Reversible and Irreversible Spacetime Thermodynamics for General Brans-Dicke Theories

0.44 MB·English

by  Goffredo Chirco

Reversible and Irreversible Spacetime Thermodynamics
for General Brans-Dicke Theories
Goffredo Chirco, Christopher Eling, and Stefano Liberati
SISSA, Via Bonomea 265, 34136 Trieste, Italy and INFN Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy
(Dated: January 21, 2011)
WederivetheequationsofmotionforPalatiniF(R)gravitybyapplyinganentropybalancelaw
TdS = δQ+δN to the local Rindler wedge that can be constructed at each point of spacetime.
Unlike previous results for metric F(R), there is no bulk viscosity term in the irreversible flux δN.
Both theories are equivalent to particular cases of Brans-Dicke scalar-tensor gravity. We show that
the thermodynamical approach can be used ab initio also for this class of gravitational theories
and it is able to provide both the metric and scalar equations of motion. In this case, the presence
of an additional scalar degree of freedom and the requirement for it to be dynamical naturally
implyaseparatecontributionfromthescalarfieldtotheheatfluxδQ. Therefore,thegravitational
flux previously associated to a bulk viscosity term in metric F(R) turns out to be actually part of
1 the reversible thermodynamics. Hence we conjecture that only the shear viscosity associated with
1 Hartle-Hawking dissipation should be associated with irreversible thermodynamics.
0
2
PACSnumbers: 04.70.Dy,04.20.Cv,04.62.+v
n
a
J
I. INTRODUCTION necessarytogeneralizethelocalequilibriumconditionto

0 a more general entropy balance law. The new local equi-
2
libriumcondition,TdS =δQ+δN (thegeneralizedClau-
In the past decade, there has been significant inter-
siusrelation),containsextrairreversibleentropyproduc-
] est in the idea of gravity as the thermodynamics of a
c tiontermsduetothesourcecontributionsforthehorizon
quantum theory associated to some underlying micro-
q area/entropyevolutionwhicharequadraticinexpansion
scopic structure of spacetime [1–6]. The first striking
-
r hint in this direction was given by Jacobson [1], who de- andshearofthenullgeodesicbundlecomprisingthehori-
g zon.
rived the Einstein equation as an equilibrium equation
[
of state, starting from the thermodynamical properties Already interpreted in [3] as dissipative effects, these

3 of the vacuum. The key ingredients of the derivation terms were linked to purely gravitational/internal de-
v were the possibility to characterize the Minkowski vac- grees of freedom of the theory [5]. In particular, the
5
uum perceived by an uniformly accelerated observer as entropy production terms associated with the quadratic
0
4 a canonical thermal state (the Unruh effect), together shear were recognized to be the equivalent, for the local
1 with the idea that the local acceleration horizon, associ- horizon setting, of the tidal heating term associated to
. ated with the former observer, could be analogous to a dissipation of the black hole horizon’s perturbations via
1
tinypieceofablackholeeventhorizon, therebyallowing gravitational fluxes. Similarly, the entropy production
1
0 for an area-entropy proportionality assumption. terms associated with the quadratic expansion seemed
1 In this context, the coupling between geometry and to be an extra purely scalar dissipative contribution.
v: matter was then provided by demanding a local equilib- However, the scalar dissipative contribution seems to
i rium Clausius relation, TdS = δQ, as the main tool to be formally local, as a derivative of a scalar field at a
X
relate the variation of entropy of the vacuum fields dS point. Being local, this term would be frame indepen-
r (horizon area deformation) to the perturbative effects of dent,therebyitwouldexistforanyobserver(accelerated
a
alocalfluxofmatterenergyδQthroughtheacceleration or inertial) in the local patch of spacetime and would al-
horizon. waysendupdescribingthedynamicsoftheglobalspace-
More recently, in the attempt to understand whether time. Therefore,aninterpretationofthistermasadissi-
andhowthethermodynamicalderivationoftheEinstein pative contribution would imply waves of the scalar field
equation could be generalized to allow for the higher would be dissipative in the spacetime. This is inconsis-
curvature terms expected by the effective field theory, tentwiththefactthatclassicalgravitationaltheoriesare
it was realized that the thermodynamical approach can time reversal invariant.
be naturally generalized from an equilibrium to a non- In order to address this problem, we start in this work
equilibrium thermodynamical setting [3, 4]. Indeed, it by extending the thermodynamical approach, previously
was pointed out in [3] that in a natural extension of applied to metric F(R) gravity, and work in the Pala-
the framework to F(R) gravity theory, the horizon en- tini formalism, where the connection is a priori an in-
tropy proportionality to a function of the Ricci scalar dependent variable from the metric. In this theory, the
necessarily led to a break down of the thermodynamical connection is not a propagating degree of freedom and
equilibrium. In order to recover the F(R) gravity field can be eliminated by an algebraic relation. As a conse-
equation from the thermodynamical prescription, it was quence, unlike the metric F(R) case, we show that no
2
extra entropy production terms is required in order to Here κ is an arbitrary constant with dimensions [L]−1
get the equations of motion from the generalized local (weworkinunitswhere(cid:126)=c=k =1)associatedwith
B
equilibrium condition. This is in agreement with what thenormalizationofthetimelikeKillingvector∂ . These
τ
one would expect from the reasoning proposed in [5] re- coordinatesonlycovera“wedge”subregionofthespace-
gardingtheinterpretationofthesetermsaslinkedtothe time,intermsofglobalMinkowskicoordinatestherange
dynamicalgravitationaldegreesoffreedomofthetheory. where z > |t|. The timelike Killing flow χµ = (∂ )µ is
τ
However, both metric and Palatini F(R) gravity are equivalent to a continuous boost in the z direction. The
equivalent, at the classical level, to Brans-Dicke theories respective boost time parameter τ is proportional to the
with particular values of the Dicke constant. With this proper time along the worldlines of a uniformly acceler-
motivation, we then consider an entropy density that is ated observer, defined by the ξ =const hyperbolas. The
ascalarspacetimefunction, promotingtheinverseofthe surface ξ =0 is the null surface z =t, which acts similar
Newton constant to be an independent scalar field. In to future event horizon of a black hole since the points
thisway,wecanshowthatthethermodynamicderivation “inside” are causally disconnected from the accelerated
in this case can capture both the field equations of the observers.
