Gödel and Gödel-type universes in Brans–Dicke theory

Citation for this paper:

Agudelo, J.A., Nascimento, J.R., Petrov, A.Y., Porfírio, P.J. & Santos, A.F. (2016).

Gödel and Gödel-type universes in Brans–Dicke theory. Physics Letters B, 762,

96-101.

http://dx.doi.org/10.1016/j.physletb.2016.09.011

UVicSPACE: Research & Learning Repository

_____________________________________________________________

Faculty of Science

Faculty Publications

_____________________________________________________________

Gödel and Gödel-type universes in Brans–Dicke theory

J.A. Agudelo, J.R. Nascimento, A.Yu. Petrov, P.J. Porfírio, A.F. Santos

2016

©2016 The Authors. Published by Elsevier B.V. This is an open access article under

the CC BY license (

http://creativecommons.org/licenses/by/4.0/

). Funded by

SCOAP3

This article was originally published at:

http://dx.doi.org/10.1016/j.physletb.2016.09.011

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Gödel

and

Gödel-type

universes

in

Brans–Dicke

theory

J.A. Agudelo

a

,

J.R. Nascimento

b

,

A.Yu. Petrov

b

,

∗

,

P.J. Porfírio

b

,

A.F. Santos

a

,

c

a_{InstitutodeFísica,UniversidadeFederaldeMatoGrosso,78060-900,Cuiabá,MatoGrosso,Brazil}

b_{DepartamentodeFísica,UniversidadeFederaldaParaíba,CaixaPostal5008,58051-970,JoãoPessoa,Paraíba,Brazil} c_{DepartmentofPhysicsandAstronomy,UniversityofVictoria,3800FinnertyRoadVictoria,BC,Canada}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received12August2016 Accepted9September2016 Availableonline14September2016 Editor:M.Cvetiˇc

In thispaper, conditionsfor existenceof Gödeland Gödel-type solutionsin Brans–Dicke (BD)scalar– tensor theoryand their main features are studied. Theconsistency ofequations ofmotion, causality violationand existenceofCTCs(closedtime-like curves)areinvestigated.Therolewhichcosmological constantandMachprincipleplaytoachievetheconsistencyofthismodelisstudied.

©2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

Among the known exact solutions of Einstein field equations (EFEs) gravity, the Gödel and Gödel-type metrics [1–3] play the specialrole.Itwasshownwithintheusualgeneralrelativity(GR) thatthesesolutionsdescriberotatinguniverses,allowforthe exis-tenceofclosedtime-likecurves(CTCs)andshowthattheEinstein theoryofgravityisnotcompletelycompatiblewithMachprinciple (MP)1[4–8].

Atthesametime,thegeneralrelativityencountersseveral fun-damentalproblems,suchasitsnon-renormalizabilityatthe quan-tumlevelandtheneedofexplanationforthecosmicacceleration. Tosolvetheseproblems,differentalternativegravitytheorieswere proposed(forareview onthesetheories,see[9,10]).Therefore,it is interesting to studythe behavior ofthe Gödel andGödel-type solutionswithinthesemodels,lookingfortheconsistencyofthese metricswithin such theories,andortheir corresponding physical interpretations. Such studies, includingdiscussion of problems of causality, the existence of CTCs and correspondence with GR in the respective limit, were performed through verification of the compatibility of resulting equations of motion in several gravity modelsincludingforexample f

(

R

)

gravity,Horava–Lifshitzgravity andbumblebeegravity[11–16].

*

Correspondingauthor.

E-mailaddresses:jaar@fisica.ufmt.br(J.A. Agudelo),jroberto@fisica.ufpb.br

(J.R. Nascimento),petrov@fisica.ufpb.br(A.Yu. Petrov),pporfirio@fisica.ufpb.br

(P.J. Porfírio),alesandroferreira@fisica.ufmt.br(A.F. Santos).

1 _{Accordingtothisprinciple, theabsolute accelerationdoesnotexist,butthe} accelerationrelativetodistantcosmic matterdistribution,whilesuchmatter de-terminesinertialandgeometricalpropertiesofmatterandspace–time,respectively, canoccur.

OnemoreexampleofanalternativegravitymodelistheBrans– Dicke(BD)gravitywhichwillbetakenasthemainsubjectofthis paper.Here,weshalldiscussthebehavior andmainpropertiesof GödelandGödel-typesolutionsinthistheory,oneofthefirstand well-known scalar–tensortheories,builtupto betotally Machian andreducingto theGRina limitingcase[17–19].Todothis,we usethemattersourcecomposedbyaperfectfluidandan electro-magneticfield.Withinourstudy,weverifythecausalityfeaturesof the possiblesolutions.Further,we findonecompletelycausal so-lutioncorrespondingtotheemptyspacecase.Wenotethatearlier the

ω

→ ∞

limitofBDtheorywastreatedin[20].

