Maximum Entropy Principle for Self-gravitating Perfect Fluid in Lovelock Gravity

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by  Li-Ming Cao

ICTS-USTC-13-01
MaximumEntropy PrincipleforSelf-gravitatingPerfect FluidinLovelock Gravity
Li-Ming Cao∗, Jianfei Xu† and Zhe Zeng‡
Interdisciplinary Center for Theoretical Study
University of Science and Technology of China
Hefei, Anhui 230026, China
3 We consider a static self-gravitating system consisting of perfect fluid with isometries of an (n − 2)-
1
dimensionalmaximallysymmetricspaceinLovelockgravitytheory.Astraightforwardanalysisofthetime-time
0
componentoftheequationsofmotionsuggestsageneralizedmassfunction.Tolman-Oppenheimer-Volkofflike
2
equation is obtained by using this mass function and gravitational equations. We investigate the maximum

n entropy principle in Lovelock gravity, and find that this Tolman-Oppenheimer-Volkoff equation can also be
a deducedfromthesocalled“maximumentropyprinciple”whichisoriginallycustomizedforEinsteingravity
J theory.Thisinvestigationmanifestsadeepconnectionbetweengravityandthermodynamicsinthisgeneralized

1 gravitytheory.
2

]
c I. INTRODUCTION
q
-
r The relation among gravity, thermodynamicsand quantum theory has been studied for abouthalf a century. Starting from
g Einstein equations and appropriate definitions of mass, angular-momentumand charges, people have found the four laws of
[
black hole mechanics[1]. Once the quantumfield on the spacetime of a black hole is considered, it is foundthat black hole

2 behaveslike a blackbodywitha temperaturewhichis proportionalto thesurfacegravityatthe horizonofthe blackhole[2].
v Then,theblackholemechanicsisnaturallyinterpretedasblackholethermodynamicswithanentropywhichisonequarterofthe
5 areaofthehorizon[3]. Sincetheseseminalworks,thethermodynamicsofblackholesandotherpossiblespacetimesinvarious
9
gravitytheorieshavebeenwidelydiscussed,andnowpeoplebelievethattheblackholeisarealthermodynamicalsystem.
8
Actually, for a given gravity theory, assuming some spacetime which contains horizon(s) and has a good asymptotical be-
0
havior, then, based on the related gravitationalequations, we could study the correspondingconservedcharges, the geometry
.
1 ofthehorizonandsomepossiblequantumeffects, andthendevelopthemechanicsorthermodynamicsofthisspacetime. For
0
example, the thermodynamics of stationary black holes in general differmorphism invariant gravity theory has been estab-
3
lished [4](referencestherein). This is a traditionalway to study the thermodynamicsof the spacetime since the beginningof
1
: 1970s.Ontheotherhand,somepeoplebelievethatthegravitationalequations(orapartofthegravitationalequations)canalso
v
be derived from some fundamentalrelations in thermodynamics. Once the logic is turned around, some problems arise: For
i
X instance,in thetraditionaldiscussionofthermodynamicsofspacetime,thegloballydefinedconservedcharges(suchasADM
mass) suggeststhat it is quite difficultto extractusefullocalinformationfrom the law of thermodynamics. So, to solve such
r
a problem,onehastointroducesomelocallyorquasilocallydefinedthermodynamicquantitiesandconsiderthelocalorquasilocal
versionof thethermodynamicrelations. Thegravitationalequations(orsome partofthese equations),suchasEinstein equa-
tions,arereallypossibletobederivedfromtheselocalorquasilocalthermodynamicquantitiesandassociatedthermodynamic
relations. Infact,basedonthedefinitionoflocalRindlerhorizons,JacobsonshowsthattheEinsteinequationscanbederived
fromthe Clausiusrelation[5, 6]. Inthecase withenoughisometries, basedona particularquasilocalmass, i.e., theso called
Misner-Sharpenergy[7, 8] (a typical quasilocalenergy), one can also derivea partof Einstein equationfromthe first law or
Clausiusrelationofthermodynamics[9–12].