metric and the scalar field for these theories. WhenrestrictedtotheRindlerwedge,theusualglobal
Inthisscalar-tensorframework,theextraentropycon- Minkowski vacuum state |0(cid:105) in quantum field theory
tribution associated with the quadratic expansion in the turns out to be equivalent to a Gibbs thermal state with
metricF(R)appearsasaseparatecontributionfromthe an Unruh-Tolman temperature [7, 8]
scalar field to the reversible heat flux δQ. Therefore, the
κ
gravitational flux previously associated to a bulk viscos- T = , (2)
2π
ityterminmetricF(R)isunderstoodasacomponentof
the reversible sector in the thermodynamical formalism. and a thermal entanglement entropy [9]
In this picture, we are able to generalize the thermody-
namical approach to general Brans-Dicke scalar-tensor S =−Trρlnρ. (3)
ent
theories of gravity.
We start in section II with a review of the spacetime As said, the factor κ is an arbitrary rescaling factor for
thermodynamicsapproachdevelopedforEinsteingravity the proper time τ, and as such it can always be set to
inanditsextensiontometricF(R)theory. InsectionIII, one. Hence from this point forward we will usually work
we reconsider the thermodynamical formalism for Pala- with the dimensionless “temperature” 1/2π. Remark-
tini gravity. We show how the equilibrium condition, ably,theentanglemententropyassociatedwiththefields
which was just an auxiliary condition in the metric for- in the Rindler wedge scales not like the volume of the
malism, now implies the field equation associated with wedge, but instead with the area of the Rindler horizon
the variation of the Lagrangian with respect to the in- boundary, just like the Bekenstein-Hawking entropy of a
dependent connection. Then, we derive the equation of black hole. On the other hand, in quantum field theory
motion from the local equilibrium assumption, where no itisalsoquadraticallyultraviolet(UV)divergentandwe
additional bulk viscosity term is needed in the analy- will assume a regulator has been introduced in order to
sis. InsectionIV,theinterpretationofthebulkviscosity render it finite.
term is reconsidered through the equivalence of metric Now a general spacetime is a curved manifold, but the
andPalatiniversionsofF(R)toparticularcasesofscalar equivalence principle implies that the neighborhood of
tensor theories. This provides the background to gener- any point p is approximately flat. In this flat patch
alize the argument to a general Brans-Dicke theory in of spacetime, one can always construct a local Rindler
section V. We conclude with a summary and discussion wedge system. Let us be more precise about this con-
of future work. struction. The key quantity we need is a localized ver-
sion of the boost Killing vector χµ, vanishing at p. This
object will allow us to define a local temperature and
II. THERMODYNAMICS OF SPACETIME AND eventually a notion of heat flow. There are no Killing
METRIC F(R) GRAVITY vectors for an arbitrary spacetime, but Killing’s equa-
tion 2∇ χ = 0 can be solved locally around p up to
(µ ν)
Herewewillfirstreviewthethermodynamicsofspace- O(x3) using Riemann normal coordinates xµ, where
time approach originally developed for Einstein gravity
[1] and its extension to metric F(R) gravity [3] (also see gµν =ηµν +O(x2). (4)
[5]) as this will be the basis for our results in the fol-
In the neighborhood of each point in spacetime we
lowing sections. The foundation of the approach is in
think of the thermal system as the local Minkowski
thethermodynamicalpropertiesoftheflatRindlerspace-
vacuum |0(cid:105) restricted to the local Rindler wedge. Note
time, which are independent of gravity. In the Rindler
that the full Minkowski patch, where there is no local
coordinates Yµ =(τ,ξ,x,y), a flat manifold has the line
horizon and no coarse graining, has zero temperature
element
and entropy and is not the thermal system in question.
ds2 =g dYµdYν =κ2ξ2dτ2−dξ2−dx2−dy2. (1) To probe the behavior of this system out of equilibrium
µν
3
one constructs a local causal horizon at p as follows. whichisperfectlythermalizedbythehorizonsystemand
Choose a spacelike 2-surface patch B including p, and therefore can be thought of as heat. It is assumed δQ is
choose one side of the boundary of the causal past of B. theflowofboostmatterenergycurrentTMχν acrossthe
µν
Near p, this boundary is a congruence of null geodesics horizon, in terms of the affine kµ this is
orthogonal to B. These comprise the horizon. At p, B δQ (cid:90) √
agrees with the tangent plane B preserved by the flow =2π TM kµkν(−λ)dλ hd2x, (8)
p T µν
of the local Killing vector (Fig. 1).
√
where histhecross-sectionalareaelementonthecausal
horizon.
B The additional δN is an irreversible internal entropy
p
production term, or uncompensated heat. This term
p arises in a slower approach to equilibrium. Using linear
B constitutive relations between fluxes of momentum in a
fluidandthethermodynamic“forces”givenbygradients
of a fluid velocity, it was argued in [3] that the entropy
productioncanbeexpressedintermsofthesquaredshear
σˆ and expansion θˆof the flow
µν
δN 2η ξ
= σˆ σˆµν + Bθˆ2, (9)
FIG. 1. The 2-surface B agrees at p with the tangent plane T T µν T
B preserved by the flow of the local Killing vector.
p
where η and ξ are shear and bulk viscosities respec-
B
tively.