It is remarkable that the BD scalar field can be interpreted withindifferentcosmologicalcontexts,mainlywithinmodeling the very earlyrapidexpansion periodknownasinflation [21,22]. Ad-ditionally,thisfield can be identifiedwiththe dilatonwithin the stringtheorycontext.Therefore,theBDmodelcouldbetreatedas alowenergylimitofsomeunifiedandmoregeneraltheory[23].

However,itisknownthat theacceleratedexpansioncannotbe describedwithinapureBDgravity.So,wemustdevelopits possi-bleextensionsliketheinclusionofcosmologicalconstant(fixedor possessingdifferentdependencies),scalarfieldpotentialsor func-tions of the scalar curvature [24–26]. Thus, we review with the special attention the structure and solubility of resulting Gödel and Gödel-type field equations within BD theory, realistic cases and possible consequences. The role played by the cosmological

constant and Mach principle as essential components for model

coherence andcompatibilityisexaminedusingtheanalogousand well-knownresultsinGR.

Thiswork isstructuredasfollows.InSection2,a briefreview offundamentalideasrelatedtoprinciplesandpropertiesofGödel andGödel-typeuniversesarepresented.Similarly,inSection3,the http://dx.doi.org/10.1016/j.physletb.2016.09.011

0370-2693/©2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

Brans–Dicke theory basics has been presented. In Section 4, the simpleGödeluniverseinBD-



modelisstudied.TheSection5is devotedtostudyoftheGödel-typeuniverseinBD-



model.Inthe Section6,conclusionsandremarksarepresented.

2. GödelandGödel-typeuniverses

Westart our paper witha briefreview of main propertiesof GödelandGödel-typesolutionsofEFEs.

2.1.Gödelcase

ThesimplestEFEsexactsolutionthatallows CTCsistheGödel metric[1].Thissolutioniscompatiblewithincoherentmatter dis-tributionatrestandcanbedescribedbythelineelementlooking like ds2

=

a2



(

dx0

)

2

− (

dx1

)

2

+

e 2x1 2

(

dx 2

₎

2

_{− (}

_dx3

₎

2

₊

_2ex1

(

dx0dx2

)



,

(1)

wherea2 isapositiveconstant.Thissolution,fora

=

0,is consis-tentonlyifthecosmologicalconstantdiffersfromzero.Therefore, consideringan energy–momentumtensor ofthepressureless rel-ativistic fluid, T μν

=

ρ

vμ vν ,where

ρ

is the matter density and

vμ isits4-velocity,itissimpletocheckthat

Rμν

= −

1

a2vμvν

,

R

=

1

a2

,

(2)

sothattheEFEscanbewrittenintheform

Rμν

+ ( −

1

2R

)

gμν

= −

8

π

G Tμν

,

(3)

whichimpliesthat,inthesystemofunitswithc

=

1,the cosmo-logicalconstantandmatterdensityare



= −

1

2a2

,

ρ

=

1

8

π

Ga2

.

(4)

Itisworthwhiletomentionsomespecificandimportant prop-ertiesofthissolution.Weseethattheenergy–momentumtensor isthe sameasthat one corresponding to theEinstein static uni-verse,henceEFEshavetwodifferentsolutionsforthesamematter

content, which, from a purely Machian viewpoint, seems to be

totally contradictory, since matter distribution should determine thespace–time geometryuniquely [27]. Thus, theGödel solution showsthat GRhasnot satisfiedMachprinciplecompletelyvia its fieldequations.

Additionally,thisspecialsolutiondescribesa rotationalcosmic behavior,whichcanbeseenclearly,comparing(1)withthemetric correspondingtoaflatspacewithcylindricalcoordinatesr,

ϕ

and

z,rotatingaroundz-axiswithangularvelocity

ω

,thatis

ds2

= (

1

−

ω

2_r2

c2

)

c

2_dt2

₋

_dr2

₋

_r2_d

_ϕ

2

₋

_dz2

₋

₂

_ω

_r2_dtd

_ϕ

_,

₍₅₎ whichisanalogoustotheGödelsolution(1)throughanatural cor-respondence

(

x0

_,

_x1

_,

_x2

_,

_x3

₎

_{→ (}

_t

_,

_r

_{, ϕ,}

_z

₎

_{.Now,inordertodescribe} quantitativelythisrotationaldynamics,onecan introducethe fol-lowingconstantsconstructedonthebaseofthe4-velocity:



β

=

c

βμνγ

√

−

gaμνγ

,

aμνγ

=

vμ

∂

γvν

,

(6)

wherethe4-velocityisgivenbythevector

vμ

= (

a

,

0

,

aex

1

,

0

),

vμ

= (

1

/

a

,

0

,

0

,

0

),

(7)

so,the



β is thevorticity vector andaμνγ isa completely anti-symmetrictensorcharacterizingtheorthogonalityofgeodesic tra-jectorieswithintheGödelsolution[28].

