research article study of antigravity in an () model and...

Research ArticleStudy of Antigravity in an 119865(119877)Model and in Brans-DickeTheory with Cosmological Constant

V K Oikonomou1 and N Karagiannakis2

1 Department of Theoretical Physics Aristotle University of Thessaloniki 54124 Thessaloniki Greece2 Polytechnic School Aristotle University of Thessaloniki 54124 Thessaloniki Greece

Correspondence should be addressed to V K Oikonomou voikophysicsauthgr

Received 23 May 2014 Revised 23 July 2014 Accepted 24 July 2014 Published 17 August 2014

Academic Editor Cosimo Bambi

Copyright copy 2014 V K Oikonomou and N Karagiannakis This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We study antigravity that is having an effective gravitational constant with a negative sign in scalar-tensor theories originatingfrom 119865(119877) theory and in a Brans-Dicke model with cosmological constant For the 119865(119877) theory case we obtain the antigravityscalar-tensor theory in the Jordan frame by using a variant of the Lagrange multipliers method and we numerically study the timedependent effective gravitational constant As we will demonstrate by using a specific 119865(119877)model although there is no antigravityin the initial model it might occur or not in the scalar-tensor counterpart mainly depending on the parameter that characterizesantigravity Similar results hold true in the Brans-Dicke model

1 Introduction

During the last two decades our perception about theuniverse has changed drastically owing to the discoveredlate time acceleration that our universe has Particularly itcan be thought as one of the most striking astrophysicalobservations with another striking observation being theverification of the inflating period of our universe Actuallymoving from time zero to present time inflation came firstwith the late time acceleration occurring at present epochOne of the greater challenges in cosmology is to model thislate time acceleration in a self-consistent way According tothe new Planck telescope observational data for the presentepoch the universe is consistently described by the ΛCDMmodel according to which the universe is nearly spatially flatand consists of ordinary matter (sim49) cold dark matter(sim268) and dark energy (sim683) The dark energy isactually responsible for late time acceleration and currentresearch on the field is mostly focused on this issue

One of the most promising and theoretically appealingdescriptions of dark energy and late time acceleration issues isprovided by the 119865(119877)modified theories of gravity and relatedmodifications For important review articles and papers onthe vast issue of 119865(119877) theories the reader is referred to

[1ndash19] and references therein For some alternative theories tomodified gravity that model dark energy see [5 20ndash24] Themost appealing characteristic of modified gravity theoriesis that what is actually changed is not the left hand sideof the Einstein equations but the right hand side Latetime acceleration then requires a negative 119908 fluid whichcan be consistently incorporated in the energy momentumtensor of these theories This feature naturally appears in119865(119877) theories and also late time acceleration solutions ofthe Friedmann-Robertson-Walker equations naturally occurin these theoretical frameworks [1ndash19 25] In additioninflation the first accelerating period of our universe is alsoconsistently described by some 119865(119877) theories rendering thelatter a very elegant and economic description of natureat large scales where general relativity fails to describephenomena consistently Particularly the possibility to the-oretically describe in a consistent and elegant way early-time inflation and late-time acceleration in 119865(119877) gravity wasexplicitly demonstrated in the Nojiri-Odintsov model [25]For studies on specific solutions in several strong curvedbackgrounds see [26ndash31] Remarkable possibilities likemodified gravitational theories with nonminimal curvature-matter couplingwere given in [32ndash35] and references therein

Hindawi Publishing CorporationJournal of GravityVolume 2014 Article ID 625836 10 pageshttpdxdoiorg1011552014625836

2 Journal of Gravity

In principle every consistent generalization of generalrelativity inevitably has to be confronted with the successesof general relativity Since general relativity is a successfuldescription of nature in strong gravitational environmentsthere exist a large number of constraints that need to besatisfied in order that an 119865(119877)modified gravity theory can beconsidered as viableThe constraints to be satisfied aremainlyimposed from local tests of general relativity for examplefrom planetary and star formation tests and moreover fromvarious cosmological bounds In addition since each 119865(119877)

theory has a Jordan frame scalar-tensor gravitational theorycounterpart with 120596 zero and a potential the scalarons ofthis counterpart theory must be classical in order to ensurequantum-mechanical stability (see [1ndash6])

In theories of modified gravity a longstanding debatabletheoretical problem exists related to Jordan and Einsteinframes [36 37] since the physics coming out from thetwo frames can be quite different in principle In view ofthis we will focus on the physics of Jordan frame anddemonstrate that it is possible to have antigravity [38ndash41]For the possibility of antigravity regimes in scalar-tensortheories consult [38ndash40] and for antigravity in 119865(119877) theoriessee [41] In this paper we will study antigravity regimescoming from 119865(119877) theories and from Brans-Dicke theoriesin the Jordan frame In reference to the 119865(119877) theories we willfind the Jordan frame antigravity scalar-tensor counterpartusing a modified method of the Lagrange multipliers aswe will see in the following sections The interesting featureabout these theories is that although the 119865(119877) theory has noantigravity the resulting Jordan frame scalar-tensor theorymay or may not have antigravity We exemplify this bynumerically working out an example In the case of Brans-Dicke antigravity we introduce by hand an antigravity termand numerically solve the cosmological equations and as wewill demonstrate similar results hold true that is antigravitymay exist or not depending on the parameters of the theory

This paper is organized as follows in Section 2 we brieflyrecall the essentials of 119865(119877) theories in Section 3 we getto the core of the paper and introduce a modification ofthe Lagrange multipliers method in order to get antigravityfrom 119865(119877) theories Accordingly we apply the technique toone quite known 119865(119877) model and present the result of ouranalysis The study of antigravity is performed in Section 4and the conclusions follow in the end of the paper

2 General Features of119865(119877) Dark Energy Models in the JordanFrame

In this section in order to maintain the paper self-containedwe briefly review themain features of 119865(119877) gravity theories inthe Jordan frame in the theoretical framework of the metricformalism For an important stream of review papers andarticles see [1ndash19] and references therein

The geometrical background of the manifolds used hereis pseudo-Riemannian and is described locally by a Lorentzmetric (the FRWmetric in our case) in addition to a torsion-less symmetric and metric compatible affine connection

the so-called Levi-Civita connection In such a geometricbackground the Christoffel symbols are

Γ119896

120583] =1

2119892119896120582(120597120583119892120582] + 120597]119892120582120583 minus 120597120582119892120583]) (1)

and the Ricci scalar becomes

119877 = 119892120583](120597120582Γ120582

120583] minus 120597]Γ120588

120583120588minus Γ120590

120590]Γ120590

120583120582+ Γ120588

120583120588119892120583]Γ120590

120583]) (2)

The 119865(119877) theories of modified gravity are described by amodification of the Einstein-Hilbert action with the four-dimensional action being equal to

S =1

21205812int1198894119909radicminus119892119865 (119877) + 119878119898 (119892120583] Ψ119898) (3)

where 1205812= 8120587119866 and 119878119898 is the matter action containing

the matter fields Ψ119898 For simplicity in this section it will beassumed that the form of the 119865(119877) theory that will be usedis 119865(119877) = 119877 + 119891(119877) and in addition the metric formalismframework will be used Varying the action (3) with respectto the metric 119892120583] we get the following equations of motion

1198651015840(119877) 119877120583] (119892) minus

1

2119865 (119877) 119892120583] minus nabla120583nabla]119865

1015840(119877)

+ 119892120583]◻1198651015840(119877) = 120581

2119879120583]

(4)

In the above equation 1198651015840(119877) = 120597119865(119877)120597119877 and also 119879120583] is theenergy momentum tensor

The most striking feature of the 119865(119877) modified gravitytheories is that what actually changes in reference to the usualEinstein-Hilbert gravity equations is the right hand side ofthe Einstein equations and not the left which remains thesame Indeed the equations of motion (4) can be cast in thefollowing form

119877120583] minus1

2119877119892120583]

=1205812

1198651015840 (119877)(119879120583] +

1

120581[119865 (119877) minus 119877119865

1015840(119877)

2119892120583] + nabla120583nabla]119865

1015840(119877)

minus 119892120583]◻1198651015840(119877) ])

(5)