We arrange χµ so that it is future pointing on the
The entropy balance law (7) we postulated requires a
causal horizon. Therefore on the horizon generator
corresponding change in the entanglement entropy asso-
through p we have (up to the ambiguities at O(x3))
ciated with the local Rindler wedge. In [1] a constant,
χµ = −λkµ, where kµ ips the affinely parameterized null universalUVcut-offαwithunitsof[L]−2 waspostulated
tangent vector kµ. This affine parameter obeys the for-
tomaketheentropydensitysfiniteandthetotalentropy
mula
justproportionaltothehorizoncross-sectionalarea. Im-
λ=−e−v, (5) posing the entropy balance law ultimately yielded two
results, which we will briefly summarize. The Einstein
where v is the “Killing time” parameter defined via equations (with an undetermined cosmological constant
χµ∇µv = 1 characterizing the time flow of the local and the Newton constant GN = (4α)−1) are associated
wedge system. with reversible changes in the global spacetime. This
The choice of how B is warped in the spacetime de- is consistent with the time reflection symmetry of Ein-
termines the horizon expansion and shear, and therefore stein’s theory. In the irreversible sector, associated with
the deviation of the system from equilibrium. Using (5) thehorizon,theshearviscositywasfixedtohavetheuni-
onecanshowtherelationshipbetweentheexpansionand versal ratio η/s = 1/4π [3, 4]. The shear viscosity term
shear of the local horizon defined in terms of v and cor- coincideswiththeHartle-Hawkingtidalheatingtermand
responding quantities in λ is thereforeappearstobeassociatedwiththepurelytenso-
rial gravitational degrees of freedom in the theory [5].
θˆ=−λθ σˆ =−λσ. (6)
However, one can imagine that the entanglement en-
TheKillingquantitiesautomaticallyvanishbecausepisa tropy density has a more complicated structure in gen-
fixedpointofthelocalboostflow,consistentwithourno- eral. Forexample, theonlyknowntheoryofgravitycon-
tionoflocalthermodynamicequilibriumthere. Whether sistent with the strong equivalence principle is Einstein
the affine quantities are zero or not in turn determines gravity. The strong equivalence principle implies that
the rate at which this equilibrium is approached. For gravity is purely geometrical. Physics (with gravity in-
example, when σ = 0, σˆ ∼ e−2v, while for σ (cid:54)= 0, cluded)isthesameinanylocallyflatregionofspacetime,
|p |p
σˆ ∼e−v. which means the UV cutoff is constant, GN is universal,
The basic idea is to impose a general entropy balance and there are no extra gravitational fields. However, in
law on this local system. When the thermal density ma- more general theories of gravity, the strong equivalence
trix ρ associated with the quantum fields in the wedge principle is not satisfied, suggesting one should consider
is perturbed, the change in the entanglement entropy an entanglement with a spacetime dependent cutoff and
should be related to the change in mean energy via the as a result, more general entropy functionals.
entropy balance law In [3] it was assumed the entropy density is a general
function of the Ricci scalar f(R), so that the total en-
dS =δQ/T +δN/T. (7) tropy is
We assume the change in the mean energy of the system (cid:90) √
S =α hf(R)d2x, (10)
isduetoafluxintotheunobservableregionofspacetime,
4
Inthiscasewewillreviewthedetailsofhowtheentropy where Φ is an arbitrary function. Local energy-
balance law is imposed. The variation of this entropy in momentum conservation requires ∇µTM = 0 and
µν
terms of the affine parameter λ has the form thereforegivesusaconditiontosolveforΦ. However,as
was shown in [3], with the additional kinetic term (16)
δS =α(cid:90) √h(cid:18)fθ+ df (cid:19)dλd2x. (11) present, there is a contradiction and one cannot solve
dλ for Φ. In this case, conservation of energy-momentum
seemstorequirethistermtocontributetotheirreversible
To compare with the heat flux in (8) we perform a series entropy production. Put differently, one must have the
expansion in λ around the point p, identifications
(cid:90) √ (cid:20)(cid:18) df (cid:19) αf
δS =α h θf + + η = (18)
dλ 4π
(cid:18) df dθ d2f df (cid:19)(cid:21) 3αf
+λ θdλ +fdλ + dλ2 +fθ2+θdλ . (12) ξB = 4π . (19)
p
to cancel out the kinetic and shear squared terms. Note
The first three terms in the O(λ) piece come from the
that since s=αf, the ratio η/s=1/4π still. The purely
derivative of the integrand in (11), while the last two
reversible changes are now associated with the equation
termsareassocia√tedwiththederivativeofthetransverse
volume element h. (cid:18) (cid:19)
2π
We now match this expansion with the heat flux and fRµν −∇µ∇νf +Φgµν = α TMµν, (20)
irreversible heat at lowest orders in λ as required by the
entropy balance law. Because we are dealing with quan-
Imposing local energy conservation and then using the
tities evaluated at p, equating the integrals (12) and (8)
contracted Bianchi identity and the commutator of co-
is equivalent to equating the integrands. Since there is variant derivatives, one finds that Φ = (cid:3)f − 1F, where
no zeroth order part of these fluxes (they must vanish at 2
F(R)istheLagrangianofthetheoryanddefinedhereby
theequilibriumpointp,seeforexample(8)),wehavethe
the equation f =dF/dR. As a result (20) now coincides
equilibrium condition that
with the equation of motion derived from the action
(cid:18) (cid:19)
θf + ddfλ p =0. (13) Imet = 4απ (cid:90) √−gd4x(F(R)+Lmatt). (21)
Onecanalwaysconstructa2-surfaceBatptosatisfythis
condition. NowconsidertheO(λ)termsof(7). First,we
III. PALATINI F(R) GRAVITY
replace dθ/dλ with the right hand side of the Raychaud-
huri equation
In this section we will address whether the formalism
dθ 1 ofPalatinigravityiscompatiblewiththethermodynamic
=− θ2−σ σµν −R kµkν, (14)
dλ 2 µν µν approach. In the Palatini formalism, one treats the con-
nection as an independent variable a priori. The Rie-
Comparing the integrands, we find at p
mann tensor R , constructed out of this connection,
µνρλ
3 is now therefore also independent of the metric.