2.2. Gödel-typecase

Itisawell-known resultthatallGödel-typemetrics,i.e., homo-geneous space–timesexhibitingvorticity,characterized by



,and a givenvalueofm parameter2 _{[29,30],canbe rewrittenin} cylin-dricalcoordinatesas

ds2

=

dt2

+

2H

(

r

)

dtd

ϕ

−

G

(

r

)

d

ϕ

2

−

dr2

−

dz2

,

(8)

wherethefunctionsG

(

r

)

andH

(

r

)

mustobeytherelations

H

(

r

)

D

(

r

)

=

2

,

D

(

r

)

D

(

r

)

=

m 2

_,

(9)

the prime denotes the derivative withrespect to r. The solution ofEqs.(9)canbe dividedinthreedifferentclassesofGödel-type metricsintermsofm2: i)hyperbolicclass:m2

>

0, H

(

r

)

=

2



m2

[

cosh

(

mr

)

−

1

],

D

(

r

)

=

1 msinh

(

mr

),

(10)

ii)trigonometricclass:

−

μ

2

₌

_m2

_<

_0,

H

(

r

)

=

2



μ

2

[

1

−

cos

(

μ

r

)

],

D

(

r

)

=

1

μ

sin

(

μ

r

),

(11)

iii)linearclass:m2

₌

_0,

H

(

r

)

= 

r2

,

D

(

r

)

=

r

.

(12)

Thecasem2

₌

₂

_

2 _{isa particularcaseofthehyperbolicclass} which corresponds to Gödel solution [1]. It satisfies the relation

m2

= −

2



=

κρ

=

2



2,where



isthecosmologicalconstant,

ρ

is thematter density,



istherotation and

κ

=

8

π

G,withG being

thegravitationalconstant.

Aninteresting aspect ofGödel-typesolutionsisthe possibility forexistenceofCTCs.ThecircledefinedbyC

= {(

t

,

r

,

θ,

z

)

;

t

=

t0

,

r

=

r0

,

θ

∈ [

0

,

2

π

],

z

=

z0

}

is a CTC if G

(

r

)

becomes negative for a rangeofrc values(r1

<

rc

<

r2) [2],whererc isthe critical ra-dius,theminimalvalueofr allowingforexistenceofCTCs.Forthe hyperbolicclass(m2

>

0)thecriticalradiusis

sinh2



mrc 2



=



4



2 m2

−

1



−1

,

(13)

such that it is valid on the range ofparameters, 0

<

m2

<

4



2, andconsequentlythereexistsone non-causalregionwhenr

>

rc. On theother hand,the range,m2

_≥

₄

_

2_{, doesnot present CTCs,} i.e.,theregioniscompletelycausal,forinstance,thelimitingcase

m2

₌

₄

_

2 _impliesr

c

→ ∞

.The linearclass(m2

=

0) presentsone non-causal region,r

>

rc, suchthat thecriticalradius isgivenby

rc

=

1

/ 

.Thetrigonometricclass(m2

= −

μ

2

<

0)presentsan in-finite sequenceof alternating causalandnon-causal regions [14]. SoispossibletohaveCTCsforallthreeclasses.

Additionally,forthesakeofthesimplicity,wechoosethebasis3

2 _−∞<m2_=1/a4_<∞. 3 _{IndicesA,B,C ,}

ds2

=

η

A B

θ

A

θ

B

=



θ

0



2

−



θ

1



2

−



θ

2



2

−



θ

3



2

,

(14)

wherethe1-forms

θ

A

₌

_eA

α dxα aregivenby

θ

0

=

dt

+

H

(

r

)

d

ϕ

,

θ

1

=

dr

,

θ

2

=

D

(

r

)

d

ϕ

,

θ

3

=

dz

.

(15)

Withthisbasisinthetangentspace[5],itispossibletocompute important quantities such as the Ricci scalar R and the Einstein tensorGA B,obtaining R

=

2

(

m2

− 

2

),

(16) and G00

=

3



2

−

m2

,

G11

=

G22

= 

2

,

(17) G33

=

m2

− 

2

.