Therefore we get an additional contribution for the energymomentum tensor coming from the term

119879eff120583] =

1

120581[119865 (119877) minus 119877119865

1015840(119877)

2119892120583]

+ nabla120583nabla]1198651015840(119877) minus 119892120583]◻119865

1015840(119877) ]

(6)

It is this term that actually models the dark energy in 119865(119877)

theories of modified gravity Taking the trace of (4) westraightforwardly obtain the following equation

3◻1198651015840(119877) + 119877119865

1015840(119877) minus 2119865 (119877) = 120581

2119879 (7)

Journal of Gravity 3

where 119879 stands for the trace of the energy momentum tensor119879 = 119892

120583]119879120583] = minus120588+3119875 and additionally 120588 and 119875 stand for the

matter energy density and pressure respectivelyThere exists another degree of freedom in 119865(119877) theories

as can be easily seen by observing (7) This degree of free-dom is actually a scalar degree of freedom called scalarondescribed by the function 1198651015840(119877) with (7) being the equationof motion of this scalar field In a flat Friedmann-Lemaitre-Robertson-Walker spacetime the Ricci scalar is equal to

119877 = 6 (21198672+ ) (8)

with119867 being the Hubble parameter and the ldquodotrdquo indicatingdifferentiation with respect to time The cosmological equa-tions of motion are given by the following set of equations

31198651015840(119877)119867

2= 1205812(120588119898 + 120588119903)

+

(1198651015840(119877) 119877 minus 119865 (119877))

2minus 3119867

1015840(119877)

(9a)

minus21198651015840(119877) = 120581

2(119901119898 +

4

3120588119903) + 119865

1015840(119877) minus 119867

1015840(119877) (9b)

with 120588119903 and 120588119898 standing for the radiation and matter energydensity respectively Thereby the total effective energy den-sity and pressure of matter and geometry are [1ndash6]

120588eff =1

1198651015840 (119877)[120588119898 +

1

1205812(1198651015840(119877) 119877 minus 119865 (119877) minus 6119867

1015840(119877))]

(10a)

119901eff =1

1198651015840 (119877)[119901119898 +

1

1205812( minus 1198651015840(119877) 119877 + 119865 (119877)

+ 41198671015840(119877) + 2

1015840(119877)) ]

(10b)

where 120588119898 119875119898 denote the total matter energy density andmatter pressure respectively

3 Antigravity in 119865(119877) Models

The possibility of antigravity sectors in 119865(119877) theories wasfirstly pointed out in [41] and also in various scalar-tensormodels in [38ndash40] In most cases a passing from antigravityto a gravity regime always occurs with a singularity existingat the transition between these two different gravitationalregimes At the transition the effective gravitational constantand also several invariants of the geometry such as the Weylinvariant become singular quantities [38ndash41] In the presentpaper we are interested in studying the time dependence ofthe effective gravitational constant and see how this behavesfor both an 119865(119877) theory related antigravity scalar-tensormodel and an antigravity version of the Brans-Dicke modelwith cosmological constant In reference to 119865(119877) theorieswe will explicitly demonstrate in the next subsection howto find the antigravity scalar-tensor theory in the Jordanframe By doing so we will have at hand an antigravity scalar-tensor theory with a potential term and we will explicitly find

how the scalar field along with the gravitational constantand the energy density behaves for various values of themodel dependent and cosmological variables Then we studythe Brans-Dicke model in which we will make a by handmodification in order to render it an antigravity modelAs we will see in both cases there exist several gravity-antigravity regimes depending on the values of the modeldependent and cosmological variables Moreover for the119865(119877)model although themodel per se has no antigravity thecorresponding scalar-tensor model gives rise to antigravityregimes However there exist values of the variables forwhich the model describes gravity regimes In the followingsubsections we will study in detail these models

31 A General Way to Obtain Antigravity Scalar-TensorModels from 119865(119877) Models It is a quite well-known fact thatscalar-tensor theories are equivalent to 119865(119877) theories Inthe literature one starts from an 119865(119877) theory and ends upto a nonminimally coupled scalar-tensor theory and morespecifically to a Brans-Dicke theory with 120596BD equal to zeroThis is practically the Lagrange multipliers method (see [1ndash6]and particularly Nojiri and Odintsov [1] and De Felice andTsujikawa [2])

In this paper we will use a variant but quite similarmethod to obtain an antigravity scalar-tensor theory startingfrom a given 119865(119877) theory Consider the general 119865(119877) theorywith matter which is described by the action

S = int1198894119909radicminus119892119865 (119877) + 119878119898 (119892120583] Ψ119898) (11)

Introducing an auxiliary field 120594 which acts as a Lagrangemultiplier the action (11) becomes

S = int1198894119909radicminus119892 (119865 (120594) + 119865120594 (120594) (119877 minus 120594))

+ 119878119898 (119892120583] Ψ119898)

(12)

with119865120594(120594) being the first derivative of the function119865(120594)withrespect to 120594 By varying the action (12) with respect to 120594 weobtain

119865120594120594 (120594) (119877 minus 120594) = 0 (13)Given that 119865120594120594(120594) = 0 which is actually true for most viable119865(119877) theories wemay conclude that119877 = 120594 Hence the action(12) actually recovers the initial 119865(119877) gravity action (11) Wedefine

120593 minusB = 119865120594 (120594) (14)and the action of (12) is expressed as a function of the field 120593in the following way

S = int1198894119909radicminus119892 [(120593 minusB) 119877 minus 119880 (120593)] + 119878119898 (119892120583] Ψ119898) (15)

Comparing the nonminimal coupling term (120593 minusB)119877 to thecorresponding term (116120587119866)119877 of the standard Einstein-Hilbert action we get the relation for the effective gravita-tional constant

119866eff =1

16120587 (120593 minusB) (16)


It is easy to see that if 120593(119905) minus B = 119865120594(120594) lt 0 there emergesantigravity The potential term 119880(120593) is equal to

119880 (120593) = 120594 (120593) (120593 minusB) minus 119865 (120594 (120593)) (17)

where the function 120594(120593) is directly obtained by solving thealgebraic equation (14) with respect to 120594 so that 120594 is anexplicit function of 120593 Therefore as result starting from an119865(119877) theory and using the technique we just presented oneobtains Jordan frame antigravity scalar-tensor theories

32 The Model 119865(119877)=119877minus119877minus119901 with 119901 a Positive Integer As anapplication of themethodwe just presented let us use a viable119865(119877) model a modified version of which is quite frequentlyused in 119865(119877) cosmology [1ndash6] The model has the followingform as a function of the curvature scalar 119877

119865 (119877) = 119877 minus 119877minus119901 (18)

with 119901 being some positive integer number This form of the119865(119877) function ensures that the first derivative of the 119865(119877)

functionwith respect to119877 is positive definite for119877 ge 119877119863 with119877119863 being the final de-Sitter attractor solution of the theorythat is

119889119865 (119877)

119889119877gt 0 (19)

Condition (19) assures that no antigravity occurs for the 119865(119877)model [1ndash6] However as we will demonstrate antigravitymight occur in the Jordan frame scalar-tensor model Theaction corresponding to the 119865(119877) action (18) is the following

S = int1198894119909radicminus119892 (119877 minus 119877

minus119901) + 119878119898 (119892120583] Ψ119898) (20)

Using the Lagrange multipliers method we introduced in theprevious section we obtain the corresponding scalar-tensorantigravity theory with the Jordan frame action being equalto

S = int1198894119909radicminus119892 [(120593 minusB) 119877 minus 119880119865(119877) (120593)] + 119878119898 (119892120583] Ψ119898)

(21)

The potential 119880119865(119877)(120593) for the present 119865(119877)model is equal to

119880119865(119877) (120593) = (119901

120593 minusB minus 1)

1(119901+1)

(120593 minusB)

minus (119901


1(119901+1)

+ (119901


minus(119901(119901+1))

(22)

Having action (21) at hand along with potential term (22) wecan study the antigravity scalar-tensor model in a straight-forward way By varying action (21) with respect to themetric and the scalar field we get the Einstein equations that

describe the cosmic evolution of the antigravity 119865(119877)-relatedscalar-tensor model Assuming a flat FRWmetric of the form