α(−fRµνkµkν +kµkν∇µ∇νf − 2fθ2−ϕσµνσµν) Our starting point will be an entropy density which is
an arbitrary function of the independent Ricci scalar
=2π(−TM kµkν −2ησ σµν −ξ θ2)(.15)
µν µν B
(cid:90) √
Notethatinwritingthisequationweusedd/dλ=kµ∇µ S =α hf(R)d2x. (22)
and the affine geodesic equation kµ∇ kν = 0. Using
µ
(13), the expansion can be expressed as a kinetic term
If the thermodynamic approach works in this case, we
for f,
expect now two equations, which are the equations of
3 3 motion following from the variation of the Lagrangian
fθ2 = kµkν∇ f∇ f. (16)
2 2f µ ν
α (cid:90) √
I = −g(F(R)+L (g ,ψ)) (23)
If we consider this kinetic term to be a part of the P 4π matt µν
reversible sector and demand the equation hold at any
point p and for all null vectors kµ, we get with respect to gµν and Γλµν. Note that in the Palatini
formalism the matter part of the action is assumed not
(cid:18) (cid:19)
3 2π to depend on the independent connection.
fR −∇ ∇ f + ∇ f∇ f +Φg = TM ,
µν µ ν 2f µ ν µν α µν Palatini F(R) gravity theory has been discussed ex-
(17) tensivelyoverthepastdecadeasanalternativetheoryof
5
gravity [10]. Here we pause briefly to review the proper- II. This is worth doing not only for checking the validity
ties of this theory. Defining f = dF/dR as before, the of the above discussed expectation (no-bulk viscosity as-
equations of motion are sociatedtoF(R))butalsoasafirststeptowardsthegen-
eralization of the spacetime thermodynamics formalism
(cid:18) (cid:19)
1 2π to the broader class of metric-affine theories of gravity.
f(R)R − F(R)g = T (24)
µν 2 µν α µν In the neighborhood of each point, spacetime is still
locally flat and we still can construct the boost Killing
√ √ vector χµ since the Killing equation L g =0 does not
∇¯σ( −gf(R)gσ(µδν)λ−∇¯λ( −gf(R)gµν)=0, (25) depend on the connection. On the loχcaµlνhorizon χµ is
still a null generator. However, in the presence of an
where ∇¯ represents the covariant derivative defined with
independentconnection,thereisapriorianambiguityin
respect to the independent connection. The connection
whetherχµ isageodesicwithrespecttotheindependent
equation is equivalent to the more compact condition
connection or the metric one. We have either
√
∇¯ ( −gf(R)gµν)=0. (26)
λ χν∇¯ χµ =κ¯χµ (31)
ν
Notethatwhenf isequaltoaconstantthisequationre-
or
ducestotheusualmetriccompatibilityconditionforgµν.
ThereforeweseethetextbookequivalenceofthePalatini
χν∇ χµ =κχµ, (32)
and metric formalisms of GR. In general however, (26) ν
implies the conformally related metric
andthecorrespondingchoicesfortheaffineparametriza-
tion
g¯ =f(R)g (27)
µν µν
χµ =−κ¯λ¯(cid:96)µ (33)
is compatible with the connection. Imposing this con-
χµ =−κλkµ. (34)
dition, one can relate the Ricci tensor and scalar con-
structed from g¯ to the metric quantities
µν Each vector above is affinely parameterized with respect
3 1 to either the independent or the Levi-Civita connection:
Rµν =Rµν + 2f2(∇µf(cid:48))(∇νf) (cid:96)ν∇¯ν(cid:96)µ = 0 or kν∇νkµ = 0. Therefore, in the entropy
change δS, one must consider changes with respect to
1 1
− (∇ ∇ − g (cid:3))f (28) either affine parameter. A priori the different Clausius
f µ ν 2 µν
relations could yield two different equations of motion.
3 1 3
R=R+ (∇ f)(∇µf)+ (cid:3)f. (29) In the next two subsections we will consider variations
2f2 µ f with respect to λ¯ and λ in turn.
Theseequationscanbesubstitutedinto(24)toyield[10]
(cid:18)2π(cid:19) 1 1. Variation using independent connection
fG = T − g (fR−F(R))
µν α µν 2 µν
1 3 1 Firstweconsidertheheatflux. WeexpresstheKilling
+f(∇µ∇ν −gµν(cid:3))f − 2f2[(∇νf)(∇νf) field in terms of the affine (cid:96)µ using the formula χµ =
−λ¯(cid:96)µ, where we scale the κ¯ =1 as usual.1 Therefore,
1
− g (∇f)2]. (30)
2 µν δQ (cid:90) √
=2π TM (cid:96)µ(cid:96)ν(−λ¯)dλ¯ hd2x. (35)
Solving the trace of (24) for R in terms of T, one can T µν
completely eliminate the connection as an independent
In the expression above, differently from Eq. (8), we use
variableandreducethesystemtooneequationofmotion
the affine null vector (cid:96)µ with respect to the independent
that looks like GR with a modified source. Following
connection and the null geodesic bundle comprising the
the reasoning of [5] one then should expect that no bulk
horizon is now parametrized by λ¯. On the other hand,
viscositytermappearsinthiscase,asno-extradynamical
as the matter only feels the metric g , the relevant vol-
gravitationaldegreeoffreedomwithrespecttothemetric √ µν √
appears in Palatini F(R) gravity. ume element is still given by g, reducing to h on the
A. Thermodynamic formalism
1 Itisworthnoticingherethatthereisanambiguityaboutwhichκ
(barredorunbarred)wouldactuallyappearintheTolman-Unruh
With the connection now an independent variable and
temperatureT,seeEqn. (2). Iftheunbarredκischosen,dueto
themetricnolongeraprioricompatiblewithit,wewant thecouplingofmatterfieldsonlytometric,thenweassumethe
toconsidertheeffect(ifany)onthebasicsofthethermo- ratioofthetwosurfacegravitiescanbescaledtounitywithout
dynamics of spacetime formalism introduced in Section lossofgenerality.