Forstaticself-gravitatingfluidsystems, maximumentropyprinciplecanbeusedtodeduceapartoftheEinsteinequations,
i.e, Tolman-Oppenheimer-Volkoff(TOV)equation. Thisprincipleisfirst appliedbyCocke [13]to the self-gravitatingperfect
fluidsphereandimprovedbySorkin,WaldandZhang(SWZ) [14]. InthediscussionofSWZ,onlythetime-timecomponent
oftheEinsteinequationsisused.However,SWZ’sdiscussionwasrestrictedtoradiation.Recently,GaohasgeneralizedSWZ’s
discussiontoanarbitraryperfectfluid,andhassuccessfullygottheusualTOVequationinEinsteintheorybytheprincipleof
∗e-mailaddress:[email protected]
†e-mailaddress:[email protected]
‡e-mailaddress:[email protected]
2
maximumentropy[15]. Inthesediscussions,theexistenceofa“massfunction”m(r)isquiteimportant. Actually,itisjustthe
Misner-Sharpenergyapplyingto this static system [8]. This mass functionis veryspecial, and it only exists in some special
gravitytheories,suchasEinsteingravityandLovelockgravity[16].
The maximumentropyprincipleclearly and profoundlyrevealsthe deep connectionbetween thermodynamicsand gravity.
However, at present, only Einstein gravity theory is considered. Our question is: Whether this principle is valid or not in
moregeneralgravitytheories? Inthispaper,wewillconsiderastaticself-gravitatingsystemconsistingofaperfectfluidwith
isometriesofan(n 2)-dimensionalmaximallysymmetricspaceinLovelockgravitytheory.Firstly,generalizedTOVequation
−
in this theory is presented, then, we show that the maximum entropy principle can also be applied to this self-gravitating
perfect fluid system: Correct TOV equation can be obtained by a similar deduction as in [15]. Our results show that the
implicitrelationship betweengravity and thermodynamicsin Einstein gravitystill exists in Lovelocktheory, which manifests
thepossibilityofturningthelogicaroundtousethermodynamicstodescribethemoregeneralgravitationalphenomena.
This paper is organized as follows: In Sec.2, we give a brief review of the Lovelock gravity. In Sec.3, starting from the
equations of motion of the Lovelock gravity, we introduce a generalized mass function for the system with isometries of an
(n 2)-dimensionalmaximallysymmetricspace.Basedonthismassfunction,generalziedTOVequationisobtained.InSec.4,
−
weapplythemaximumentropyprincipletothissystemtoderivetheTOVequation. Section5isdevotedtotheconclusionand
discussion.
II. LOVELOCKGRAVITY
Lovelockgravity[16]isanaturalgeneralizationofgeneralrelativityinhigherdimensionsinthesensethattheequationsof
motion of Lovelock gravity do not contain more than second order derivatives with respect to metric, as the case of general
relativity.Actually,withthedevelopmentofstringtheoryandsupergravityoverthepastyears,Lovelockgravityhasdrawnalot
ofattention. ItiswellknownthatEinsteingeneralrelativitynaturallyarisesfromstringtheories. Ascorrectionsfrommassive
statesofstringtheoriesandfromloopexpansionsinstringtheories,thehigherordercurvaturetermsinlovelocktheoryappear
in the low-energyeffectiveactionof stringtheories[17–21]. So itis importantto investigatetheeffectsofthese higherorder
terms. Forexample,theseeffectshavebeenmanifestedinblackholesolutions[22,23].
TheLagrangiandensityofLovelockgravityconsistsofthedimensionallyextendedEulerdensitieswhicharegivenby
[n/2]
L= α L , (2.1)
i i
i=0
X
whereα sareconstantsandL saretheEulerdensitiesof2i-dimensionalmanifoldsdefinedby
i i
1
L = δa1b1...aibiRc1d1 ...Rcidi . (2.2)
i 2i c1d1...cidi a1b1 aibi
Thesymbolδdenotesatotallyanti-symmetricproductofKroneckerdeltasdefinedas
δa1...ap =p!δa1...δap . (2.3)
b1...bp [bp bp]
TheactionforLovelockgravityisgivenby
[n/2]
1
S = dnx√ g α L +S . (2.4)
2κ2 − i i matter
n Z i=0
X
Followingfromthevariationofthisaction,thegravitationalequationiswritteninacompactform
[n/2]
= α G(i) =κ2T , (2.5)
Gab i ab n ab
i
X
where
1
G(i)f = δfa1b1...aibiRc1d1 ...Rcidi (2.6)
e −2i+1 ec1d1...cidi a1b1 aibi
isdeducedfromthevariationofL , andT istheenergy-momentumtensorformatterfieldsobtainedfromthematteraction
i ab
S . Inthecasewhereonlyα andα arenonvanishing,wegetusualEinsteingravitywiththecosmologicalconstant.Ifα
matter 0 1 2
isalsononvanishing,wegettheGauss-Bonnetgravitytheory.