1198891199042= minus119889119905

2+ 1198862(119905)sum

119894

1198891199092

119894 (23)

the cosmological equations are equal to

3 (120593 minusB)1198672= 120588 + 119880119865(119877) (120593) minus 3119867 (24a)

minus2 (120593 minusB) = 120588 + 119875 + minus 119867 (24b)

minus119877 + 2119889119880119865(119877) (120593)

119889120593= 0 (24c)

where denotes differentiation of the scalar field function120593(119905) with respect to the time variable 119905

=119889120593

119889119905 (25)

In addition 119875 = 119908120588 and also the continuity equation formatter stemming from 119879

120583]120583

= 0 holds true

120588 + 3119867 (1 + 119908) 120588 = 0 (26)

From (21) it easily follows that the effective gravitationalconstant of the Jordan frame scalar-tensor theory is equal to

119866eff (119905) =1

16120587 (120593 (119905) minusB) (27)

We numerically solved the cosmological equations(24a) (24b) and (24c) and in Figures 1 and 2 we present theresults which we will now analyze in detail As a generalcomment let us note that depending on the value of theantigravity parameter B the Jordan frame scalar-tensortheory may or may not have antigravity Therefore althoughwe started with an 119865(119877) theory with no antigravity solutionsthe Jordan frame counterpart exhibits antigravity for somevalues of the parameter B In order for the time dependentfunctions 120593 120588 and 119866eff to vary smoothly we chose the initialconditions to be

120588 (1) = 1 120593 (1) = 1 (1) = 0 119905 times 119867 (1) sim 1

(28)

which are similar to those used in [42] (check also [37]) Wealso performed the following rescaling for time

119905 = 1 997888rarr 10minus46sec (29)

in favor of the simplicity of the plots The above initialconditions and time scaling are used for all the plots inthis paper The results obtained by the numerical analysisare qualitatively robust towards the change of the initialconditions meaning that the only thing that changes is notthe whole phenomenon but the exact time point whenthe singularity occurs in all cases the transition singularityoccurs long before the beginning of inflation at 119905 = 10

10rarr

10minus36 sec In Figure 1 we provide plots of the scalar field


0 1 2 3 4

00

05

10

Time

120593(t)

(a)

0 1 2 3 4

0

5

10

Geff(t)

Time

minus5

minus10

(b)

00 05 10 15 20 25 30

0

1

2

3

4

Time

120588(t)

(c)

Figure 1119865(119877)model time dependence of the scalar field 120593(119905) (a) the effective gravitational constant119866eff(119905) (b) and thematter energy density120588(119905) (c) for 119908 = 13 119901 = 3 andB = 1

0 1 2 3 4

02

04

06

08

10

Time

Geff(t)

(a)

0 1 2 3 4

0

5

10

Time

Geff(t)

minus5

minus10

(b)

Figure 2 119865(119877) model the effective gravitational constant 119866eff(119905) as a function of time for nonrelativistic matter 119908 = 0 with B = 0001119901 = 2 (a) andB = 1 119901 = 2 (b)

120593(119905) the energy density 120588(119905) and the effective gravitationalconstant 119866eff(119905) as a function of the time 119905 with the time axisproperly rescaledWe have chosen the numerical values to be119908 = 13 119901 = 3 andB = 1 that is in a radiation dominateduniverse The same behavior however is observed forB = 1

and different values for 119908 Therefore we observe that the

parameterB critically affects the antigravity behavior In thepresent case the occurring antigravity can be seen in theright part of Figure 1 as can be seen there appears a gravitydominated period for 0 lt 119905 lt 17 and after the singularityat 119905 = 17 antigravity occurs In Figure 2 we present the timedependence of the effective gravitational constant 119866eff(119905) for


two different values ofB namelyB = 0001 (a) andB = 1

(b) We assumed a universe filled with nonrelativistic matterthat is 119908 = 0 and also 119901 = 2 As we can see in this casefor B = 0001 there is no antigravity and conversely forB = 1 there is This is the expected behavior of the Jordanframe theory since as B increases the possibility that theterm (120593(119905) minus B) becomes negative increases depending ofcourse on the initial conditions and on the other parametersrsquovalues

The model we studied in this section is similar to the onestudied in [41] in which case the antigravity scalar-tensormodel was the following

119878 = int1198891199094radicminus119892[

1 minus 1205932

12119877 minus

1

2119892120583]120597120583120593120597]120593 minus 119869 (120593)] (30)

The corresponding 119865(119877) gravity action following the tech-nique presented in [41] is easily found to be

119878 = int1198891199094radicminus119892119865 (119877) (31)

where 119865(119877) stands for

119865 (119877) =119890120578(120593(119877))

12(1 minus 120593

2(119877)) 119877 minus 119890

2120578(120593(119877))119869 (120593 (119877)) (32)

Moreover the real function 120578(120593) satisfies

(1 + 21205932) 1205781015840(120593)2minus 41205781015840(120593) minus 4 = 0 (33)

and as a result the kinetic term of the scalar field vanishesThis antigravity model clearly provides us with regimesgoverned by a negative gravitational constant for some valuesof the scalar field 120593 clearly indicating a highly nonsmoothbig crunch-big bang transition in the theoretical context of[41]

Before we close this section we discuss an importantissue Reasonably it can be argued that since the effectivegravitational constant119866eff(119905) diverges at some time this couldimply some sort of instability of the 119865(119877) theory Indeedthis is true to some extent Actually the singularity of thegravitational constant is a spacetime one since spacetimegeometric invariants like the Kretschmann scalar 119877119886119887119888119889119877

119886119887119888119889

seriously diverge In a mathematical context this singularityis also a naked Cauchy horizon not ldquodressedrdquo by some eventhorizon which in turn would imply the loss of predictabilityand also signal a spacetime singularity Therefore it is betterif these singularities occur in the very early universe As forthe issue of stability of the initial 119865(119877) theory this is aninvolved question since the quantum mechanical stabilityof the 119865(119877) theory is examined in the Einstein frame andnot in the Jordan frame [1] In the case of an occurringsingularity the Einstein frame is not consistently definedsince this singularity also introduces another singularity inthe scalar field redefinition necessary for the definition of thecanonical transformation in the Einstein frame (see the bookof Faraoni for more details on this [36]) A very thoroughanalysis of the stability of a similar to ours scalar-tensormodel was studied in [43] (see equation (1) of [43]) in which

case the model can exhibit antigravity if the nonminimalcoupling term becomes negative The model in [43] can beidentical to our Brans-Dickemodel if the potential is zero andthe nonminimal coupling contains terms of the order of sim 120593

4 Antigravity in Brans-Dicke Models

As we saw in the previous section even though we startedfrom an 119865(119877) theory with no antigravity the antigravityJordan frame action may or may not have antigravity solu-tions In this section we will study a minor modificationof the Brans-Dicke model with cosmological constant Theantigravity termwill be introduced by hand and will be of theform (120593 minus B)119877 with B being the extra term introduced byhand The general action in the Jordan frame that describesa general Brans-Dicke model with cosmological constantpotential 119880(120593) and matter is

119878 = int1198891199094radicminus119892[

1

2120593 (119877 minus 2Λ) minus

120596BD120593

119892120583]120597120583120593120597]120593 minus 119880 (120593)]

+ int1198891199094radicminus119892119871matter

(34)

In the following we will assume that initially the scalarpotential119880(120593) is zero and also that the cosmological constantis positive and has the value Λ = 10

minus49GeV4 The antigravitymodel we will study is obtained from the original Brand-Dicke model with cosmological constant (34) if we modifyby hand the action in the following way

119878 = int1198891199094radicminus119892[

1

2(120593 minusB) 119877 minus

120596BD120593

119892120583]120597120583120593120597]120593 minus 120593120582]


(35)

The term 120593120582 acts as a potential term and hence we haveat hand an antigravity Brans-Dicke model with potential119880BD(120593) = 120593Λ By varying (35) with respect to the metric andthe scalar field we obtain the Einstein equations describingthe cosmological evolution of the antigravity Brans-Dickemodel which for a flat FRWmetric are equal to

