Horizon entropy and higher curvature equations of state

0.68 MB·English

by  Raf Guedens

Horizon entropy and higher curvature equations of state
Raf Guedens∗, Ted Jacobson†,a, Sudipta Sarkar‡,a,b
aCenter for Fundamental Physics, University of Maryland, College Park, MD 20742-4111, USA and
bInstitute of Mathematical Sciences, Chennai, India
(Dated: January 4, 2012)
The Clausius relation between entropy change and heat flux has previously been used to derive
Einstein’s field equations as an equation of state. In that derivation the entropy is proportional to
theareaofalocalcausalhorizon,andtheheatistheenergyfluxacrossthehorizon,definedrelative
toanapproximateboostKillingvector. Weexamineherewhetherasimilarderivationcanbegiven
forextensionsbeyondEinsteingravitytoincludehigherderivativeandhighercurvatureterms. We
review previous proposals which, in our opinion, are problematic or incomplete. Refining one of
these,weassumethatthehorizonentropydependsonanapproximatelocalKillingvectorinaway
that mimics the diffeomorphism Noether charge that yields the entropy of a stationary black hole.
We show how this can be made to work if various restrictions are imposed on the nature of the
horizon slices and the approximate Killing vector. Also, an integrability condition on the assumed
2 horizon entropy density must hold. This can yield field equations of a Lagrangian constructed
1 algebraically from the metric and Riemann tensor, but appears unlikely to allow for derivatives of
0 curvature in the Lagrangian.
2
PACSnumbers: 04.70.Dy04.50.Kd,04.20.Fy
n
a
J
I. INTRODUCTION boundary, (ii) the bifurcation surface lies to the past of
2
theterminalpointpand,(iii)theentropydependsonthe

] The notion of horizon entropy and thermodynamics approximateKillingvector. Inparticular,ithasthesame
c dependenceontheapproximateKillingvectorashavethe
was first discovered for black holes in general relativity
q
NoetherchargesassociatedwithaLagrangian,inanalogy
- (GR) and quickly generalized to other sorts of horizons.
r with the Wald entropy [2, 3] for stationary black holes.
Theoriginofthisthermodynamicbehaviorcanbetraced
g
Such dependence will be referred to as “Noetheresque”,
[ to local physics, and in a sense arises from the nature of
the vacuum. This led to the observation that the Ein- butbyitselfdoesnotmaketheentropyaNoethercharge.
2 Itmakessomesensethattheentropydependsontheap-
stein equation can be derived as an equation of state for
v proximate Killing vector, because the latter determines
local causal horizons in the neighborhood of a point ‘p’
5
thenotionofstationarityandentersthedefinitionofthe
in spacetime, by imposing the Clausius relation between
1
heat flux. However, at present we can offer no statistical
2 theirentropychangeandtheenergyfluxacrossthem[1].
interpretation for this form of the entropy.
6 Inthispaperweexamineapproachestogeneralizingthis
2. equationofstatederivationtoallowforhigherderivative The equation of state we can derive is consistent with
1 contributions to the entropy and field equations. The local energy-momentum conservation only if the leading
1 Einstein-HilbertLagrangianisonlythelowestorderterm ordertermintheentropysatisfiesanintegrabilitycondi-
1 (otherthanthecosmologicalconstant)inaderivativeex- tion. Thisconditionissatisfiediftheentropyarisesfrom
v: pansionofgenerallycovariantactionsforametrictheory, variation of a generally covariant function with respect
i andthepresenceofhigherderivativetermsispresumably to curvature. In other words, the entropy coincides with
X
inevitable. Severalapproachestoincludinghigherderiva- a Noether charge associated with a (particular type of)
r tivetermsintheentropyandequationofstatehavebeen gravitational Lagrangian. The need for such an integra-
a
tried. In our view none have fully succeeded, except in bility condition was anticipated in Ref. [1], since it was
thecaseofL(R)theories. Thosetheoriesarespecialhow- knownthattheentropyofstationaryblackholehorizons
ever, since they are trivially related to general relativity has this form[3–5]. We have not been able to ascertain
coupled to a scalar field by a field dependent conformal whether this is the only way to satisfy the integrability
rescaling of the metric. Here we will explain the prob- condition, and in particular whether field equations for
lems that arise with previous proposals, and propose a Lagrangiansinvolvingderivativesofcurvaturecanbeob-
solution that adopts aspects of some of the proposals. tained.