3
III. TOVEQUATIONSINLOVELOCKGRAVITYTHEORY
A. staticperfectfluidinLovelockgravity
Letusconsideraperfectfluidsystemwithsymmetriescorrespondingtotheisometriesofan(n 2)-dimensionalmaximally
−
symmetric space in Lovelock gravity. Additionally, we assume the system is static (an additional isometry). The spacetime
of the system automaticallyhas such isometries. By choosingcoordinates t,r,zi with i =,1, ,n 2, the metric of the
{ } ··· −
spacetimeisgivenby
ds2 = e2Φ(r)dt2+e2Ψ(r)dr2+r2γ dzidzj, (3.1)
ij
−
where γ dzidzj is the metric of an (n 2)-dimensionalmaximally symmetric manifold. Assuming the energy-momentum
ij
−
tensoroftheperfectfluidisgivenby
T =ρu u +p(g +u u ),
ab a b ab a b
whereua isatimelikevectorfield whichisthetangentoftheworldlineofthefluidelement,then, wecanfurtherchoosethe
timecoordinatetsothatu = eΦ(dt) . Actually,uaisjustthevectorfieldforthestaticobserver(whichisalsothecomoving
a a
−
observerofthefluid). Theenergydensityρisjustmeasuredbythisobserver.
ItiseasytofindthatthenontrivialcomponentsoftheRiemanntensorofsuchspacetimeisgivenby
e−2Ψ
Rtr = 2e−2Ψ(Φ′2 Φ′Ψ′+Φ′′), Rti = Φ′δi ,
tr − − tj r j
k e−2Ψ e−2Ψ
Rij = − δi δj , Rri = Ψ′δi . (3.2)
kl r2 k l rj − r j
wherek = 0, 1correspondstothesectionalcurvatureofthemaximallysymmetricmanifold. Aftersubstitutingtheseresults
±
intoEq.(2.5),theequationsofmotionforthissystemaregivenby:
[n/2]
κ2ρ= 1 d αi(n−2)! rn−1−2i k e−2Ψ i , (3.3)
n rn−2dr( 2(n 2i 1)! − )
i=0 − −
X (cid:0) (cid:1)
whichcomesfrom t =κ2Tt,and
Gt n t
[n/2] iα (n 2)! e−2ΨΦ′ k e−2Ψ i−1
κ2p= i − −
n (n 2i 1)! r r2
i=0 − − (cid:18) (cid:19)
X
[n/2] α (n 2)! k e−2Ψ i
i − − , (3.4)
− 2(n 2i 2)! r2
i=0 − − (cid:18) (cid:19)
X
which is given by r = κ2Tr. Obviously, equation r = κ2Tr is trivially satisfied. In Einstein gravity theory without
Gr n r Gt n t
cosmologicalconstant,Eq.(3.3)suggeststhatwecandefineamassfunctionm(r)as
r
m(r):=ω ρxn−2dx, (3.5)
k
Z0
suchthattheleftofEq.(3.3)canbeexplainedasthechangeofthismassalongtheradialdirection,i.e.,m′ = ρω rn−2. Here,
k
“′”denotesthederivativerespecttor,andω isthevolumeofthemaximallysymmetricmanifoldwiththesectionalcurvature
k
k,i.e.,
ω := dn−2z√γ.
k
Z
Actually,Eq.(3.3)suggeststhatthismassfunctionisnothingbutthesocalledMisner-Sharpenergy[7,8]insidethespherewith
radiusr:
(n 2)ω
m(r)= − k rn−3 k e−2Ψ . (3.6)
2κ2 −
n
(cid:0) (cid:1)
Ofcourse,hereweareconsideringastaticfluidinn-dimension,sothisisgeneralizedversionoftheMisner-Sharpenergyinthis
higherdimensionalEinsteingravitytheory(forinstance,see[11]andreferencestherein).