3 (120593 minusB)1198672= 120588(119898)

+120596BD2

()2+ 120593Λ minus 3119867 (36a)

minus2 (120593 minusB) = 120588(119898)

+ 119875(119898)

+ 120596BD()2+ minus 119867 (36b)

+ 3119867 +1

2120596BD[minus119877 + 2Λ] = 0 (36c)

In the following we will take 120596BD = 12 As in the previouscase the effective gravitational constant varies with time inthe Jordan frame model and its value is given by

119866eff (119905) =1

16120587 (120593 (119905) minusB) (37)

We have solved numerically the cosmological equations(36a) (36b) and (36c) and as a general remark let us note


Time0 1 2 3 4

10

12

14

16

18

20

22120593(t)

(a)

Time0 2 4 6 8

0

1

2

Geff(t)

minus1

(b)

Time0 1 2 3 4

00

05

10

15

20

120588(t)

(c)

Figure 3 Brans-Dicke model with cosmological constant time dependence of the scalar field 120593(119905) (a) the effective gravitational constant119866eff(119905) (b) and the matter energy density 120588(119905) (c) for 119908 = 13B = 1 and the cosmological constant Λ = 10

minus49

Time0 2 4 6 8

0

2

4

6

Geff(t)

minus2

minus4

(a)

Time0 2 4 6 8

00

05

10

15

Geff(t)

(b)

Figure 4 Brans-Dicke model with cosmological constant the effective gravitational constant 119866eff(119905) as a function of time for nonrelativisticmatter 119908 = 0 with Λ = 10

minus49 andB = 2 (a) andB = 1 (b)

that the model has both gravity and antigravity solutionsdepending on the values of the parameters and specifically onthe value of the antigravity parameter B In Figures 3 and 4we have presented the results of our numerical analysis forvarious parameter values and we now discuss them in detailIn Figure 1 appears the time dependence of the scalar field

120593(119905) the energy density 120588(119905) and the effective gravitationalconstant 119866eff(119905) where again we have properly rescaled thetime axis The numerical values we used in Figure 3 are 119908 =

13B = 1 andΛ = 10minus49 Changing the value of119908 does not

drastically affect the solutions which crucially depend on thevalue of the antigravity parameter B As can be seen from


the time dependence of the effective gravitational constant119866eff(119905) in Figure 3 antigravity occurs along with a singularitybetween the transition from gravity to antigravity This latterfeature is quite common in antigravity models (see eg [38ndash41]) Accordingly in Figure 4 we have provided the plots ofthe effective gravitational constant as a function of time for119908 = 0 Λ = 10

minus49 and B = 2 (1) for (a) and (b) plotsObviously for B = 2 (a) a complex antigravity patternoccurs while for B = 1 (b) there is no antigravity at allThis result validates our observation that antigravity cruciallydepends on the values of theB

5 A Brief Discussion

Before closing this section we discuss a last issue of someimportance It is generally known that a general 119865(119877) theorywith the method of Lagrange multipliers can be transformedto a Brans-Dicke theory with 120596BD = 0 and nonzero potentialIndeed it is easy to see this and we demonstrate it shortlyConsider a general 119865(119877) theory described by the followingaction


We introduce an auxiliary field 120594 which actually is theLagrange multiplier Using this field the action (38) becomes

S = int1198894119909radicminus119892 (119865 (120594) + 119865120594 (120594) (119877 minus 120594)) + 119878119898 (119892120583] Ψ119898)

(39)

with 119865120594(120594) being the first derivative of the function 119865(120594)

with respect to 120594 Varying the action (39) with respect to theauxiliary field 120594 we get

119865120594120594 (120594) (119877 minus 120594) = 0 (40)

Recalling that 119865120594120594(120594) = 0 which actually holds true for mostviable 119865(119877) theories we get 119877 = 120594 Therefore the action (39)recovers the initial 119865(119877) gravity action (39) If we define

120593 = 119865120594 (120594) (41)

then the action appearing in (39) becomes actually a functionof the field 120593 as can be seen below

S = int1198894119909radicminus119892 [120593119877 minus 119880 (120593)] + 119878119898 (119892120583] Ψ119898) (42)

The scalar potential term 119880(120593) is equal to the followingexpression

119880 (120593) = 120594 (120593) 120593 minus 119865 (120594 (120593)) (43)

Solving the algebraic equation (41) with respect to120594will actu-ally give us in closed form the function 120594(120593) (at least in mostcases) as a function of120593Therefore it is a straightforwardwayto obtain a Brans-Dicke theory with 120596BD zero and nonzeropotential by starting from a general 119865(119877) theory A questionnaturally springs to mind that is whether it is possible to

have any sort of coincidence between119865(119877) gravity and Brans-Dickewith a nonzero potential and zero120596BD and the answer isactually yes but only when the potential of the Brans-Dicke isexactly the one of (43) Now one has to be cautious howeverbecause this coincidence is ldquoonewayrdquo onlymeaning that if westart with the Brans-Dicke theory with 120596BD = 0 and we tryto find the corresponding 119865(119877) theory by using a conformaltransformation then we may end up with a different 119865(119877)theory which we denote for example 119891(119877) This requires amuch deeper study that extends beyond the purpose of thispaper and we defer this interesting issue to a near futurework However the reader is referred to the method in fourdimensions used by the authors in [41] There it can be seenthat when starting from a general scalar-tensor theory weend up with a certain class of 119865(119877) theories determined by aconstraint which the scalar field has to obey It is not obvioushowever that starting from a Brans-Dicke theory with 120596BD =

0 and nonzero potential we will end up to the original 119865(119877)theory we started with We hope to answer this issue in afuture paper

6 Conclusions

In this paper we studied antigravity in scalar-tensor theoriesoriginating from 119865(119877) theories and also antigravity in theBrans-Dicke model with cosmological constant In the caseof the 119865(119877) theories we used a variant of the Lagrangemultipliersmethod leading to antigravity scalar-tensormodelin the Jordan frame with 120596 = 0 and a scalar potentialWe applied the technique and studied numerically the time-dependence of the gravitational constant As we exemplifiedalthough the initial 119865(119877) model has no antigravity guaran-teed by the condition 119865

1015840(119877) gt 0 the scalar-tensor Jordan

frame counterpartmay ormaynot have antigravityThis latterfeature strongly depends on the parameters of the theory andparticularly on the antigravity parameterB In the case of theBrans-Dicke model with cosmological constant we studieda by-hand introduced antigravity modification of the modelin the Jordan frame The numerical analysis of the cosmo-logical equations showed that the model exhibits antigravitydepending on the numerical values of the parameters andparticularly on theB antigravity parameter like in the 119865(119877)model case In both cases there exist regimes in the cosmicevolution in which either gravity or antigravity prevails andwhen going from antigravity to gravity and vice versa asingularity occurs like inmost antigravity contexts [38ndash41] Itis worth searching theoretical constructions in which such asingularity is avoidedThis would probably require some sortof singular conformal transformations between frames orsome singularity of the Lagrangian a task we hope to addressin the near future

Finally it is worth discussing the results and also thecosmological implications of our resultsThemain goal of thispaper was to demonstrate all possible cases in which anti-gravity might appear in modified theories of gravity As weexplicitly demonstrated in the case of119865(119877) theories althoughthe initial Jordan frame119865(119877) theory had no antigravity (recallthe condition 119865

1015840(119877) gt 0 which actually guarantees this)


antigravity might show up when the Jordan frame equivalenttheory is considered modified in the way we explicitlyshowed in the text This is one of the new and notableresults of this paper In the case of Brans-Dicke modelintroducing by hand a term that causes antigravity thenantigravity might or might not appear in the resulting theoryThe latter depends strongly on the value of the antigravityparameterB In principle antigravity is a generally unwantedfeature in modified theories of gravity and thus it can beconsidered less harmful if it occurs in the very early universeprior to inflation Indeed this is exactly what happens in allthe cases we explicitly demonstrated in the text Howeverantigravity is rather difficult to detect experimentally unlessthere exists somemechanismof creation of a primordial blackhole during the antigravity regime that could retain someinformation in terms of some sort of gravitational memory[44] The evaporation of this black hole could reveal thevalue of the gravitational constant at the time it was createdA well posed question may be to ask how such a compactgravitational object could be created in an antigravity regimeThe answer to this could be that antimatter behaves somehowdifferent in antigravity regimes so it could probably playa prominent role in such a scenario However we have toadmit that this is just a speculation since after antigravityoccurs the universe experiences a gravitational regimewith aspacetime singularity at themoment of transitionWe cannotimagine how a compact gravitational object (if any) couldreact under such severe conditions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Nojiri and S D Odintsov ldquoIntroduction to modified gravityand gravitational alternative for dark energyrdquo InternationalJournal of Geometric Methods in Modern Physics vol 4 no 1pp 115ndash145 2007