6
horizon. In this sense, we ca√n write the relevant horizon Demanding this equation hold for all null vectors (cid:96)µ, at
volume element as dΣν =(cid:96)µ hd2xdλ¯. each point p, and using the fact that (38) implies
Alongthesameparameterλ¯,wenowconsiderageneral
g¯ =fg , (45)
variationoftheentropy. Thishasthesameformas(11), µν µν
but we express it in a slightly different way,
we arrive at the metric compatibility equation (26),
(cid:90) √ √
δS = hfθ¯d2x, (36) ∇¯λ(f −ggµν). (46)
Remarkably,theequilibriumcondition,whichwasjustan
where auxiliary condition in the metric formalism, now implies
√ √ the field equation associated with the variation of the
1 d h 1 df dln(f h)
θ¯= √ + = . (37) Lagrangian with respect to the independent connection.
h dλ¯ f dλ dλ¯ Now we go back to the entropy balance law and con-
tinue to next order in λ¯. We find
This suggests that the new expansion measuring the
product of f and the transverse element is the relevant (cid:90) (cid:112) dθ¯
δS =α h¯ λ¯d2x. (47)
one. Thus, we make the transformation dλ¯
(cid:112)h¯ =f(R)√h. (38) Using the Raychaudhuri equation this can be re-
expressed as
so that in terms of this new variable, the entropy is an (cid:90) (cid:112)
area entropy δS =−α h¯Rµν(cid:96)µ(cid:96)νλ¯d2x, (48)
S =α(cid:90) (cid:112)h¯d2x. (39) in terms of the Ricc√i tensor constructed fro√m g¯µν. N√ote
the appearance of h as opposed to the h¯ = f h.
ImposingtheClausiusrelation,andmatchingbothsides,
Imposingtheentropybalanceequationandmatchingor-
we find
der by order in λ¯ we find the zeroth order equilibrium
(cid:18) (cid:19)
condition, which can be expressed as 2π
fR +Φg = TM. (49)
(cid:112) √ µν µν α µν
θ¯=0→(cid:96)µ∇¯ ( h¯)=(cid:96)µ∇¯ (f(cid:48) h)=0. (40)
µ µ
The f in front of the Ricci tensor has reappeared to ac-
countforthemismatchbetweentheeffectivevolumeele-
Note that we could have performed the above conformal
ment and the metric volume element felt by matter flux.
transformation also in metric F(R) gravity, but in that
In the Palatini formalism, ∇¯µT (cid:54)= 0, but the usual
case the equilibrium condition involves the Levi-Civita µν
metric conservation law ∇µT =0 holds. Imposing this
connection. On the other hand, the above formula in µν
condition yields
termsoftheindependentconnectionisreminiscentofthe
metric compatibility condition (26).
∇ Φ=−f∇µR −R ∇µf. (50)
First, note that the vanishing of the expansion θ¯ can ν µν µν
be expressed as The main problem is to calculate the g covariant diver-
genceoftheg¯Riccitensor. WeknowthatR isrelated
d (cid:113) d (cid:113) µν
dλ¯ det(g¯µνeµaeνb)= dλ¯ det(g¯µν)det(eµaeνb)=0. (41) tfaoctRoµrνf1v/i2a. aThceoncfoonrnmeactliotrnarneslfaotrimngattihoencowviathriacnotndfoerrmivaa-l
Thequantitieseµ arebasisvectorsinthecross-sectionof tives with respect to the two metrics has the form
a
the horizon and the index a runs over the two transverse 1
directions. Since these basis vectors by construction are Γσµν =−δ(σµ∇¯ν)lnf(cid:48)+ 2g¯µνg¯σδ∇¯δlnf(cid:48). (51)
Lie transported along the horizon, we must have that
Using the formula (see e.g. [11])
d √
dλ¯ −g¯=0. (42) ∇µGµν =−∇µlnfRµν (52)
anddefiningG =R −1g R,weultimatelyfindthat
But since µν µν 2 µν
ddλ¯√−g¯= 12g¯µνddλ¯g¯µν =0, (43) ∇νΦ=−12f∇¯νR=−21∇¯νF, (53)
so that Φ = −1/2F +const. as we expect. Therefore in
this condition implies that
theirreversiblesector,thereisnoneedforabulkviscosity
d √ term, and the shear viscosity remains the same as in GR
( −g¯g¯µν)=0. (44)
dλ¯ and metric F(R) gravity.
7
2. Variation using Levi-Civita connection where Y is some unknown function. Then this term can
be manipulated into total derivative terms plus a term
multiplying ∇ f:
Now we re-consider the same problem, but working ν
instead with quantities defined with respect to the Levi-
1 1 1 1
Civitaconnection. Henceweconsidertheaffinelyparam- − f(R)∇ R=− ∇ f + ∇ (fY)+ Y∇ f. (60)
2 ν 2 ν 2 ν 2 ν
eterizedtangenttobekµ. Therepresentationoftheheat
flowandtheentropychangeisexactlythesameasforthe Astotalderivatives,thefirsttwopiecescontributetothe
metric F(R) case in Section II. As a result, the analysis solution for Φ, which now agrees with the set of terms
shows that we have an equation similar to (17) proportionaltog inthesingleequationofmotion(30).
µν
The last term above combines with the terms propor-
(cid:18) (cid:19)
fR −∇ ∇ f + 3 ∇ f∇ f +Φg = 2π TM , tional to ∇νf in (58). Demanding that term be zero as
µν µ ν 2f µ ν µν α µν a type of consistency or integrability condition implies
(54)
but instead of the fully metric-derived object f(R), now Y =R−R= 3 ∇ f∇µf + 3(cid:3)f, (61)
we have f(R). The Ricci tensor that appears explicitly 2f2 µ f
is constructed from the metric and comes from the Ray-
which is exactly the relationship between the two Ricci
chaudhuri equation in terms of metric compatible vari-
scalars in (29) derived from the conformal transforma-
ables.
tion.