4
IngeneralLovelockgravitytheory,wecanalsodefineamassfunctionm(r)fortheperfectfluid,whichisgivenby
[n/2]
m(r)= ωk αi(n−2)! rn−1−2i k e−2Ψ i . (3.7)
2κ2 (n 1 2i)! −
n i=0 − −
X (cid:0) (cid:1)
SoEq.(3.3)justmeans
ρω rn−2 =m′. (3.8)
k
This is the same as the case in Einstein gravity theory. It should be emphasized here: To get Eq.(3.8), only the time-time
componentofthegravitationalequationsisrequired.
MoregeneraldefinitionoftheMisner-SharpenergyinLovelockgravitytheorycanbefoundinreferences[24]and[25]. An
explicitformofthisenergyinVaidyaspacetimehasbeenstudiedin[26]. However,thedefinitionofthemassfunction(3.7)is
enoughtodiscusstheprobleminthispaper.
TogettheTOVequation,letusconsiderBianchiidentity(orlooselyspeaking,energy-momentumconservationequation). It
iseasytofind
∂ b ∂ b ∂ b
aT =(ρ+p) ua u + p=0. (3.9)
ab a b b
∂r ∇ ∂r ∇ ∂r ∇
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
Forthevectoruaofthestaticobserver,wehave(∂/∂r)bua u =Φ′. Soweget
a b
∇
p′
Φ′ = . (3.10)
−ρ+p
BysubstitutingthisexpressionintoEq.(3.4),weobtain
p′ [n/2] iα (n 2)! i−1
i − rn−1−2ie−2Ψ k e−2Ψ
ρ+p( (n 1 2i)! − )
Xi=0 − − (cid:16) (cid:17)
[n/2]
= κ2prn−2 αi(n−2)! rn−2−2i k e−2Ψ i , (3.11)
− n − 2(n 2i 2)! −
i=0 − −
X (cid:0) (cid:1)
or
∂m p′ ∂m
= pω rn−2+m′ Ψ′ . (3.12)
k
∂Ψρ+p − − ∂Ψ
h i
Here∂m/∂ΨdenotesthederivativeofthemrespecttoΨwithrfixedasaparemeter.Eq.(3.12)isgeneralizedTOVequationin
Lovelockgravitytheory.Infollowingsubsection,twoexamplesaregiventounderstandthisgeneralexpression.
B. Examples
1. Einsteingravity
Ifwesetα = 2Λ= (n 1)(n 2)λandα = 1,thenwegetEinsteingravitytheory. Inthiscase,thegeneralizedmass
0 1
− − −
function(3.7)becomes
(n 2)ω
m= − krn−3 λr2+(k e−2Ψ) . (3.13)
2κ2 −
n
(cid:2) (cid:3)
Sowehave
∂m (n 2)ω
= 2m+ − krn−3(λr2+k), (3.14)
∂Ψ − κ2
n
and
∂m 1 (n 2)ω
m′ Ψ′ = (n 3)m+ − kλrn−1 . (3.15)
− ∂Ψ r − κ2
(cid:20) n (cid:21)
5
Thus,wegettheTOVequationsinn-dimensionalEinsteingravitywithacosmologicalconstant:
dp
= (ρ+p) F [p,m(r),r], (3.16)
E
dr − ·
whereF [p,m(r),r]isanalgebraiccombinationofp,m(r)andr,whichhastheform
E
κ2p+(n 2)λ ω rn−1+(n 3)κ2m(r)
F [p,m(r),r]= n − k − n . (3.17)
E r (n 2)(λr2+k)ω rn−3 2κ2m(r)
(cid:2) − (cid:3) k − n
Infourdimensionwithoutcosmologicalconstant(n(cid:2)=4andλ=0),wegettheusualTOV(cid:3)equationforthesystem,i.e.,
dp m(r)+4πpr3
= (ρ+p) . (3.18)
dr − ·( r[r 2m(r)] )
−
Actually,inthiscase,onlythecasewithk =1shouldbeconsidered.Sowehaveω =4π.BychoosingG=1,wegetκ2 =8π
k 4
andthenobtaintheresultabove.
2. Gauss-Bonnetgravity
ForGauss-Bonnetgravitytheory,we canalsogettheexplicitformof∂m/∂Ψin termsofm(r). Inthiscase, we setα =
0
2Λ=(n 1)(n 2)λ,α =1andα =α,andget
1 2
− − −
r2
e−2Ψ =k+ (1 f), (3.19)
2(n 3)(n 4)α ±
− −
where
(n 3)(n 4)κ2m(r)
f = 1 4(n 3)(n 4)λ+8α − − n . (3.20)
s − − − (n 2) ωkrn−1
−
Sothesolutionhastwobranches.However,onlythenegativeonein(3.19)canreducetoEinsteingravitytheoryinlargerlimit.