[2] A De Felice and S J Tsujikawa ldquo119865(119877) theoriesrdquo Living Reviewsin Relativity vol 13 p 3 2010

[3] T P Sotiriou and V Faraoni ldquof(R) theories of gravityrdquo Reviewsof Modern Physics vol 82 p 451 2010

[4] S Nojiri and S D Odintsov ldquoUnified cosmic history inmodified gravity from 119865(119877) theory to Lorentz non-invariantmodelsrdquo Physics Reports vol 505 pp 59ndash144 2011

[5] K Bamba S Capozziello S Nojiri and S D Odintsov ldquoDarkenergy cosmology the equivalent description via differenttheoretical models and cosmography testsrdquo Astrophysics andSpace Science vol 342 no 1 pp 155ndash228 2012

[6] S Capozziello and M de Laurentis ldquoExtended Theories ofGravityrdquo Physics Reports vol 509 no 4-5 pp 167ndash321 2011

[7] S Capozziello S Nojiri S D Odintsov and A Troisi ldquoCos-mological viability of 119865(119877)-gravity as an ideal fluid and itscompatibility with a matter dominated phaserdquo Physics LettersB vol 639 no 3-4 pp 135ndash143 2006

[8] S Nojiri and S D Odintsov ldquoModified gravity with ln R termsand cosmic accelerationrdquo General Relativity and Gravitationvol 36 p 1765 2004

[9] S Nojiri and S D Odintsov ldquoModified f(R) gravity consistentwith realistic cosmology frommatter dominated epoch to darkenergy universerdquo Physical Review D vol 74 Article ID 0860052006

[10] S Tsujikawa ldquoObservational signatures of f(R) dark energymodels that satisfy cosmological and local gravity constraintsrdquoPhysical Review D vol 77 Article ID 023507 2008

[11] S Nojiri and S D Odintsov ldquoUnifying inflation with ΛCDMepoch in modified 119865(119877) gravity consistent with solar systemtestsrdquo Physics Letters B vol 657 pp 238ndash245 2007

[12] A A Starobinsky ldquoDisappearing cosmological constant in f(R)gravityrdquo JETP Letters vol 86 no 3 pp 157ndash163 2007

[13] S M Carroll V Duvvuri M Trodden and M S Turner ldquoIscosmic speed-up due to new gravitational physicsrdquo PhysicalReview D vol 70 Article ID 043528 2004

[14] O Bertolami and R Rosenfeld ldquoThe higgs portal and an unifiedmodel for dark energy and darkmatterrdquo International Journal ofModern Physics A vol 23 no 30 article 4817 2008

[15] A Capolupo S Capozziello and G Vitiello ldquoDark energycosmological constant and neutrino mixingrdquo InternationalJournal of Modern Physics A vol 23 no 31 pp 4979ndash49902008

[16] P K S Dunsby E Elizalde R Goswami S Odintsov and DS Gomez ldquoΛCDM universe in f (R) gravityrdquo Physical Review Dvol 82 Article ID 023519 2010

[17] E I Guendelman and A B Kaganovich ldquoExotic low densityfermion states in the two measures field theory neutrino darkenergyrdquo International Journal of Modern Physics A vol 21 p4373 2006

[18] G Cognola E Elizalde S Nojiri S D Odintsov L Sebastianiand S Zerbini ldquoClass of viable modified f(R) gravities describ-ing inflation and the onset of accelerated expansionrdquo PhysicalReview D vol 77 no 4 Article ID 046009 2008

[19] S K Srivastava ldquoGravitational origin of phantom dark energyand late cosmic accelerationrdquo International Journal of ModernPhysics A vol 22 p 1123 2007

[20] S Capozziello V F Cardone S Carloni and A TroisildquoCurvature quintessence matched with observational datardquoInternational Journal of Modern Physics D vol 12 no 10 pp1969ndash1982 2003

[21] S Capozziello ldquoCurvature quintessencerdquo International Journalof Modern Physics D vol 11 pp 483ndash492 2002

[22] P J E Peebles and B Ratra ldquoThe cosmological constant anddark energyrdquo Reviews of Modern Physics vol 75 no 2 pp 559ndash606 2003

[23] V Faraoni ldquoSuperquintessencerdquo International Journal of Mod-ern Physics D vol 11 p 471 2002

[24] E J Copeland M Sami and S Tsujikawa ldquoDynamics of darkenergyrdquo International Journal of Modern Physics D vol 15 no11 pp 1753ndash1935 2006

[25] S Nojiri and S D Odintsov ldquoModified gravity with negativeand positive powers of curvature Unification of inflation andcosmic accelerationrdquo Physical Review D vol 68 Article ID123512 2003

[26] J P Morais Graca and V B Bezerra ldquoGravitational field ofa rotating global monopole in f (R) theoryrdquo Modern PhysicsLetters A vol 27 no 31 Article ID 1250178 2012

[27] M Sharif and S Arif ldquoStatic cylindrically symmetric interiorsolutions in f(R) gravityrdquo Modern Physics Letters A vol 27 no25 Article ID 1250138 12 pages 2012


[28] S Asgari and R Saffari ldquoVacuum solution of a linear red-shift based correction in f(R) gravityrdquo General Relativity andGravitation vol 44 no 3 pp 737ndash750 2012

[29] K A Bronnikov M V Skvortsova and A A StarobinskyldquoNotes on wormhole existence in scalar-tensor and 119865(119877) grav-ityrdquoGravitation and Cosmology vol 16 no 3 pp 216ndash222 2010

[30] E V Arbuzova and A D Dolgov ldquoExplosive phenomena inmodified gravityrdquo Physics Letters Section B vol 700 no 5 pp289ndash293 2011

[31] C-C Lee C-Q Geng and L Yang ldquoSingularity phenomena inviable f(R) gravityrdquo Progress of Theoretical Physics vol 128 no2 pp 415ndash427 2012

[32] T Harko F S N Lobo S Nojiri and S D Odintsov ldquoF(R T)gravityrdquo Physical Review D vol 84 no 2 Article ID 0240202011

[33] O Bertolami C G Boehmer T Harko and F S N Lobo ldquoExtraforce in 119891(119877) modified theories of gravityrdquo Physical Review Dvol 75 Article ID 104016 2007

[34] Z Haghani T Harko F S N Lobo H R Sepangi andS Shahidi ldquoFurther matters in space-time geometry119891(119877 119879 119877120583]119879

120583]) gravityrdquo Physical Review D vol 88 Article

ID 044023 2013[35] M Sharif and M Zubair ldquoStudy of Bianchi I anisotropic model

in f(RT) gravityrdquo Astrophysics and Space Science vol 349 no 1pp 457ndash465 2014

[36] V Faraoni Cosmology in Scalar-Tensor Gravity Kluwer Aca-demic Publishers Amsterdam The Netherlands 2004

[37] Y Fujii and K-I Maeda The Scalar-Tensor Theory of Gravita-tion Cambridge University Press Cambridge UK 2004

[38] PCaputa S SHaque JOlson andBUnderwood ldquoCosmologyor catastrophe A non-minimally coupled scalar in an inhomo-geneous universerdquo Classical and Quantum Gravity vol 30 no19 2013

[39] I Bars S H Chen P J Steinhardt and N Turok ldquoAntigravityand the big crunchbig bang transitionrdquo Physics Letters B vol715 no 1ndash3 pp 278ndash281 2012

