 9 u9 . u u9 . u 0 0 . y . y u u . 9 0 00 0 . y . y . u u 00 0 0 . y . y . u uZWW

W[WWW\

Where M refers to column and to row Chapter 7: The Energy-Momentum Tensor

P.160: Perfect Fluids Alternative derivation

The most general form of the stress energy tensor is 5$% S$S% #$% (7.8) In the local frame we know that

5$% P 0 0 00 0 00 0 00 0 0 Q (7.6) and S$ 1,0,0,0 Then we choose the metric with negative signature

$% P1 0 0 00 .1 0 00 0 .1 00 0 0 .1Q

This we can use to find the constants and 5 SS 5 SS . 0 n n R . and . Which leaves us with the most general form of the stress energy tensor for a perfect fluid for a metric with

negative signature 5$% S$S% . #$% (7.11) If we instead choose the metric with positive signature

$% P.1 0 0 00 1 0 00 0 1 00 0 0 1Q




5 SS . 5 SS 0 n n R and Which leaves us with the most general form of the stress energy tensor for a perfect fluid for a metric with

negative signature 5$% S$S% #$% (7.12) P.161, Example 7-2: The Einstein tensor and Friedmann-equations for the Robertson Walker

metric

The Ricci scalar:

0 $% 0$% .0 0HH 0II 0J J .0 '' 0 H'H' 0 I'I' 0 J 'J' .0 HH . 0 II . 0 J J 0 HH 0 HIHI 0 HJ HJ 0 II 0 IHIH 0 IJ IJ 0 J J 0 J HJH 0 J IJI .20 HH . 20 II . 20 J J 20 HIHI 20 HJ HJ 20 IJ IJ 2 MyM 2 MyM 2 MyM 2 Mu 9 M9 2 Mu 9 M9 2 Mu 9 M9 {aya Mu 9 ka9 |


2$% 0$% . 12 $% 0 (4.48) 2 0 . 12 0 0 HH 0 II 0 J J 12 .20 HH . 20 II . 20 J J 20 HIHI 20 HJ HJ 20 IJ IJ 0 HIHI 0 HJ HJ 0 IJ IJ 3 {Mu 9 ka9 | 2H 0H . 12 H0 0 H'' 0 2I 2J 2H 2IH 2J H 2I 2HI 2J I 2J 2HJ 2IJ 0 2HH 0HH . 12 HH0 0 HH 0 HIHI 0 HJ HJ . 12 .20 HH . 20 II . 20 J J 20 HIHI 20 HJ HJ 20 IJ IJ 0 HH 0 HIHI 0 HJ HJ 0 HH 0 II 0 J J . 0 HIHI . 0 HJ HJ . 0 IJ IJ 0 II 0 J J . 0 IJ IJ . MyM . MyM . Mu 9 M9 . {2MyM Mu 9 M9 | 2II 0II . 12 II0 0 II 0 IHIH 0 IJ IJ . 12 .20 HH . 20 II . 20 J J 20 HIHI 20 HJ HJ 20 IJ IJ 0 II 0 IHIH 0 IJ IJ 0 HH 0 II 0 J J . 0 HIHI . 0 HJ HJ . 0 IJ IJ




0 HH 0 J J . 0 HJ HJ . MyM . MyM . Mu 9 M9 . {2MyM Mu 9 M9 | 2J J 0J J . 12 J J 0 0 J J 0 J HJH 0 J IJI . 12 .20 HH . 20 II . 20 J J 20 HIHI 20 HJ HJ 20 IJ IJ 0 J J 0 J HJH 0 J IJI 0 HH 0 II 0 J J . 0 HIHI . 0 HJ HJ . 0 IJ IJ 0 HH 0 II . 0 HIHI . MyM . MyM . Mu 9 M9 . {2MyM Mu 9 M9 |


2$% VWWWXWWWY3 {Mu 9 ka9 | 0 0 0

0 . {2MyM Mu 9 M9 | 0 00 0 . {2MyM Mu 9 M9 | 00 0 0 . {2MyM Mu 9 M9 |ZWW

W[WWW\


The Friedmann equations:

Given the Einstein equation ( if 2 1: 2$% . $% 845$% (7.14) and the stress-energy tensor: 5$% 84 P 0 0 00 0 00 0 00 0 0 Q (7.16)

You can find the Friedmann- equations

VWWWXWWWY3 {Mu 9 ka9 | 0 0 0

0 . {2MyM Mu 9 M9 | 0 00 0 . {2MyM Mu 9 M9 | 00 0 0 . {2MyM Mu 9 M9 |ZWW

W[WWW\

. P.1 1 1 1 Q 84 P 0 0 00 0 00 0 00 0 0 Q

R 3M9 Mu 9 84 (7.17) 2 MyM 1M9 Mu 9 .84 (8.18)




P.161, Example 7-2: The Einstein tensor for the Robertson Walker metric Alternative version.