Theseresultswillbeusedinthenextsections. 3. Brans–Dicketheory

TheBrans–Dicke(BD)theoryisthefirstandthebestmotivated model introduced within the context of scalar–tensor gravity. It represents itself as a natural extension forthe general relativity andwas originally proposed to be totally compatible withMach ideas and the weak equivalence principle (WEP) [23,31]. Within thistheory,inertialmassesof bodiesandparticles are treatedas consequences oftheir interactions withsome cosmic field rather than fundamentalconstants[32].

Originally,BransandDickesuggestedthattheaction ofa new gravitytheoryshouldbesimilartotheEinstein–Hilbertactionbut includinganadditionalnon-minimalscalarfieldcoupling:

S

=

 √

−

g



φ

R

−

ω

φ

∂

μ

φ∂

μ

_φ

₊

₁₆

_π

_L

m



d4x

,

(18)

where R is the scalar curvature,

φ

is a scalar field treatedas a somegeneralizationofthegravitationalconstant(

φ

∝

G−1),which measuresits scalelocally.Furtherwewillrefer toitasto theBD field. Also,

L

m is the matter Lagrangian which does not depend on

φ

,so,

∂

φ

L

m

=

0.Finally,the

ω

isadimensionlessconstant rep-resentingitselfastheuniquefreeparameterinthetheory.Varying thisactionwithrespectto

φ

andgμν ,wearriveattheoriginalBD fieldequationslookinglike

2

ω

φ

φ −

ω

φ

2

∂

μ

φ∂

μ

_φ

₊

_R

₌

₀

_,

₍₁₉₎ Rμν

−

1 2gμνR

=



8

π

φ



Tμν

+

ω

φ

2



∂

μ

φ∂

ν

φ

+ −

1 2gμν

∂

ρ

φ∂

ρ

_φ



+

1

φ



∇

ν

(∂

μ

φ)

−

gμν

φ



,

(20)

withthecovariantd’AlembertianoperatoractsontheBDfieldas

φ = ∇

μ

(∂

μ

φ)

=

∂

μ

√

−

g

∂

μ

φ

√

₋

g

.

(21)

MultiplyingtheEq.(20)bytheinversemetricgμν ,weget

R

= −



8

π

φ



T

+

ω

φ

2

∂

ρ

φ∂

ρ

_φ

₊

3

φ

φ,

(22)

whichwecancombinewiththeEq.(19),obtaining

φ =



8

π

3

+

2

ω



T

.

(23)

Thisequation isevidently consistentwiththeMachprinciple, be-causeofthedirectrelationshipbetweenmattercontent character-izedbyT ,andtheBDfield

φ

characterizingtheinertialproperties ofthegravity.Itisimportanttoemphasizethat,despitethe

mat-ter and the BD field

φ

seem to be decoupled in the action of

thetheory,sincetheycorrespondtodifferentcontributionsinthe Lagrangian, they turn out to be strongly related because of this equation. Additionally,asaconsequenceofthefactthat the mat-ter Lagrangian does not depend on

φ

,there is no possibility for spontaneousmattercreationcausedbyBDfield,sincetheenergy– momentumtensorofmatterobeysthe

φ

-independentequation

∇

νTμν

=

0

,

(24)

hencesatisfying theWEP.

Now,weplantostudytheconsistencyoftheGödeland Gödel-typesolutionswithinBDmodel.

4. GödeluniverseinBrans–Dickegravity

Consideration of the Gödel solution within the BD gravity is equivalent to suggesting the possibility to have a non-stationary

Gödel solution, since the BD field

φ

should depend at least on

the time t [17]. From now on we suppose that the scalar field

φ

depends either on the time t or on z coordinate. These

de-pendencies havecertain physicalinterpretations,forexample,the

t dependence is motivated by cosmologicalreasons whereas the

z dependence –by the axial symmetry characterizingthe metric ofGödel.

TostudytheGödeluniverseinBDgravity, onecanrewritethe fieldequation(20)as Rμ_ν

−

1 2

δ

μ νR

=



8

π

φ



Tμ_ν

+

ω

φ

2



∂

μ

φ∂

ν

φ

−

1 2

δ

μ ν

∂

ρ

φ∂

ρ

φ



+ φ

−1

_∇

ν

∂

μ

φ

− δ

μν

φ

,

(25)

and assume the energy–momentum tensor and4-velocity of the

mattertobegivenby

Tμν

=

ρ

vμvν

,

vμ

= (

1

a

,

0

,

0

,

0

),

vμ

= (

a

,

0

,

ae

x

_,

₀

_).