Inthefollowingdiscussion,onlythiscasewillbeconsidered.Aftersomecalculation,weget
∂m (n 2)ω r2f
= − krn−3f k , (3.21)
∂Ψ κ2n " − 2(n−3)(n−4)α#
and
∂m 1 m(r)
m′ Ψ′ = 2(n 5)κ2 +(n 1)(n 2)λω rn−2
− ∂Ψ 2κ2 − n r − − k
n
h i
(n 2)ω rn−2
− k (n 3)(n 4)(n 5)λ+1 f . (3.22)
−2(n 3)(n 4)ακ2 − − − −
− − nh i
Sowegettheexplicitformof∂m/∂Ψandm′ (∂m/∂Ψ)Ψ′intermsofthemassfunctionm(r),andthenthegeneralizedTOV
−
equationinGauss-Bonnetgravitytheory:
dp
= (ρ+p) F [p,m(r),r], (3.23)
GB
dr − ·
with
r (n 3)(n 4)(n 5)λ+(1 f) (n 3)(n 4)α
F [p,m(r),r]= − − − − + − −
GB
f 2(n 3)(n 4)αk+r2(1 f) n 2
(cid:2) − − − (cid:3) −
2κ2pω rn−1+2(n 5)κ2m(r)+(n 1)(n 2)λω rn−1
n (cid:2)k − n (cid:3)− − k . (3.24)
× ω rn−2f 2(n 3)(n 4)αk+r2(1 f)
k
− − −
Obviously,F [p,m(r),r]isexplicitlyexpressedinterm(cid:2)sofm(r),randp.Thisissimilar(cid:3)totheoneinEinsteingravitytheory.
GB
Inprinciple,forgeneralLovelockgravitytheorieswithn 10,wealsohavethisconclusion. Fortheseperfectfluidsystems,
≤
see[27,28]forrelevantdiscussions.
6
IV. GENERALIZEDTOVEQUATIONFROMMAXIMUMENTROPYPRINCIPLE
Inprevioussection,wehavegetthegeneralizedTOVequationinLovelockgravitybyusinggravitationalequations.Recently,
in [15], Gao has used the maximum entropy principle for self-gravitating perfect fluid in Einstein theory to deduce the TOV
equationtorevealtheimplicitrelationshipbetweengravityandthermodynamics.Themaximumentropyprinciplecangivepart
ofinformationsofgravitationalequationsinEinsteintheory,i.e.,theTOVequation. Inthissection,weshowthatsuchkindof
implicitconnectionalsoexistsinLovelocktheory.Ourworkcovertheresult[15]exceptthecasewithcharges.
A. Localthermodynamicrelationsofperfectfluid
We begin with a brief introductionof local thermodynamicrelationsof perfectfluid. The perfectfluid system satisfies the
familiarfirstlaw
1 p µ
dS = dE+ dV dN. (4.1)
T T − T
TheentropyS,energyE andparticlenumberN areallextensivevariables. Theirdensityvariablesaredenotedbys, ρandn
respectively.ApplyingEq. (4.1)toanunitvolume,onecaneasilyfindthefollowingdensityrelations
1 µ
ds= dρ dn. (4.2)
T − T
In the followingsubsection, we will treats asthe functionof two independentvariables(ρ,n), i.e., we will use generalstate
equation s = s(ρ,n) without imposing any additional constraint between ρ and n. It is also easily to find the fundamental
thermodynamicsrelation
Ts+µn=ρ+p (4.3)
bytheextensivepropertyoftheentropy(Eulerrelation).
B. Maximumentropyprincipleofperfectfluid
Now, we apply the maximumentropy principle to this self-gravitatingperfectfluid in Lovelockgravity. The metric of the
spacetime for this system is still given by Eq.(3.1). However, the situation is quite differentfrom the discussion in previous
section: here,onlytime-timecomponentofgravitationalequationsEq.(2.5)(orEq.(3.3))willbeusedasaconstrainttofindthe
TOVequation.OthercomponentsofEq.(2.5)mightbeviewedasunknowntous.