We need to solve for the unknown function Φ. In the
Wehavederived(albeitsomewhatindirectly)theequa-
metrictheorythisleadstoacontradictionandonehadto
tions of motion for Palatini F(R) gravity when the con-
cancelthekinetic∇ f∇µf byintroducingabulkviscos-
µ
nectioniseliminatedasaindependentvariable. Thefact
ity (or equivalently, move it into the irreversible sector),
that equations of motion derived in Sections IIIA1 and
butherewehavetoconsiderthepresenceoftwodifferent
IIIA2 are equivalent can be seen a posteriori from the
curvaturetensors. Takingacovariantdivergenceof(54),
conformal relationship between g¯ and g . This im-
we find that µν µν
plies that the Killing vector χµ is geodesic with respect
∇ Φ=−(∇µf)R −f∇µR +∇µ∇ ∇ f to both the independent and metric connections. There-
ν µν µν µ ν
fore, we have showed that the thermodynamic approach
3 3
+ ∇ f∇µf∇ f − (cid:3)f∇ f canbeextendedtoencompassthePalatiniformalismand
2f2 µ ν 2f ν
Palatini F(R) gravity. In this case no additional bulk
3
− ∇µf∇ ∇ f. (55) viscosity term is needed in the analysis.
2f µ ν
Next,weusetheBianchiidentity,∇µR = 1∇ Randa
µν 2 ν IV. SCALAR-TENSOR REPRESENTATIONS
contractedversionofthecommutatorofcovariantderiva-
tives
ItiswellknownthatboththemetricandPalatiniver-
sions of F(R) gravity are equivalent to particular scalar-
∇µ∇ V −∇ ∇µV =RτV , (56)
ν µ ν µ ν τ tensor theories. First consider the metric F(R) action
(21). One can treat f(R) ≡ ∂F as an auxiliary field ϕ
whereVµ ≡∇µf tore-expressthesecondandthirdterms and assume F(cid:48)(cid:48)(R) (cid:54)= 0 for a∂llRR. Then one can take
on the right hand side above. In addition, note that the
thepotentialV(ϕ)astheLegendretransformofF(R)so
last term can be re-expressed as
that R=V(cid:48)(ϕ). Therefore one can rewrite the action in
3 3 the equivalent form
− ∇µf∇ ∇ f =− ∇ (∇µf∇ f)=
2f µ ν 4f ν µ
α (cid:90) √
−∇ (cid:18) 3 ∇µf∇ f(cid:19)− 3 ∇ f∇µf∇ f. (57) Iω=0 = 4π d4x −g(ϕR+V(ϕ)+Lmatt). (62)
ν 4f µ 4f2 µ ν
ThisistheJordanframerepresentationofaBrans-Dicke
Combining these results, we obtain
scalar-tensortheorywiththeDickecouplingconstantset
(cid:18) (cid:19) to ω =0. The corresponding equations of motion are
3 1
∇ Φ=∇ (cid:3)f − ∇ f∇µf − f(R)∇ R
ν ν 4f µ 2 ν R=V(cid:48)(ϕ) (63)
(cid:18) 3 3 (cid:19) (cid:18)2π(cid:19)
− ∇ f∇µf + (cid:3)f ∇ f.(58) ϕG =∇ ∇ ϕ+ TM −g (cid:3)ϕ
4f2 µ 2f ν µν µ ν α µν µν
1
Here, the key term is the −1/2f(R)∇ R. We can intro- − g V(ϕ). (64)
ν 2 µν
duce the following ansatz
Theseequationsalsosimplyfollowfromthemetricequa-
R=R−Y, (59) tion of motion (20) with the identification f ≡ϕ. Hence
8
in the scalar-tensor representation the “bulk viscosity” Using the contracted Bianchi identity and the commuta-
term has the form [5] tor of covariant derivatives, we are left with
δNbulk =α(cid:90) d2xdλλ√h23ϕkµkν∇µϕ∇νϕ. (65) ∇νΦ=∇ν((cid:3)ϕ)− 21∇ν(ϕR)+ 21R∇νϕ. (72)
Now in order to solve this equation, we must impose the
The procedure is the same for the Palatini action (23)
following “integrability condition” on the last term of
and one finds
the right hand side of the previous equation. Namely,
α (cid:90) √ we assume it can be expressed as the derivative of some
Ipal = 4π d4x −g(ϕR+V(ϕ)+Lmatt). (66) function,chosentohavetheform 12 V(ϕ),i.e. 12R∇νϕ=
1 ∇ V(ϕ). Therefore, we have the condition that
2 ν
Using the relationship between R and R found earlier in
(29) we can express this action (up to surface terms) as dV
=R. (73)
dϕ
α (cid:90) √ 3
I = d4x −g(ϕR+ ∇ ϕ∇µϕ
ω=−3/2 4π 2ϕ µ Meanwhile, the solution for Φ is
+V(ϕ)+L ), (67)
matt 1 1
Φ=(cid:3)ϕ− ϕR+ V(ϕ)+Λ, (74)
2 2
which is the Brans-Dicke theory with ω =−3/2.
To apply the thermodynamic formalism, much of the and the reversible equation becomes
previous analysis can be carried over, just with the iden-
1
tification ϕ = f. However, it is initially unclear how ϕR −∇ ∇ ϕ− ϕRg +g (cid:3)ϕ
the equations of motion for the scalar field can emerge µν µ ν 2 µν µν
(cid:18) (cid:19)
outofthisanalysis. Tostart, weassumetheholographic 1 2π
+ g V(ϕ)= TM , (75)
entropy has the form 2 µν α µν
(cid:90) √ wherewehaveabsorbedthecosmologicalconstantΛinto
S =α hϕd2x. (68)
the potential V(ϕ). This is exactly the set of field equa-
tionsfortheω =0theory. Thescalarequationofmotion
Supposewefollow([3])andcancelouttheexpansionterm is an integrability condition we must impose for consis-
by treating it as a part of the irreversible sector. Then tency with the conservation of local energy-momentum.
we arrive at Suppose,ontheotherhand,thatwedonotintroducea
bulk viscosity term. Then equation describing reversible
(cid:18) (cid:19)
2π changes is
ϕR −∇ ∇ ϕ+Φg = TM , (69)
µν µ ν µν α µν
(cid:18) (cid:19)
3 2π
ϕR −∇ ∇ ϕ+ ∇ ϕ∇ ϕ+Φg = TM .