₍₂₆₎

As thesimplestexample,we assume theBDscalar tobe only timedependent,

φ

= φ(

t

)

,whichcorrespondstothecosmologically interestingsituation(indeed,suchachoicereflectsthefactthatthe Universeishomogeneousandisotropic)onefindsthecomponents oftheequation(25)intheform

(

0

,

0

)

:

1 2a2

=



8

π

φ



ρ

−

ω

2a2

 ˙

φ

φ



2

,

(

i

,

i

)

: −

1 2a2

=

ω

2a2

 ˙

φ

φ



2

+

1 a2

 ¨

φ

φ



,

(27)

(

0

,

2

)

:

1 a2

=



8

π

φ



ρ

,

(

1

,

2

)

= (

2

,

1

)

:

 ˙

φ

φ



=

0

,

(28) wherei

=

1

,

2

,

3.

However,thissystemturnsouttobeinconsistentexceptofthe trivialcase.Indeed,fromtheequationforthecomponent

(

1

,

2

)

we obtain:

φ (

t

)

=

C

,

(29)

whereC isan arbitraryconstant,thus,the BDscalarturns outto be trivial. Therefore we conclude that forthe case

φ

= φ(

t

)

, the Gödelmetricinapure BDmodelrepresentsitselfonlyasatrivial solution, withthe BD scalar is reduced just to a constant, thus, theBDtheoryisreducedtotheusualEinsteingravity.Thenatural questionnowis–whethertheBDgravitycanbeextended,andthe Gödelmetriccanbegeneralized,toachievetheconsistencyforthe nontrivialBDscalar?Toanswerthisquestion,wecanconsiderthe Gödel-typemetric originally proposed in [30] and introduce the cosmological constant. In this context, we will consider another possibilityforthe

φ

field,thatis,

φ

= φ(

z

)

.

5. Gödel-typesolutioninBD-



gravity

TheactionoftheBD-



theory[33]canbewrittenas

S

=

1 16

π

 √

−

g



φ (

R

−

2

)

−

ω

φ

∂

μ

φ∂

μ

_φ

₊

₁₆

_π

_L

m



d4x

.

(30)

Forthisstudy,weusethetangentspacetomakecalculations sim-pler.Thusthefieldequationscanbewrittenas

GA_B

− δ

A_B



=



8

π

φ



TA_B

+

ω

φ

2



∂

A

φ∂

B

φ

−

1 2

δ

A B

∂

C

φ∂

C

φ



+ φ

−1



∇

B

∂

A

φ

− δ

AB

φ



,

(31) where GA B

=

eμ₍A)e ν (B)Gμν

,

TA B

=

eμ₍A)e ν (B)Tμν

,

(32) and

η

A B

=

eμ_(A)e₍νB)gμν

,

∂

A

=

e μ (A)

∂

μ

,

∇

B

=

eν(B)

∇

ν

.

(33)

Now, we will add to our matter content an electromagnetic

fieldalignedonz-axisanddependentof z,suchachoiceproduces thefollowingnon-vanishingcomponentsofelectromagnetictensor inframe(15)

F(0)(3)

= −

F(3)(0)

=

E

(

z

),

F(1)(2)

= −

F(2)(1)

=

B

(

z

),

(34)

withthesolutionsoftheMaxwellequationsare

E

(

z

)

=

E0cos

[

2

(

z

−

z0

)

],

B

(

z

)

=

E0sin

[

2

(

z

−

z0

)

],

(35)

where E0 is the amplitude of the electric and magnetic fields.

Hence, the non-zero components of the energy–momentum

ten-sorfortheelectromagneticfieldare

T₍(₀ef₎₍)₀₎

=

T₍(₁ef₎₍)₁₎

=

T₍(₂ef₎₍)₂₎

=

E 2 0 2

,

T (ef) (3)(3)

= −

E2₀ 2

.

(36)

Asaconsequence,thenewenergy–momentumtensorisgivenby

Tμν

=

ρ

vμvν

+

Tμν(ef)

.

(37)

Next,wewillfindthesolutionsforthecases

φ (

t

)

and

φ (

z

)

.

5.1.

φ

= φ(

t

)

Inthiscasethed’Alembertianoperatorgetstheform

φ =

η

A B



_∂

B

(∂

A

φ)

−

wCA B

(∂

C

φ)



,

φ =



D2

₋

_H2 D2



¨φ,

(38)

wherewC_{A B}aretheRiccicoefficientsofrotation.Thusthediagonal componentsoftheequations(31)are

(

0

,

0

)

3



2

−

m2

−  =



8

π

φ



ρ

+



4

π

φ



E2₀

+

ω

2

 ˙

φ

φ



2

+

ω

2

 ˙

φ

φ



2 H2 D2

+

¨φ

φ

H2 D2

,

(

1

,

1

)

− 

2

−  = −



4

π

φ



E2₀

−

ω

2

 ˙

φ

φ



2

+

ω

2

 ˙

φ

φ



2 H2 D2

−

¨φ

φ

D2

−

H2 D2

,

(39)