We havetoemphasizehere: Generally,togettheTOVequation,onehastouseothercomponents(suchastheradial-radial
component(3.4))ofthegravitationalequation(2.5). Ifthemaximumentropyprincipleplusthetime-timecomponentconstraint
equationreallyimpliestheTOVequation,wecanconcludethatthemaximumentropyprincipledoescontainpartofinformation
ofgravitationalequationsinLovelockgravitytheory,andthentherereallyexistssomeimplicitrelationshipbetweengravityand
thermodynamics. ThissuggeststhatwemayalsoturnthelogicaroundtoexplainthephenomenaofLovlockgravitybyusing
thermodynamics.
Byusingthetime-timecomponentofthegravitationalequation,i.e.,Gt = κ2ρasaconstraint,thedensityρcanbeviewed
t n
asafunctionofthemassmbytherelationEq.(3.8),i.e.
m′
ρ= ,
ω rn−2
k
where m is the Misner-Sharp energy which is given in Eq.(3.7). On any t = costant hypersurface, the total entropy of this
systemis
R
S =ω s ρ m′(r) ,n(r) eΨ(m(r))rn−2dr, (4.4)
k
Z0 h (cid:0) (cid:1) i
where s is the entropy density of the perfect fluid and R is the radius of the system. Since Ψ is determined by m and r by
an algebraic equation through the definition of the Misner-Sharp energy Eq.(3.6), we can regard Ψ as a function of m and r
algebraically,i.e.,Ψ=Ψ(m,r). However,forsimplicity,wehaveomittedthedependenceofΨuponrinEq.(4.4)above.
The entropy of an isolated system never decreases, and the entropy reaches the maximum when the isolated system is in
equilibrium. We hope to find the appropriate m(r) and n(r) to make the entropy maximum, and we consider the entropy is
7
maximumunderfollowingconditions: (i). TheparticlenumberN isfixedinthesystem. (ii). TheMisner-Sharpmassm(R)is
fixed.Thesetwoconditionswillbeusedsoon.
ThetotalparticlenumberN satisfies
R
N =ω n(r)eΨ(m(r))rn−2dr, (4.5)
k
Z0
wheren(r)isthenumberdensity(donotconfusewiththedimensionn). ThefixingofthetotalparticlenumberN isadesired
conditionformaximumentropyprinciple,soaccordingtothestandardmethodofLagrangemultipliers,theequationofvariation
becomes
δS+λδN =0, (4.6)
whereλisaLagrangemultiplier. Thetotalmassm(R)isfixedandthemassinsidether = 0sphere,i.e.,m(0),mustbezero,
whichisalsofixed,sotheirvariationshavetosatisfy
δm(0)=δm(R)=0. (4.7)
Additionally,wesupposethenumberdensitiesatr =0andr=Rarealsofixedunderthevariation,i.e.
δn(0)=δn(R)=0. (4.8)
Bydefiningthe“totallagrangian”
L(m,m′,n)=s ρ(m′),n eΨ(m)rn−2+λneΨ(m)rn−2, (4.9)
theconstrainedEuler-Lagrangeequationsaregiven(cid:0)by (cid:1)
∂L
=0, (4.10)
∂n
d ∂L ∂L
=0. (4.11)
dr∂m′ − ∂m
Eq.(4.10)yields
∂s
+λ=0. (4.12)
∂n
Byusingtherelation(4.2),wehave
µ
+λ=0. (4.13)
− T
whichshowsthatµ/T mustbeaconstantfortheself-gravitatingfluid.