for some undetermined function Φ. To determine Φ we µν µ ν 2ϕ µ ν µν α µν
can demand the local conservation of matter-energy as (76)
usual. Imagine that we know the action for the matter This has the same form as (17), but now that ϕ is an
fieldspresent. ItisafunctionalofI (g ,ψ),whereψ independent field, we can repeat the analysis above to
matt µν
represents some arbitrary matter. Using the diffeomor- solve for the unknown Φ function and find the scalar
phism invariance of this action and assuming the matter equation of motion as an integrability condition.
fields satisfy their equation of motion δI /δψ = 0, Taking a covariant divergence of (76), we find that
matt
one can easily show the following conservation equation
holds2 ∇νΦ=−(∇µϕ)Rµν −ϕ∇µRµν
3
+∇µ∇ ∇ ϕ+ ∇ ϕ∇µϕ∇ ϕ
∇µTMµν =0. (70) µ ν 2ϕ2 µ ν
3 3
Imposing this equation we find the following equation −2ϕ(cid:3)ϕ∇νϕ− 2ϕ∇µϕ∇µ∇νϕ. (77)
for Φ,
As before, we can use the Bianchi identity and the com-
∇ Φ=−(∇µϕ)R −ϕ∇µR +∇µ∇ ∇ ϕ. (71) mutator of covariant derivatives (56) along with the for-
ν µν µν µ ν
mula (57) in Section (IIIA2). Combining these results,
we obtain
(cid:18) (cid:19)
1 3
2 In general, the matter part of the action can also depend on ∇νΦ=∇ν (cid:3)ϕ− 2ϕR− 4ϕ∇µϕ∇µϕ
the scalar field: Imatt(g,ψ,ϕ). Then the matter stress ten-
s√or is not conserved: ∇µTMµν = 12Tϕ∇νϕ, where Tϕ = +(cid:18)1R− 3 ∇ ϕ∇µϕ− 3 (cid:3)ϕ(cid:19)∇ ϕ. (78)
( −g)−1δImatt/δϕ. 2 4ϕ2 µ 2ϕ ν
9
Wenowimposetheintegrabilityconditiononthesecond Therefore in this case the scalar field is algebraically re-
term as before lated to the matter sources and does not propagate.
dV 3 3 Wenowarguetheclearlinkbetweentheadditionalflux
dϕ =R+ 2ϕ2∇µϕ∇µϕ− ϕ(cid:3)ϕ, (79) termassociatedwiththehorizonexpansionθandaprop-
agatingscalardegreeoffreedomindicatesthattheprevi-
which allows us to solve for Φ. The resulting metric field ous“bulkviscosity”interpretationin[3,5]wasincorrect.
equation is First,letusgobackandconsidertheshearsquaredterm.
1 3 We think of this term as indicating a channel for hori-
ϕRµν − 2ϕRgµν −∇µ∇νϕ+ 2ϕ∇µϕ∇νϕ zondissipation,givenagravitytheoryorequivalentlyan
entropy functional. In GR, this channel is sourced by
3
+(cid:3)ϕg − ∇ ϕ∇µϕg a flux of gravitational perturbations across the horizon
µν 4ϕ µ µν
(specifically, a perturbation of the electric part of the
(cid:18) (cid:19)
1 2π Weyl tensor) and gives rise to the Hartle-Hawking tidal
+ V(ϕ)g = TM . (80)
2 µν α µν heating term [12–15]
Therefore we arrive at the equations of motion for Pala-
1 (cid:90) √
tini F(R) theory in ω = −3/2 scalar-tensor representa- δN = σˆ σˆµνdv hd2x. (84)
shear 8πG µν
tion. N
The above expression generalizes in metric F(R) in a
V. GENERAL BRANS-DICKE THEORIES AND similar way, acquiring only an overall f(R) factor.
“BULK VISCOSITY” AS A HEAT FLUX Note that this does not mean gravitational waves are
dissipative. While the spacetime dynamics is completely
In the above section we showed how the thermody- conservative, dissipation only exists for the thermody-
namicapproachcanbeusedtoderivethefieldequations namical horizon system. Furthermore, in the thermody-
forbothmetricandPalatiniF(R)gravitypurelyintheir namical argument, the shear at p only depends on the
scalar-tensorrepresentations. However,theentropyfunc- warping of B and therefore is completely independent of
tional (68) holds also for a general Brans-Dicke theory, the spacetime geometry.
which has the action Gravitational waves have no local stress tensor, and
(cid:90) √ α ω correspondinglytheHartle-Hawkingtermhasanon-local
Igen = −gd4x[4π(ϕR− ϕ∇µϕ∇µϕ character as the integral over the horizon of an object
constructedoutofderivativesofthenullnormalkµ. This
+V(ϕ))+L ]. (81)
matt termisconsistentwithanirreversibleflux,whichinnon-
Previously we were only able to derive the equations of equilibrium thermodynamics [16, 17] is positive definite
motion for the special cases ω = 0 and ω = −3/2, de- andconstructedoutofderivativesofthestatefunctionals
pending on whether “bulk viscosity” term is needed. In of the system.
particular, for the ω =−3/2 case equivalent to Palatini, Itwasarguedin[5]thatwhentheadditionalscalarde-
no such term was needed to complete the analysis. gree of freedom is present, there is a new “gravitational”
Whether or not we need an additional term appears channelavailablefordissipatingenergy(e.g. forrelaxing
to be directly related to the existence of an additional horizon perturbations). However, it is unclear if the re-
propagatingscalardegreeoffreedominF(R)andscalar- lated term (65) should be associated to some irreversible
tensor gravity, as was first hypothesized in [5]. Since in branch of the thermodynamic equations. The problem
Palatini F(R) the connection is only an auxiliary field, with this interpretation can be realized by analyzing the
one would not identify it with any additional propagat- form of the “bulk viscosity” term itself given in (65):
ingdegreeoffreedom. Itispossibletomoreclearlyshow Unlike the shear squared term, it does not depend on
this distinction between an auxiliary field and a dynam- derivatives of kµ and both the integral over the horizon
ical, propagating one in the scalar-tensor representation. and the arbitrary kµ vectors can be peeled off. Hence,
Consider, for example, the ω = 0 theory (any general ω like the other terms with this structure, it has a local
will do). The trace of the metric field equation (75) and interpretation,whichisconsistentwithfactthatascalar
the scalar integrability condition (73) yield field has a local stress tensor. After the kµ vectors are
peeled off, local terms at p are frame independent. They
(cid:18) (cid:19)
dV 2π
3(cid:3)ϕ+2V(ϕ)−ϕ = TMµ, (82) existforanyobserver(acceleratedorinertial)inthelocal
dϕ α µ patch of spacetime and always end up describing the dy-
namics of the global spacetime. Indeed, the expansion is
sothepropagationofthescalarisdeterminedbythemat-
nolongeranarbitraryquantitydefiningthelocalhorizon
ter sources, as usual. On the other hand, in the special
system, but instead fundamentally linked to the deriva-
case where ω =−3/2 the same procedure yields
tive of the scalar field on the spacetime. Therefore, if we
(cid:18) (cid:19)
dV 2π insist that this term is irreversible, it would imply waves
2V(ϕ)−ϕ = TMµ. (83)
dϕ α µ of the scalar field would be dissipative in the spacetime.