The solution differs in several ways from the original
Besides thelackof a statisticalinterpretation, and the
derivation for GR, among which are: (i) the entropy is
dependence of the entropy on the local Killing vector,
compared on two horizon slices that share a common
a strange feature of this approach is that in the case of
GR the entropy on a general horizon slice differs from
the area at the same order as the relevant area changes
in the Clausius relation. So this approach seems to lose
∗[email protected]
†[email protected] contact with some of the original statistical motivation
‡[email protected] for the local Clausius relation. On the other hand, it is
2
quiteanalogoustothefirstlawofblackholemechanicsin heat is the energy flux conjugate to the spacetime sym-
generalized gravity theories. Therefore it is not clear to metry that translates along the horizon generators. The
us whether this approach is purely formal, or maintains work term δW in the usual first law of thermodynamics
some significance as true thermodynamics of spacetime. has no analog in the relation (1). Although “first law” is
not an appropriate name, we will use it here since it is
entirely standard terminology.
II. THE EINSTEIN EQUATION OF STATE Close to a black hole horizon the Hawking tempera-
ture becomes the Unruh temperature (cid:126)a/2π for station-
WebeginwithareviewofthederivationoftheEinstein ary, uniformly accelerated observers, and the Clausius
equation as an equation of state, which emerges from relationtakesonalocalform(howlocaldependsonhow
black hole thermodynamics as follows. General relativ- slowly the changes take place) whose validity requires
ity and quantum field theory in a black hole background that the Einstein equation hold, as mentioned above.
implytheso-called“firstlawofblackholethermodynam- Moreover,thereisgoodreasontobelievethisappliesnot
ics” [6–8], just to black holes but to any causal horizon. (See e.g.
[10] for a review of the arguments, and [11] for further
dM −Ω dJ =T dS , (1)
H H BH discussion and clarification.)
whereM istheblackholemass,Ω theangularvelocity The idea of Ref. [1] (see also [12] for a slight reformu-
H
of the horizon, J the angular momentum, T = (cid:126)κ/2π lation) was that, conversely, the Einstein equation can
H
with surface gravity κ is the Hawking temperature, and be derived, as an equation of state, by requiring that
S = A/(4(cid:126)G) with horizon area A is the Bekenstein- the Clausius relation hold for the entropy of sufficiently
BH
Hawking entropy. This relation can be viewed as a com- smallpatchesofalllocalcausalhorizons(LCHs)inspace-
parison between two stationary black holes, but it also time. Any such horizon H is defined as the boundary of
holds for small, slowly time-dependent changes of a sin- the past of a patch of (D−2)-dimensional spacelike sur-
gle black hole. Its validity in that setting hinges on the face, where D is the spacetime dimension. (LCHs will
fact that the evolution of horizon area is governed by bedefinedpreciselyinwhatfollows; fornowitsufficesto
spacetimecurvature,whichinturnislinkedviaEinstein’s remain somewhat vague.) In that derivation the heat is
equation to the energy flux. Specifically, it relies on the taken as the boost energy flux, defined with respect to
relation R kakb = 8πGT kakb, where R is the Ricci an approximate local boost Killing vector field ξa as
ab ab ab
tensor, T is the energy momentum tensor of matter,
ab (cid:90)
and ka is a 4-vector tangent to the horizon generating δQ= (−T ξa)dHb, (4)
ab
null geodesics.