(

2

,

2

)

− 

2

−  = −



4

π

φ



E2₀

−

ω

2

 ˙

φ

φ



2

−

ω

2

 ˙

φ

φ



2 H2 D2

−

¨φ

φ

,

(

3

,

3

) 

2

−

m2

−  =



4

π

φ



E2₀

−

ω

2

 ˙

φ

φ



2

+

ω

2

 ˙

φ

φ



2 H2 D2

−

¨φ

φ

D2

−

H2 D2

,

andthenon-diagonalcomponentsare

(

0

,

1

)

H H  2D2

˙φ

φ

=

0

,

(

0

,

2

)

H D

ω

 ˙

φ

φ



2

+

¨φ

φ



=

0

,

(40)

(

1

,

2

)

H _D

₋

_{H D} D

ω

 ˙

φ

φ



2

+

¨φ

φ



=

0

.

A direct inspection of the component

(

0

,

1

)

implies that

φ

should be constant (we note that H cannot be constant since

it is fixed fromthe requirement of the space–time homogeneity

Eq.(9)).Therefore, inthiscasetheGödel-typesolutions inBD-



gravity reduce totheGR solutionsforone well-motivatedmatter whosesolutioniswellknown[30].

5.2.

φ

= φ(

z

)

Inthiscasethed’Alembertianoperatoractson

φ

as

φ = −φ



_,

₍₄₁₎

andthenon-zerocomponentsofthefieldequationare

(

0

,

0

)

3



2

−

m2

−  =



8

π

φ



ρ

+



4

π

φ



E2₀

+

ω

2



φ



φ



2

+

φ



φ

,

(

k

,

k

) 

2

+  =



4

π

φ



E2₀

−

ω

2



φ



φ



2

−

φ



φ

,

(42)

(

3

,

3

) 

2

−

m2

−  =



4

π

φ



E2₀

−

ω

2



φ



φ



2

.

Thesefieldequationsimplytherelations

4



2

−

m2

=



8

π

φ



(

ρ

+

E2₀

),

m2

+

2



= −

φ



φ

.

(43)

Discussingtheseequations(43),itispossibletodifferthreecases, cf.[29,30]:

(i)If

ρ

=

0 and E0

=

0 thecondition 4



2

=

m2 isfound,since the solution belongs to the hyperbolic class and is completely causal.Wenote thatthissolutionis consistentwiththeequation ofmotionofthescalarfield

−φ



=

1

3

+

2

ω

(

8

πρ

+

2

φ).

(44)

(the traceofthe energy–momentumtensoris T

=

ρ

(it doesnot depend of E0) which in this case gives zero), so, the equation for

φ

yields theexponentialsolution

φ (

z

)

=

C1ekz

+

C2e−kz,with

k

=



−

2

3+2ω , this form of the solution is valid when

 <

0. If

 >

0 we get

φ (

z

)

=

C3cos

(

kz

)

+

C4sin

(

kz

)

, wherek

=

ik. Us-ingEqs. (43)–(44)them parameteris relatedwithk throughthe relation m2

+

2



=

2



3

+

2

ω

=⇒

m2 4

+  = −

k 2



₁

₊

ω

2



.

(45)

We note that



plays an important role since the parameters of the metricm2 _and

_

2_{,as well asthe field}

_φ

_{, can be written in} termsofit.Thenthiscase(thatis,thevacuumsolution)represents one completelycausal solutionof theGödel-typeuniverse inthe BD-



formalism.Thesamesolution(m2

₌

₄

_

2_{)hasbeenobtained} in GR-



context forthe massless scalar field asthe only matter source [30]. In addition, in the limit

ω

→ ∞

we can show the similarityamongBDandGRtheory(itisexpectedthatinthislimit theBDfield equationsreduce to GRfield equationsforthesame energy–momentumtensor,formorediscussions onthisissuesee [34–36]).Bytakingthislimitweget

φ (

z

)

≈ φ

0

(

1

±

kz

)

,whereC1

=

C2

= φ

0

=

1

/

G.

Usingthislimitandthevierbein,i.e.,

∂

A

=

eAμ

∂

μ,

∂

A

=

η

A B

∂

B, onecanrewriteeq.(31)as

GA B

= 

η

A B

−

1

2



σ

A B

+

O

(

1

/

√

ω

),

(46)

where

σ

(3)(3)

= −

3 and

σ

A B

=

η

A B, with A

,

B

=

3. Therefore, in thiscase, the solution for

ω

→ ∞

does not recover the vacuum Einstein field equations, asshown in[35] when the trace ofthe

energy–momentum tensor vanishes the BD theory. However, we

caninterpretthe term 1₂



σ

A B inEq.(46),asonecontributionto

the energy–momentum tensor andrecover the same completely

causalsolutionobtainedin[30]when

ω

→ ∞

.