10
This is inconsistent with the fact that classical gravita- whichcapturesanyBrans-DicketheoryifwesettheDicke
tional theories are time reversal invariant. constant ω =Ω−3/2.
We now present a new interpretation of this term as a Inthespecialcasewhenω =−3/2, Ω=0andthereis
contribution to the heat flux δQ of reversible thermody- no additional contribution to heat flux from the scalar.
namics. Let us return to the beginning of the argument This is consistent with the fact that the ϕ field is non-
and the entropy propagating(hasnokineticterm)inthisparticularcase.
Inaddition,notethatwhenω <−3/2thescalarfieldflux
(cid:90) √
S = d4x hϕ(x). (85) comesinasa“ghost”withanegativesignandthechange
in the black hole entropy can longer be positive definite
Here we have promoted the entropy density to be an due to the violation of the null energy condition. This is
independent field in the spacetime, with dimensions of consistent with the study of the classical second law for
[L]−2. In order to be consistent with the principle of Brans-Dicketheorydonein[19]andthenumericalresults
backgroundindependence,thisfieldshouldnotbeafixed of the gravitational collapse of scalar matter pulses [20],
structure,itmustbevariedlikeotherfields. Italsomust which indicated a violation of weak cosmic censorship
contribute to the total Lagrangian of the theory, i.e. when ω <−3/2.
Itisalsointerestingtomakethetransformationtothe
Lmatt(gµν,ψ)+Lscalar(gµν,ϕ) (86) so-called Einstein frame of the Brans-Dicke theory. One
makes a conformal transformation and a redefinition of
where ψ represent “ordinary” matter fields. Upon vary-
the scalar field
ing with respect to the metric, we have the usual stress
tensor of the various matter fields which do not con- g˜ =ϕg
µν µν
tribute to the horizon entropy, plus a stress tensor for
dϕ
the scalar field. The components relevant for the heat dϕ˜= (91)
ϕ
flux across the horizon are given by contraction with the
null vectors
in the action (81). The action now has the form
δQ∼kµkν(TµMν +Tµϕν). (87) I =(cid:90) (cid:112)−g˜d4x [R˜−(ω+3/2)∇˜ ϕ˜∇˜µϕ˜
Ein µ
Generally, the Lagrangian for a scalar field consists of
possible interaction terms, e.g. a mass squared term, ϕ4 +exp(−2ϕ˜)L (g˜)], (92)
matt
term, etc. These can be represented as part of a generic
scalarpotentialV(ϕ). However, inthestresstensor, this which is just Einstein gravity with the scalar field as a
term’s contribution is proportional to g , so it does not matterfieldminimallycoupledtogravity,butuniversally
µν
appearintheheatflux. Ontheotherhand,kineticterms coupled to the other matter fields3.
in the scalar field action must contribute. We assume In the thermodynamic approach, this transformation
that the action is constructed out of first derivatives of returnstheentropytojustanareainthenewconformally
the scalar field. This eliminates non-minimally coupled related metric. Working in terms of this new metric, we
k-essence models [18], where the scalar fields have non- arrive at the Einstein equations
canonical kinetic terms.
Based on dimensional analysis, the most general con- G˜µνkµkν =2πTµtoνtalkµkν, (93)
tributionofthescalartotheheatfluxhasthebasicform
where
Ω(ϕ)
δQ ∼ kµkν∇ ϕ∇ ϕ (88)
scalar ϕ µ ν δ(I (g˜,ϕ˜)+I (g˜,ϕ˜,ψ))
Ttotal ∼ scalar matt . (94)
µν δg˜µν
where Ω is some dimensionless function. Since ϕ is di-
mensionfull, one would have to introduce a new length
Inthiscasethereisnoneedforabulkviscosityterm,but
scale in order to construct a non-trivial Ω. Therefore we
inordertobeconsistentwiththeequationsofmotionwe
take Ω to be an arbitrary constant. The general form of
the heat flux is now
δQ (cid:90) √ (cid:18)Ω(cid:19)
=− d4x hλ (2πTMkµkν+ kµkν∇ ϕ∇ ϕ).
T µν ϕ µ ν 3 Note that our redefinition for the scalar field differs from the
(89) standardoneusedin[21]. Inthatcasethereisanoverallfactor
of(ω+3/2)−1/2in(91),whichnormalizesthescalarkineticterm
Following the analysis as before, the equation of motion
to the canonical unity value in the Einstein frame. Hence, in
the reversible sector is now (76), but with the extra heat
thisansatzonecannotgototheEinsteinframeintheω<−3/2
flux contribution regime, where the theory is likely to be sick as we discussed
(cid:18) (cid:19) earlier. Alsoinourcaseonecanclearlyseethatsomethinggoes
3/2−Ω
ϕR −∇ ∇ ϕ+ ∇ ϕ∇ ϕ+Φg =2πTM , wronginthesameregionofparameterspace: Eqn.(92)showsin
µν µ ν ϕ µ ν µν µν fact that for ω <−3/2 the kinetic term for the field ϕ˜ changes
(90) signleadingeffectivelytoaghostfield