The line element: 789 .79 M91 . :9 7:9 M9:97;9 M9:9 sin9 ; 7?9

Now we can compare with the Tolman-Bondi de Sitter line element, where the primes should not be mis-

taken for the derivative 7/7:. 789 79 . 9,H7:9 . 09, :7;9 . 09, : sin9 ; 7?9 And chose: 7 7 v,Hw7: M1 . :9 7: 0, :7; M:7; 0, : sin ; 7? M: sin ; 7? Comparing the two metrics we see: 7? 7?, 7; 7;, ; ;, 0, : M:, 7 7 Next we can use the former calculations of the Tolman-Bondi de Sitter metric to find the Einstein tensor

for the Robertson-Walker metric.

But first we need to find u 7, :7 v,Hw 77 vv,Hww M1 . :9 7:7: 77 {1 . :9M 7:7: | . MuM y 77 {. Mu M| 77 {. Mu M| . vMy M . Mu Mu wM9 {Mu M|9 . MyM 7, :7: v,Hw 77: vv,Hww v,Hw 7:7: 77: {1 . :9M 7:7: | . :M1 . :9 v,Hw 0u 70, :7 7M:7 Mu : 0y 7Mu :7 7Mu :7 My : 0 70, :7: 7:7: 77: M: 1 . :9M v,HwM 1 . :9v,Hw 0 77: 1 . :9v,Hw 7:7: 77: {1 . :9 M1 . :9 7:7:| 0 0u 77: Mu : 7:7: 77: Mu : Mu 1 . :9M v,Hw


Tolman Bondi de Sitter Robertson-Walker

2 109 1 . 200u u v0u w9. 200 200 099,H R 2 3 Mu 9 M9




2H .2 v0u w 0 u ,H0 R 2H 0 2HH 109 099,H . 200y . 1 . v0u w9 R 2HH . {2 My M Mu 9 M9 | 2II y . v u w9 10 0 09,H 0u u . 0y R 2II . {2 My M Mu 9 M9 | 2J J y . v u w9 10 0 09,H 0u u . 0y R 2J J . {2 My M Mu 9 M9 | P. 165, Quiz 7-3: Manipulating the Friedmann equations.

Show that the two Friedman equations 3M9 Mu 9 84 (7.17) 2 MyM 1M9 Mu 9 .84 (8.18) can be manipulated into: 77 M 77 M 0

Rewriting (7.17):

84 3M9 Mu 9 R 84M 3M Mu 9 a R 84 77 M 77 3M Mu 9 a 3Mu Mu 9 6MMuMy 3M9Mu Rewriting (7.18):

.84 2 MyM 1M9 Mu 9 R .84 77 M +2 MyM 1M9 Mu 9 / 77 M +2 MyM 1M9 Mu 9 / 3M9Mu 6MMuMy 3Mu Mu 9 3M9Mu 84 77 M .6MMuMy 3Mu Mu 9 3M9Mu

Now adding

84 77 M 84 77 M 3Mu Mu 9 6MMuMy 3M9Mu . 6MMuMy 3Mu Mu 9 3M9Mu 0 Q.E.D. Chapter 8: Killing Vectors

P.168, Example 8-1: Show that if the Lie derivative of the metric tensor with respect to vector X

vanishes ( ), the vector X satisfies the Killing equation. - Alternative version Expressions needed:

Killings equation: aB$ a$B% 0 (8.1) The Lie derivative of a (0,2) tensor: 5$% 'a'5$% 5'%a$' 5$'a%' (4.28) a'#$% 0 (4.18)

Now we can Calculate the Lie-derivative of #$% (4.28) and using (4.18) #$% B'a'#$% #'%a$B' #$'a%B'




a$#'%B' a%#$'B' a$B% a%B$ If #$% 0 this implies that a$B% a%B$ 0, which is the Killing equation. P.177, equation (8.7): Prove that We have

a'aB$ 0 %'*$ B* (8.6) Contracting with $': R $'a'aB$ ) $'v0 %'*$ B*w R aaB$ 0 %$*$ B* R aaB$ 0%*B* (8.7) P.178: Constructing a Conserved Current with Killing Vectors Alternative version:

We write a$$ 5$%a$B% and a% 5%$aB$ Now 12 a$$ 12 a% 9(5$%a$B% 5%$aB$ 5$% is symmetric so 5$% 5%$ p.155 12 5$%va$B% aB$w Killings equation a$B% aB$ 0 (8.1) 0 P.179, Quiz 8-3: Given a Killing vector the Ricci scalar satisfies : We calculate the Lie derivative:

0$% B'a'0$% 0'%a$B' 0$'a%B' (4.28) Multiplikation by #$%: #$%0$% #$%B'a'0$% #$%0'%a$B' #$%0$'a%B' Using a'#$% 0, and #$% 0: z #$%0$% B'a'v#$%0$%w #$%0'%a$B' #$%0$'a%B' z 0 B'a'0 #$%0'%a$B' #$%0$'a%B' B'a'0 0' $a$B' 0 '% a%B' B'a'0 #%'0%$a$B' #$'0%$a%B' B'a'0 0%$a$#%'B' 0%$a%#$'B' B'a'0 0%$a$B% 0%$a%B$ B'a'0 0%$a$B% a%B$ B'a'0 Now the Lie derivative of a scalar is zero 0 0, so B'a'0 0 And the answer to Quiz 8-3 is (a)

Chapter 9: Null Tetrads and the Petrov Classification

P.186, Example 9-3, and 9-4: Construct a null tetrad for the flat space Minkowski metric

The line element: 789 79 . 7:9 . :97;9 . :9 sin9 ; 7?9 The metric tensor: #$% P 1 .1 .:9 .:9 sin9 ;Q




and its inverse: #$% VWXWY 1 .1 . 1:9 . 1:9 sin9 ;ZW

[W\

The basis one forms

7 H 7: I :7; J : sin ; 7?

The null tetrad

Now we can use the basis one-forms to construct a null tetrad

12

1 1 0 01 .1 0 00 0 1 0 0 1 . HIJ

12 7 7:7 . 7::7; : sin ; 7?:7; . : sin ; 7? (9.10)

Written in terms of the coordinate basis $ 12 1, 1, 0, 0 $ 12 1, .1, 0, 0 $ 12 0, 0, :, : sin ; $ 12 0, 0, :, .: sin ; Next we use the metric to rise the indices #$$ # 1 12 12 H #$H$ #HHH .1 12 . 12 I J 0 #$$ # 1 12 12 H #$H$ #HHH .1 +. 12/ 12 I J 0 H 0 I #$I$ #III +. 1:9/ :2 . 1:2 J #$J$ #JJJ +. 1:9 sin9 ;/ : sin ;2 . 1: sin ; 2

Collecting the results




$ 12 1, 1, 0, 0 $ 12 1, .1, 0, 0 $ 12 1, .1, 0, 0 $ 12 1, 1, 0, 0 $ 12 0, 0, :, : sin ; $ 12 +0, 0, . 1: , . : sin ;/ $ 12 0, 0, :, .: sin ; $ 12 +0, 0, . 1: , : sin ;/ P.195, Example 9-5: The Brinkmann metric

The line element: 789 S, -, U7S9 27S7T . 7-9 . 7U9 The metric tensor: #$% P 11 .1 .1Q and its inverse: #$% P 11 . .1 .1Q

The basis one forms

Finding the basis one forms is not so obvious, we write:

789 S, -, U7S9 27S7T . 7-9 . 7U9 v]w9 . v_ w9 . v ^w9 . v ` w9 R 7SS, -, U7S 27T . 7-9 . 7U9 v] _ wv] . _ w . v ^w9 . v ` w9 R ] _ 7S ] . _ 7S 27T ^ 7- ` 7U

R ] 12 17S 7T 7S ] _ P 1 .1 .1 .1Q _ 12 1 . 7S . 7T 7T 12 1 . ] . 12 1 _

^ 7- 7- ^ ` 7U 7U `

The orthonormal null tetrad

Now we can use the basis one-forms to construct a orthonormal null tetrad

12

1 1 0 01 .1 0 00 0 1 0 0 1 . ]_ ^ `

12 ] _] . _ ^ ` ^ . `

12 7S7S 27T7- 7U7- . 7U (9.10)

Written in terms of the coordinate basis




$ 12 1, 0, 0, 0 $ 12 , 2, 0, 0 $ 12 0, 0, 1, $ 12 0, 0, 1, . Next we use the metric to rise the indices ] #$]$ #_]_ 1 0 0 _ #$_$ #]_] #___ 1 + 12/ . 0 12 ^ ` 0 ] #$]$ #_]_ 1 + 22/ 2 _ #$_$ #]_] #___ 1 + 12 / . + 22/ . 12 ^ ` 0 ] _ 0 ^ #$^$ #^^^ .1 12 . 12 ` #$`$ #``` .1 12 . 12

Collecting the results $ 12 1, 0, 0, 0 $ 12 0, 1, 0, 0 $ 12 , 2, 0, 0 $ 12 2, ., 0, 0 $ 12 0, 0, 1, $ 12 0, 0, .1, . $ 12 0, 0, 1, . $ 12 0, 0, .1,

The non-zero Christoffel symbols

$%' 12 ,'#$% ,%#$' . ,$#%' (4.15) %'$ #$**%' (4.16) ]]] 12 ,,S R ]]_ #_]]]] 12 ,,S ]]^ ]^] 12 ,,- R ]^_ ^]_ #_]]]^ 12 ,,- ^ ]] . 12 ,,- R ]]^ #^^^ ]] 12 ,,- ]]` ]`] 12 ,,U R ]`_

lotsofcalculationsp.1326

Documents