[40] J J M Carrasco W Chemissany and R Kallosh ldquoJourneysthrough antigravityrdquo Journal of High Energy Physics vol 1401p 130 2014

[41] K Bamba S Nojiri S D Odintsov and D Saez-GomezldquoPossible antigravity regions in F(R) theoryrdquo Physics Letters Bvol 730 pp 136ndash140 2014

[42] Y Fujii ldquoChoosing a conformal frame in scalar tensor theoriesof gravity with a cosmological constantrdquo Progress of TheoreticalPhysics vol 99 pp 599ndash622 1998

[43] M A Skugoreva A V Toporensky and S Yu ldquoGlobal stabilityanalysis for cosmological models with non-minimally coupledscalar fieldsrdquo httparxivorgabs14046226

[44] J D Barrow ldquoGravitational memoryrdquo Physical Review D vol46 no 8 pp R3227ndashR3230 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


In principle every consistent generalization of generalrelativity inevitably has to be confronted with the successesof general relativity Since general relativity is a successfuldescription of nature in strong gravitational environmentsthere exist a large number of constraints that need to besatisfied in order that an 119865(119877)modified gravity theory can beconsidered as viableThe constraints to be satisfied aremainlyimposed from local tests of general relativity for examplefrom planetary and star formation tests and moreover fromvarious cosmological bounds In addition since each 119865(119877)

theory has a Jordan frame scalar-tensor gravitational theorycounterpart with 120596 zero and a potential the scalarons ofthis counterpart theory must be classical in order to ensurequantum-mechanical stability (see [1ndash6])

In theories of modified gravity a longstanding debatabletheoretical problem exists related to Jordan and Einsteinframes [36 37] since the physics coming out from thetwo frames can be quite different in principle In view ofthis we will focus on the physics of Jordan frame anddemonstrate that it is possible to have antigravity [38ndash41]For the possibility of antigravity regimes in scalar-tensortheories consult [38ndash40] and for antigravity in 119865(119877) theoriessee [41] In this paper we will study antigravity regimescoming from 119865(119877) theories and from Brans-Dicke theoriesin the Jordan frame In reference to the 119865(119877) theories we willfind the Jordan frame antigravity scalar-tensor counterpartusing a modified method of the Lagrange multipliers aswe will see in the following sections The interesting featureabout these theories is that although the 119865(119877) theory has noantigravity the resulting Jordan frame scalar-tensor theorymay or may not have antigravity We exemplify this bynumerically working out an example In the case of Brans-Dicke antigravity we introduce by hand an antigravity termand numerically solve the cosmological equations and as wewill demonstrate similar results hold true that is antigravitymay exist or not depending on the parameters of the theory

This paper is organized as follows in Section 2 we brieflyrecall the essentials of 119865(119877) theories in Section 3 we getto the core of the paper and introduce a modification ofthe Lagrange multipliers method in order to get antigravityfrom 119865(119877) theories Accordingly we apply the technique toone quite known 119865(119877) model and present the result of ouranalysis The study of antigravity is performed in Section 4and the conclusions follow in the end of the paper

2 General Features of119865(119877) Dark Energy Models in the JordanFrame

In this section in order to maintain the paper self-containedwe briefly review themain features of 119865(119877) gravity theories inthe Jordan frame in the theoretical framework of the metricformalism For an important stream of review papers andarticles see [1ndash19] and references therein

The geometrical background of the manifolds used hereis pseudo-Riemannian and is described locally by a Lorentzmetric (the FRWmetric in our case) in addition to a torsion-less symmetric and metric compatible affine connection

the so-called Levi-Civita connection In such a geometricbackground the Christoffel symbols are

Γ119896

120583] =1

2119892119896120582(120597120583119892120582] + 120597]119892120582120583 minus 120597120582119892120583]) (1)

and the Ricci scalar becomes

119877 = 119892120583](120597120582Γ120582

120583] minus 120597]Γ120588

120583120588minus Γ120590

120590]Γ120590

120583120582+ Γ120588

120583120588119892120583]Γ120590

120583]) (2)

The 119865(119877) theories of modified gravity are described by amodification of the Einstein-Hilbert action with the four-dimensional action being equal to

S =1

21205812int1198894119909radicminus119892119865 (119877) + 119878119898 (119892120583] Ψ119898) (3)

where 1205812= 8120587119866 and 119878119898 is the matter action containing

the matter fields Ψ119898 For simplicity in this section it will beassumed that the form of the 119865(119877) theory that will be usedis 119865(119877) = 119877 + 119891(119877) and in addition the metric formalismframework will be used Varying the action (3) with respectto the metric 119892120583] we get the following equations of motion

1198651015840(119877) 119877120583] (119892) minus

1

2119865 (119877) 119892120583] minus nabla120583nabla]119865

1015840(119877)

+ 119892120583]◻1198651015840(119877) = 120581

2119879120583]

(4)

In the above equation 1198651015840(119877) = 120597119865(119877)120597119877 and also 119879120583] is theenergy momentum tensor

The most striking feature of the 119865(119877) modified gravitytheories is that what actually changes in reference to the usualEinstein-Hilbert gravity equations is the right hand side ofthe Einstein equations and not the left which remains thesame Indeed the equations of motion (4) can be cast in thefollowing form

119877120583] minus1

2119877119892120583]

=1205812

1198651015840 (119877)(119879120583] +

1

120581[119865 (119877) minus 119877119865

1015840(119877)

2119892120583] + nabla120583nabla]119865

1015840(119877)

minus 119892120583]◻1198651015840(119877) ])

(5)

Therefore we get an additional contribution for the energymomentum tensor coming from the term

119879eff120583] =

1

120581[119865 (119877) minus 119877119865

1015840(119877)

2119892120583]

+ nabla120583nabla]1198651015840(119877) minus 119892120583]◻119865

1015840(119877) ]

(6)

It is this term that actually models the dark energy in 119865(119877)

theories of modified gravity Taking the trace of (4) westraightforwardly obtain the following equation

3◻1198651015840(119877) + 119877119865

1015840(119877) minus 2119865 (119877) = 120581

2119879 (7)






119877 = 6 (21198672+ ) (8)


31198651015840(119877)119867

2= 1205812(120588119898 + 120588119903)

+

(1198651015840(119877) 119877 minus 119865 (119877))

2minus 3119867

1015840(119877)

(9a)

minus21198651015840(119877) = 120581

2(119901119898 +

4

3120588119903) + 119865

1015840(119877) minus 119867

1015840(119877) (9b)


120588eff =1

1198651015840 (119877)[120588119898 +

1

1205812(1198651015840(119877) 119877 minus 119865 (119877) minus 6119867

1015840(119877))]

(10a)

119901eff =1

1198651015840 (119877)[119901119898 +

1

1205812( minus 1198651015840(119877) 119877 + 119865 (119877)

+ 41198671015840(119877) + 2

1015840(119877)) ]

(10b)










+ 119878119898 (119892120583] Ψ119898)

(12)






119866eff =1

16120587 (120593 minusB) (16)



119880 (120593) = 120594 (120593) (120593 minusB) minus 119865 (120594 (120593)) (17)



119865 (119877) = 119877 minus 119877minus119901 (18)



119889119865 (119877)

119889119877gt 0 (19)



minus119901) + 119878119898 (119892120583] Ψ119898) (20)



(21)


119880119865(119877) (120593) = (119901


1(119901+1)

(120593 minusB)

minus (119901


1(119901+1)

+ (119901


minus(119901(119901+1))

(22)



1198891199042= minus119889119905

2+ 1198862(119905)sum

119894

1198891199092

119894 (23)


3 (120593 minusB)1198672= 120588 + 119880119865(119877) (120593) minus 3119867 (24a)


minus119877 + 2119889119880119865(119877) (120593)

119889120593= 0 (24c)


=119889120593

119889119905 (25)


120583]120583

= 0 holds true

120588 + 3119867 (1 + 119908) 120588 = 0 (26)


119866eff (119905) =1

16120587 (120593 (119905) minusB) (27)


120588 (1) = 1 120593 (1) = 1 (1) = 0 119905 times 119867 (1) sim 1

(28)


119905 = 1 997888rarr 10minus46sec (29)