In this way, we conclude that the vacuum-solution of BD-



fieldequationsforGödel-typemetricsiscompletelycausaland,in thelimit

ω

→ ∞

issimilartheGRwitha non-zero



anda spe-cificmattersource,asforinstancethescalarfield(

φ (

z

)

∝

z)used inwork[30].

(ii)If

ρ

>

0,itisnecessarytorequire ρ_φ

=

const,inordertoget solutionsconsistentwiththeEqs.(43)–(44).Inaddition,the solu-tionsarerestrictedbytheinterval,m2

_<

₄

_ω

2_{,thusitispossibleto} carryoutthefollowinganalysisofthesolutions(43):

•

0

<

m2

_<

₄

_

2 _{–solutionsofthehyperbolicclass,thereisone} non-causal regionforagivenr

>

rc givenbyEq.(13);

•

m2

=

0 –solutionsofthelinearclass,sincethereisone non-causal regionforagivenr

>

rc givenbyrc

=

1

/

ω

;

•

m2

_{= −}

_μ

2

_<

_{0 – solutions of the trigonometric class, thus}

there is an infinite number of alternating causal and

non-causal regions.

However, in this case, if

ρ

is constant, one should have

φ

=

const as well, and the situation becomes trivial reducing to the usual Einstein gravity. The possible nontrivial solutions can look

like

ρ(

z

)

=

C1

φ (

z

)

=

C1

/

Ge f f

(

z

)

, such that the decrease of the effective gravitational coupling, Ge f f

(

z

)

, implies the growth of the density (

ρ

) and reciprocally, for C1

>

0. However, for the

linear class (m2

₌

_{0) we have nontrivial solution. We choose}

ρ

=

C1cos kz,

φ

=

C2cos kz, with E0

=

0. In this case, the equa-tions(42)becomepurelyalgebraicones:



2

=

2

π

C1 C2

,

(47)



=

2

π

C1 C2 1

+

ω

.

(48)

From these equations, one can find



2 _and

_

_{as functions of}

the parameters

ω,

C1

/

C2. It is clear that the equation (44) is also consistent with thesesolutions yielding the relationk2_C

2

=

1

3+2ω

(

8

π

C1

+

2



C2

)

.

(iii) If

ρ

<

0, the condition 4



2

<

m2 takes place when

ρ

<

−

E2₀. This condition implies that there is no CTCs in the cor-responding Gödel-type space–time.However, again, a constant

ρ

implies a constant

φ

aswell, which reduces thesituation to the usual Einsteingravity,withthissolutionitselfisexcluded sinceit corresponds to m2

_≤

₂

_

2 _{(thatis, just the resultfollowing from} ourequationsat

φ

=

const),whichis incompatiblewithour con-dition4



2

<

m2.So,thissituationisinconsistent.

Atthesametime,onecannoticethatfromequationm2

₊

₂

_

₌

−

φ

φ thescalarfieldisfoundas

φ (

z

)

=

C cos

γ

z

+

D sin

γ

z

,

where

γ

2

₌

₂

_

₊

_m2_{. Also, one can see that the original Gödel} universe,m2

_{= −}

₂

_

_{,isonlypossibleifthescalarfieldisconstant.} 6. Conclusions

We discussed the Gödel and Gödel-type solutions within the context of theBD gravity. In ourstudy, theconsistency of Gödel solutionwithinBDgravitationalformalismhasbeenreviewed,and we showedtheimportanceofthenon-zerocosmologicalconstant



inorder tohavea nontrivial solution,describing theexpected valuesofdifferentparametersanalogoustotheGR.Thefield equa-tions of theBD-



formalism were solved forthe two cases

φ (

t

)

and

φ (

z

)

bothforGödelsolutionandGödel-typesolution.Forthe GödelsolutioninBD-



formalismtheconsistencyand correspon-dencewithGRwere verified.Afterwards,theGödel-typesolution with

φ (

t

)

wasconsidered,andaconditionallowingtoreducethis solution totheoriginal Gödelmetric wasdetermined by the sys-temofequations.Whenthe scalarfielddependsonlyon z,there aretwo possibilitiesdependingonthesignofthematterdensity: (i) emptycausalGödel-typeuniverse,whichcorrespondstothe ex-act solution with m2

=

4

ω

2_{, such a solution has been obtained} withtherequirementof



=

0.Additionally, weverified that,for theconstant densitycase(whichisamoreusual situationwithin thecosmologicalstudiessinceitreflectsthelarge-scale homogene-ity andisotropyofthe space), inthe limit

ω

→ ∞

, thissolution

reduces to GR with one massless scalar field, with some

impli-cations associated to cosmologicalconstant; (ii) both causal and non-causal regions are allowed. Therefore, our study also shows that theidea ofBDtheoryasa totallyMachiantheory should be revisitedanddiscussedinmoredetails.