Now,letusconsiderEq.(4.11). FromtheLagrangian,wehave
∂L ∂Ψ
= (s+λn)eΨ(m)rn−2, (4.14)
∂m ∂m
where(∂Ψ/∂m)isthederivativeofΨrespecttomwiththecoordinaterfixedasaparameter.Further,wehave
∂L ∂s ∂ρ eΨ(m)
= eΨ(m)rn−2 = , (4.15)
∂m′ ∂ρ∂m′ Tω
k
(whereEq.(4.2)andEq.(3.8)havebeenused)and
d ∂L T′ eΨ(m)
= Ψ′ . (4.16)
dr∂m′ − T Tω
k
(cid:16) (cid:17)
ByusingtheEOMofnandthefundamentalrelation(4.3),onecanwritetheEuler-Lagrangianequationofminanexplicitform
T′ ∂Ψ
= ω (ρ+p)rn−2 +Ψ′. (4.17)
k
T − ∂m
8
Theconstraint(4.13)yields
µ′ =λT′. (4.18)
Consideringthefundamentalrelationofthermodynamics(4.3)andEq.(4.2),wehave
dp=sdT +ndµ. (4.19)
Itfollowsimmediatelythat
T′
p′(r)=sT′+nµ′ =T′(s+λn)= (ρ+p). (4.20)
T
Thusweobtainanequation
p′ ∂Ψ
= ω (ρ+p)rn−2 +Ψ′. (4.21)
k
ρ+p − ∂m
Considering(∂Ψ/∂m)(∂m/∂Ψ)=1,wehave
∂m p′ ∂m
= ω (ρ+p)rn−2 Ψ′ . (4.22)
k
∂Ψρ+p − − ∂Ψ
h i
Rememberingm′ =ρω rn−2,thisequationisnothingbutthegeneralizedTOVequation(3.12)inLovelockgravitytheory.
k
HerewehavesuccessfullydeducedtheTOVequationonlyusingthemaximumentropyprincipleandthetime-timecomponent
gravitationalequationasaconstraint. Now,wecansaythatthemaximumentropyprinciplereallygivespartofinformationof
gravitationalequationsandtheimplicitconnectionbetweengravityandthermodynamicsalsoexistsinLovelockgravity.Forthe
caseswithcharges,thisdiscussioncanbeperformedstraightforwardly,andwewillnotdiscussthemhere.
V. CONCLUSIONSANDDISCUSSION
TheTOVequationisanimportantequationforself-gravitatingsystemwhichwasoriginallyderivedfromtheEinsteinequa-
tion. WehavegotthegeneralizedTOVequationinLovelockgravitytheory.Thisequationhasageneralform
dp
= (ρ+p) F[p,m(r),r]
dr − ·
with a different factor F[p,m(r),r] for a different gravity theory. By applying the maximum entropy principle to a general
self-gravitatingfluid, we have also derivedthe TOV equation of hydrostaticequilibriumonly using the time-time component
of the gravitationalconstraintequation and ordinarythermodynamicrelations. Our results show that partof the gravitational
equationsofLovelocktheorycanalsobederivedfromordinarythermodynamiclaws. Thisisadirectevidenceforthepossible
fundamentalrelationshipbetweengravitationandthermodynamics.
Itmightbeinterestingtodiscusstheprobleminamoreambitiousway:Perhaps,onemaythinkthatthetime-timecomponent
ofthegravitationalequationsisnotnecessary! ForthesamesystemdescribedatthebeginningofSec.3,wecandefineamass
functionby
R
m(r)=ω ρrn−2dr,
k
Z0
thenwehavem′ =ρω rn−2. Further,Ψisassumedasafunctionofm. Then,bytheseassumptionsandthemaximumentropy
k
principle,followingthesamereductioninSec.4,wegetEq.(4.22)orEq.(3.12).Thus,theremainderunknowninformationisjust
therelationbetweenmandΨ. Thisisdeterminedbythestructuresofvariousgravitytheories. Soevenwithoutthetime-time
componentofgravitationalequations,wecanalsogetsomeinformationabouttheTOVequation:Atleast,itshouldhasaform
likeEq.(3.12). However,withoutthedetailsofthetime-timecomponent,wecannotfixedtheambiguityinEq.(3.12)(i.e.,the
detailedrelationbetweenmandΨ). Soitseemsthattime-timecomponentofthegravitationalequationsisindispensabletoget
theTOVequation.
It is interesting to apply this principle to other generalized gravity theories and investigate whether this deep connection
betweengravityandthermodynamicsexitsornot. Anotherimportantquestionis: whetheritispossibletoapplythemaximum
entropyprincipletosomedynamicalsystem(timedependentsystem). Thesituationbecomesdifficultonceadynamicalsystem
isconsidered. Actually,wedonotknowhowtodefineanequilibriumstate inthiscase. Itmightbenecessarytoimprovethe
maximumentropyprincipletosolvesuchproblem,whichneedsfurtherdiscussion.
9
VI. ACKNOWLEDGEMENT
Thiswork is supportedbyNSFC GrantsNo.11205148and No.11235010. LMC wouldlike to thankRong-GenCai forhis
usefuldiscussionandkindlyhelp.WealsowouldliketothankSijieGaoforhisusefuldiscussionsandcomments.
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