10rarr



0 1 2 3 4

00

05

10

Time

120593(t)

(a)

0 1 2 3 4

0

5

10

Geff(t)

Time

minus5

minus10

(b)

00 05 10 15 20 25 30

0

1

2

3

4

Time

120588(t)

(c)


0 1 2 3 4

02

04

06

08

10

Time

Geff(t)

(a)

0 1 2 3 4

0

5

10

Time

Geff(t)

minus5

minus10

(b)









119878 = int1198891199094radicminus119892[

1 minus 1205932

12119877 minus

1

2119892120583]120597120583120593120597]120593 minus 119869 (120593)] (30)


119878 = int1198891199094radicminus119892119865 (119877) (31)


119865 (119877) =119890120578(120593(119877))

12(1 minus 120593

2(119877)) 119877 minus 119890

2120578(120593(119877))119869 (120593 (119877)) (32)


(1 + 21205932) 1205781015840(120593)2minus 41205781015840(120593) minus 4 = 0 (33)



119886119887119888119889





119878 = int1198891199094radicminus119892[

1

2120593 (119877 minus 2Λ) minus

120596BD120593

119892120583]120597120583120593120597]120593 minus 119880 (120593)]


(34)



119878 = int1198891199094radicminus119892[

1

2(120593 minusB) 119877 minus

120596BD120593

119892120583]120597120583120593120597]120593 minus 120593120582]


(35)


3 (120593 minusB)1198672= 120588(119898)

+120596BD2

()2+ 120593Λ minus 3119867 (36a)

minus2 (120593 minusB) = 120588(119898)

+ 119875(119898)

+ 120596BD()2+ minus 119867 (36b)

+ 3119867 +1

2120596BD[minus119877 + 2Λ] = 0 (36c)


119866eff (119905) =1

16120587 (120593 (119905) minusB) (37)



Time0 1 2 3 4

10

12

14

16

18

20

22120593(t)

(a)

Time0 2 4 6 8

0

1

2

Geff(t)

minus1

(b)

Time0 1 2 3 4

00

05

10

15

20

120588(t)

(c)


minus49

Time0 2 4 6 8

0

2

4

6

Geff(t)

minus2

minus4

(a)

Time0 2 4 6 8

00

05

10

15

Geff(t)

(b)















(39)



119865120594120594 (120594) (119877 minus 120594) = 0 (40)


120593 = 119865120594 (120594) (41)




119880 (120593) = 120594 (120593) 120593 minus 119865 (120594 (120593)) (43)




6 Conclusions










References





















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








119877 = 6 (21198672+ ) (8)


31198651015840(119877)119867

2= 1205812(120588119898 + 120588119903)

+

(1198651015840(119877) 119877 minus 119865 (119877))

2minus 3119867

1015840(119877)

(9a)

minus21198651015840(119877) = 120581

2(119901119898 +

4

3120588119903) + 119865

1015840(119877) minus 119867

1015840(119877) (9b)


120588eff =1

1198651015840 (119877)[120588119898 +

1

1205812(1198651015840(119877) 119877 minus 119865 (119877) minus 6119867

1015840(119877))]

(10a)

119901eff =1

1198651015840 (119877)[119901119898 +

1

1205812( minus 1198651015840(119877) 119877 + 119865 (119877)

+ 41198671015840(119877) + 2

1015840(119877)) ]

(10b)










+ 119878119898 (119892120583] Ψ119898)

(12)






119866eff =1

16120587 (120593 minusB) (16)



119880 (120593) = 120594 (120593) (120593 minusB) minus 119865 (120594 (120593)) (17)



119865 (119877) = 119877 minus 119877minus119901 (18)



119889119865 (119877)

119889119877gt 0 (19)



minus119901) + 119878119898 (119892120583] Ψ119898) (20)



(21)


119880119865(119877) (120593) = (119901


1(119901+1)

(120593 minusB)

minus (119901


1(119901+1)

+ (119901


minus(119901(119901+1))

(22)



1198891199042= minus119889119905

2+ 1198862(119905)sum

119894

1198891199092

119894 (23)


3 (120593 minusB)1198672= 120588 + 119880119865(119877) (120593) minus 3119867 (24a)


minus119877 + 2119889119880119865(119877) (120593)

119889120593= 0 (24c)


=119889120593

119889119905 (25)


120583]120583

= 0 holds true

120588 + 3119867 (1 + 119908) 120588 = 0 (26)


119866eff (119905) =1

16120587 (120593 (119905) minusB) (27)


120588 (1) = 1 120593 (1) = 1 (1) = 0 119905 times 119867 (1) sim 1

(28)


119905 = 1 997888rarr 10minus46sec (29)


10rarr



0 1 2 3 4

00

05

10

Time

120593(t)

(a)

0 1 2 3 4

0

5

10

Geff(t)

Time

minus5

minus10

(b)

00 05 10 15 20 25 30

0

1

2

3

4

Time

120588(t)

(c)


0 1 2 3 4

02

04

06

08

10

Time

Geff(t)

(a)

0 1 2 3 4

0

5

10

Time

Geff(t)

minus5

minus10

(b)









119878 = int1198891199094radicminus119892[

1 minus 1205932

12119877 minus

1

2119892120583]120597120583120593120597]120593 minus 119869 (120593)] (30)


119878 = int1198891199094radicminus119892119865 (119877) (31)


119865 (119877) =119890120578(120593(119877))

12(1 minus 120593

2(119877)) 119877 minus 119890

2120578(120593(119877))119869 (120593 (119877)) (32)


(1 + 21205932) 1205781015840(120593)2minus 41205781015840(120593) minus 4 = 0 (33)



119886119887119888119889





119878 = int1198891199094radicminus119892[

1

2120593 (119877 minus 2Λ) minus

120596BD120593

119892120583]120597120583120593120597]120593 minus 119880 (120593)]


(34)



119878 = int1198891199094radicminus119892[

1

2(120593 minusB) 119877 minus

120596BD120593

119892120583]120597120583120593120597]120593 minus 120593120582]


(35)


3 (120593 minusB)1198672= 120588(119898)

+120596BD2

()2+ 120593Λ minus 3119867 (36a)

minus2 (120593 minusB) = 120588(119898)

+ 119875(119898)

+ 120596BD()2+ minus 119867 (36b)

+ 3119867 +1

2120596BD[minus119877 + 2Λ] = 0 (36c)


119866eff (119905) =1

16120587 (120593 (119905) minusB) (37)



Time0 1 2 3 4

10

12

14

16

18

20

22120593(t)

(a)

Time0 2 4 6 8

0

1

2

Geff(t)

minus1

(b)

Time0 1 2 3 4

00

05

10

15

20

120588(t)

(c)


minus49

Time0 2 4 6 8

0

2

4

6

Geff(t)

minus2

minus4

(a)

Time0 2 4 6 8

00

05

10

15

Geff(t)

(b)















(39)



119865120594120594 (120594) (119877 minus 120594) = 0 (40)


120593 = 119865120594 (120594) (41)




119880 (120593) = 120594 (120593) 120593 minus 119865 (120594 (120593)) (43)




6 Conclusions










References





















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119880 (120593) = 120594 (120593) (120593 minusB) minus 119865 (120594 (120593)) (17)



119865 (119877) = 119877 minus 119877minus119901 (18)



119889119865 (119877)

119889119877gt 0 (19)



minus119901) + 119878119898 (119892120583] Ψ119898) (20)



(21)


119880119865(119877) (120593) = (119901


1(119901+1)

(120593 minusB)

minus (119901


1(119901+1)

+ (119901


minus(119901(119901+1))

(22)



1198891199042= minus119889119905

2+ 1198862(119905)sum

119894

1198891199092

119894 (23)


3 (120593 minusB)1198672= 120588 + 119880119865(119877) (120593) minus 3119867 (24a)


minus119877 + 2119889119880119865(119877) (120593)

119889120593= 0 (24c)


=119889120593

119889119905 (25)


120583]120583

= 0 holds true

120588 + 3119867 (1 + 119908) 120588 = 0 (26)


119866eff (119905) =1

16120587 (120593 (119905) minusB) (27)