Acknowledgements

This work was partially supported by Coordenação de

Aper-feiçoamento de Pessoal de Nível Superior (CAPES), andConselho

J.A. Agudelowouldlike to thankto all colleagues of Institutode Física in UFMT-Cuiabá by the shared knowledge andtime. A.F.S.

has been supported by the CNPq projects 476166/2013-6 and

201273/2015-2. The work by A.Yu.P. has been supported by the

CNPqproject303783/2015-0. References

[1]K.Gödel,Rev.Mod.Phys.21(1949)447.

[2]M.J.Rebouças,M.Novello,Astrophys.J.225(1978)719.

[3]G.F.R.Ellis,S.W.Hawking,TheLargeScaleStructureofSpace–Time,first edi-tion,CambridgeUniversityPress,London,1975.

[4]M.Schiffer,R.Adler,M.Bazin,IntroductiontoGeneralRelativity,second edi-tion,McGraw-Hill,Tokyo,1965.

[5]M.J.Rebouças,J.E.Aman,A.F.F.Teixeira,J. Math.Phys.27(1986)1370.

[6]M.P.Dabrowski,J.Garecki,Phys.Rev.D70(2003)043511.

[7]V.M.Rosa,P.S.Letelier,Phys.Lett.A370(2007)99.

[8]M.Buser,E.Kajari,W.P.Schleich,NewJ.Phys.15(2013)013063.

[9]S.Nojiri,S.D.Odintsov,Int.J.Geom.MethodsMod.Phys.04(2006)21.

[10]T.Clifton,P.G.Ferreira,A.Padilla,C.Skordis,Phys.Rep.513(2012)1.

[11]M.J.Rebouças,J.Santos,Phys.Rev.D80(2009)063009.

[12]C.Furtado,T.Mariz,J.R.Nascimento,A.Yu.Petrov,A.F.Santos,Phys.Rev.D79 (2009)124039.

[13]A.F.Santos,Mod.Phys.Lett.A28(2013)1350141.

[14]J.B.Fonseca-Neto,A.Yu.Petrov,M.J.Rebouças,Phys.Lett.B725(2013)412.

[15]J.R.Nascimento,A.Yu.Petrov,A.F.Santos,W.D.R.Jesus,Mod.Phys.Lett.A30 (2014)1550011.

[16]A.F.Santos,A.C.Ulhoa,Mod.Phys.Lett.A30(2014)1550039.

[17]R.H.Dicke,C.Brans,Phys.Rev.124(1961)925.

[18]O.Obregon,H.Dehnen,Astrophys.SpaceSci.14(1971)1677.

[19]S.Banerji,Phys.Rev.D9(1974)877.

[20]T.Clifton,P.G.Ferreira,A.Padilla,C.Skordis,Phys.Rep.513(2012)1.

[21]A.Linde,Phys.Lett.B238(1990)160.

[22]M.Artymowski,Z.Lalak,J. Cosmol.Astropart.Phys.506(2014)15.

[23]K.Maeda,Y.Fujii,TheScalar–TensorTheoryofGravitation,firstedition, Cam-bridgeUniversityPress,Cambridge,2004.

[24]T.Rador,Phys.Lett.B652(2007)228.

[25]A.Felice,S.Tsujikawa,J. Cosmol.Astropart.Phys.1007(2010)1.

[26]Y.Bisabr,Astrophys.SpaceSci.339(2012)87.

[27]R.G.Vishwakarma,Int.J.Geom.MethodsMod.Phys.1(2015).

[28]K.Gödel,Gen.Relativ.Gravit.32(2000)1149.

[29]S.N.Thakurta,A.K.Raychaudhuri,Phys.Rev.D22(1980)802.

[30]J.Tiomno,M.J.Rebouças,Phys.Rev.D28(1983)1251.

[31]C.H.Brans,Phys.Rev.125(1962)2194.

[32]S. Weinberg,Gravitation and Cosmology: PrinciplesandApplications ofthe GeneralTheory,Wiley,1972.

[33]D. Tretyakova, A. Shatskiy,I. Novikov, S. Alexeyev,Phys. Rev.D 85 (2011) 124059.

[34]C.Romero,A.Barros,Phys.Lett.A173(1992)243.

[35]V.Faraoni,Phys.Rev.D59(1999)084021.