120588 (1) = 1 120593 (1) = 1 (1) = 0 119905 times 119867 (1) sim 1

(28)


119905 = 1 997888rarr 10minus46sec (29)


10rarr



0 1 2 3 4

00

05

10

Time

120593(t)

(a)

0 1 2 3 4

0

5

10

Geff(t)

Time

minus5

minus10

(b)

00 05 10 15 20 25 30

0

1

2

3

4

Time

120588(t)

(c)


0 1 2 3 4

02

04

06

08

10

Time

Geff(t)

(a)

0 1 2 3 4

0

5

10

Time

Geff(t)

minus5

minus10

(b)









119878 = int1198891199094radicminus119892[

1 minus 1205932

12119877 minus

1

2119892120583]120597120583120593120597]120593 minus 119869 (120593)] (30)


119878 = int1198891199094radicminus119892119865 (119877) (31)


119865 (119877) =119890120578(120593(119877))

12(1 minus 120593

2(119877)) 119877 minus 119890

2120578(120593(119877))119869 (120593 (119877)) (32)


(1 + 21205932) 1205781015840(120593)2minus 41205781015840(120593) minus 4 = 0 (33)



119886119887119888119889





119878 = int1198891199094radicminus119892[

1

2120593 (119877 minus 2Λ) minus

120596BD120593

119892120583]120597120583120593120597]120593 minus 119880 (120593)]


(34)



119878 = int1198891199094radicminus119892[

1

2(120593 minusB) 119877 minus

120596BD120593

119892120583]120597120583120593120597]120593 minus 120593120582]


(35)


3 (120593 minusB)1198672= 120588(119898)

+120596BD2

()2+ 120593Λ minus 3119867 (36a)

minus2 (120593 minusB) = 120588(119898)

+ 119875(119898)

+ 120596BD()2+ minus 119867 (36b)

+ 3119867 +1

2120596BD[minus119877 + 2Λ] = 0 (36c)


119866eff (119905) =1

16120587 (120593 (119905) minusB) (37)



Time0 1 2 3 4

10

12

14

16

18

20

22120593(t)

(a)

Time0 2 4 6 8

0

1

2

Geff(t)

minus1

(b)

Time0 1 2 3 4

00

05

10

15

20

120588(t)

(c)


minus49

Time0 2 4 6 8

0

2

4

6

Geff(t)

minus2

minus4

(a)

Time0 2 4 6 8

00

05

10

15

Geff(t)

(b)















(39)



119865120594120594 (120594) (119877 minus 120594) = 0 (40)


120593 = 119865120594 (120594) (41)




119880 (120593) = 120594 (120593) 120593 minus 119865 (120594 (120593)) (43)




6 Conclusions










References





















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0 1 2 3 4

00

05

10

Time

120593(t)

(a)

0 1 2 3 4

0

5

10

Geff(t)

Time

minus5

minus10

(b)

00 05 10 15 20 25 30

0

1

2

3

4

Time

120588(t)

(c)


0 1 2 3 4

02

04

06

08

10

Time

Geff(t)

(a)

0 1 2 3 4

0

5

10

Time

Geff(t)

minus5

minus10

(b)









119878 = int1198891199094radicminus119892[

1 minus 1205932

12119877 minus

1

2119892120583]120597120583120593120597]120593 minus 119869 (120593)] (30)


119878 = int1198891199094radicminus119892119865 (119877) (31)


119865 (119877) =119890120578(120593(119877))

12(1 minus 120593

2(119877)) 119877 minus 119890

2120578(120593(119877))119869 (120593 (119877)) (32)


(1 + 21205932) 1205781015840(120593)2minus 41205781015840(120593) minus 4 = 0 (33)



119886119887119888119889





119878 = int1198891199094radicminus119892[

1

2120593 (119877 minus 2Λ) minus

120596BD120593

119892120583]120597120583120593120597]120593 minus 119880 (120593)]


(34)



119878 = int1198891199094radicminus119892[

1

2(120593 minusB) 119877 minus

120596BD120593

119892120583]120597120583120593120597]120593 minus 120593120582]


(35)


3 (120593 minusB)1198672= 120588(119898)

+120596BD2

()2+ 120593Λ minus 3119867 (36a)

minus2 (120593 minusB) = 120588(119898)

+ 119875(119898)

+ 120596BD()2+ minus 119867 (36b)

+ 3119867 +1

2120596BD[minus119877 + 2Λ] = 0 (36c)


119866eff (119905) =1

16120587 (120593 (119905) minusB) (37)



Time0 1 2 3 4

10

12

14

16

18

20

22120593(t)

(a)

Time0 2 4 6 8

0

1

2

Geff(t)

minus1

(b)

Time0 1 2 3 4

00

05

10

15

20

120588(t)

(c)


minus49

Time0 2 4 6 8

0

2

4

6

Geff(t)

minus2

minus4

(a)

Time0 2 4 6 8

00

05

10

15

Geff(t)

(b)















(39)



119865120594120594 (120594) (119877 minus 120594) = 0 (40)


120593 = 119865120594 (120594) (41)




119880 (120593) = 120594 (120593) 120593 minus 119865 (120594 (120593)) (43)




6 Conclusions










References





















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119878 = int1198891199094radicminus119892[

1 minus 1205932

12119877 minus

1

2119892120583]120597120583120593120597]120593 minus 119869 (120593)] (30)


119878 = int1198891199094radicminus119892119865 (119877) (31)


119865 (119877) =119890120578(120593(119877))

12(1 minus 120593

2(119877)) 119877 minus 119890

2120578(120593(119877))119869 (120593 (119877)) (32)


(1 + 21205932) 1205781015840(120593)2minus 41205781015840(120593) minus 4 = 0 (33)



119886119887119888119889





119878 = int1198891199094radicminus119892[

1

2120593 (119877 minus 2Λ) minus

120596BD120593

119892120583]120597120583120593120597]120593 minus 119880 (120593)]


(34)



119878 = int1198891199094radicminus119892[

1

2(120593 minusB) 119877 minus

120596BD120593

119892120583]120597120583120593120597]120593 minus 120593120582]


(35)


3 (120593 minusB)1198672= 120588(119898)

+120596BD2

()2+ 120593Λ minus 3119867 (36a)

minus2 (120593 minusB) = 120588(119898)

+ 119875(119898)

+ 120596BD()2+ minus 119867 (36b)

+ 3119867 +1

2120596BD[minus119877 + 2Λ] = 0 (36c)


119866eff (119905) =1

16120587 (120593 (119905) minusB) (37)



Time0 1 2 3 4

10

12

14

16

18

20

22120593(t)

(a)

Time0 2 4 6 8

0

1

2

Geff(t)

minus1

(b)

Time0 1 2 3 4

00

05

10

15

20

120588(t)

(c)


minus49

Time0 2 4 6 8

0

2

4

6

Geff(t)

minus2

minus4

(a)

Time0 2 4 6 8

00

05

10

15

Geff(t)

(b)















(39)



119865120594120594 (120594) (119877 minus 120594) = 0 (40)


120593 = 119865120594 (120594) (41)




119880 (120593) = 120594 (120593) 120593 minus 119865 (120594 (120593)) (43)




6 Conclusions










References





















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Time0 1 2 3 4

10

12

14

16

18

20

22120593(t)

(a)

Time0 2 4 6 8

0

1

2

Geff(t)

minus1

(b)

Time0 1 2 3 4

00

05

10

15

20

120588(t)

(c)


minus49

Time0 2 4 6 8

0

2

4

6

Geff(t)

minus2

minus4

(a)

Time0 2 4 6 8

00

05

10

15

Geff(t)

(b)















(39)



119865120594120594 (120594) (119877 minus 120594) = 0 (40)


120593 = 119865120594 (120594) (41)




119880 (120593) = 120594 (120593) 120593 minus 119865 (120594 (120593)) (43)




6 Conclusions










References





















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics











(39)



119865120594120594 (120594) (119877 minus 120594) = 0 (40)


120593 = 119865120594 (120594) (41)




119880 (120593) = 120594 (120593) 120593 minus 119865 (120594 (120593)) (43)




6 Conclusions










References





















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







References





















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



research article study of antigravity in an () model and...

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