H
Theterm“firstlaw”isactuallyamisnomerfor(1). In
thermodynamicsthatnamereferstoenergyconservation, and the temperature is taken as the Unruh boost tem-
dU =δQ+δW, where dU is the internal energy change, perature (cid:126)/2π. This is natural since the Minkowski vac-
δQ is the heat flow into and δW is the work done on the uum state of quantum fields is thermal with respect to
system. Instead, the thermodynamic nature of (1) is the theboostHamiltonianatthistemperature,andanystate
Clausius relation looksliketheMinkowskivacuumatsufficientlyshortdis-
tances. TheentropyistakentobeA/l2,wherel issome
0 0
δQ=TδS (2) UV length scale. (A compelling case can be made that
the origin of horizon entropy is quantum entanglement
between the heat flow and the entropy change. In the
of vacuum correlations across the horizon. Since this is
black hole context, energy that flows across the horizon
dominated by UV degrees of freedom, yet finite, dimen-
is,ineffect,“heat”,sinceaftercrossingthehorizonitsmi-
sional analysis suggests that the leading order contribu-
croscopic nature is effaced for an outside observer. Since
tionshouldscaleinthisway.) Thechangeinhorizonarea
the horizon is a causal barrier, it is a “perfect dissipa-
inasmallneighborhoodofapointpcanberelatedtothe
tor” [9]. The heat flux is
Ricci tensor via the Raychaudhuri equation. If the hori-
(cid:90) zon is chosen so its expansion and shear vanish at p, and
δQ=dM −Ω dJ = (−T χa)dHb, (3)
H ab ξa is chosen so it too vanishes at p, the Clausius relation
H is then seen to require T kakb = ((cid:126)/(2πl2))R kakb at
wheretheintegralisoverthehorizonandχa =∂ta+ΩH∂φa every point in spacetimeaabnd for all null ve0ctorasbka. To-
is the horizon-generating Killing vector.1 That is, the gether with energy conservation ∇aT = 0 this implies
ab
the Einstein equation
R − 1Rg −Λg =8πG T , (5)
ab 2 ab ab 0 ab
1 We (unfortunately) use spacetime signature (−+···+). The
horizon integration measure is dH = −k dVdA, where V is
b b where the value of Newton’s constant is determined by
the affine parameter along the horizon generators, kb = (∂V)b the entropy density 1/l2 to be G = l2/4(cid:126), and Λ is an
is tangent to the generators, and dA is the area element of a 0 0 0
constantV horizonslice. undetermined cosmological constant.
3
III. PREVIOUS APPROACHES TO prescription goes as follows. In the neighborhood of a
INCLUDING HIGHER DERIVATIVES spatial slice Σ of a causal horizon one constructs, by
a unique recipe of dropping selected terms in a Taylor
There have been a number of attempts to include expansion of the true metric in an adapted coordinate
higher derivative terms in the equation of state de- system, a new spacetime metric that has an exact
rived from causal horizon entropy; all these attempts boost Killing field for which Σ is a fixed point set. The
can be divided into two broad classes: The first deals entropy density of the slice Σ was taken in [14] to be
with a specific theory, L(R) gravity (see Refs. [12–14]), a scalar formed from the IBI metric associated to Σ.
while the second class studies a more general case(e.g. In particular the case with entropy density f(R) was
Refs. [15, 16]). The L(R) gravity, where the Lagrangian discussed, but it is not clear to us that all contributions
depends only on the Ricci scalar, is equivalent to gen- to the change in this entropy were taken into account.
eral relativity with an auxiliary scalar field [17, 18] and The difficulty arises because the IBI metric changes
is therefore the simplest generalization possible. Even with the horizon slice, so the R on each slice is defined
in that case, the derivation of the field equation from with respect to a different metric. Also, even if the
thermodynamics of local horizon meets considerable ob- field equation arrived at in [14] were correct, it refers to
stacles. the curvature of the IBI metric, and it remains unclear
howthisisrelatedtothecurvatureoftheoriginalmetric.
Ref.[12]consideredthecasewhenthehorizonentropy
density is proportional to an arbitrary function f(R) of
the spacetime Ricci scalar R. As was done in the deriva- Next we would like to discuss attempts to obtain the
tion of the Einstein equation, the approximate Killing fieldequationfromlocalhorizonthermodynamicsbeyond
vector ξ is chosen to vanish at a terminal point p on L(R)gravity. Ref.[15]startsfromtheWaldentropyfor-
the horizon. The Clausius relation equates the entropy mula [3] for a stationary black hole, but applied to a
change to the heat flux δQ, which is the flux of boost local horizon. A formula for the entropy of a [(D−1)-
energy current −Tabξ across the horizon. Since ξ van- dimensional] patch of a local horizon is written as a sur-
b
ishes at the terminal point, the rate of change of entropy faceintegralovertheboundaryofthepatch. Itisunclear
with respect to the affine parameter must also vanish to us why an entropy should be assigned to a (D−1)-
there. However, at a generic spacetime point p the gra- patch rather than only to a (D−2)-slice of the horizon.
dient ∇ R is non-vanishing, so the horizon entropy will Moreover, in the case of a stationary black hole horizon
a
have a nonvanishing rate of change unless the change of thisintegralvanishes,whereastheentropyofthehorizon
R is balanced by a change of horizon area. Thus it is surelydoesnotvanish. Hencethisquantitycannotbein-
necessary to adjust the terminal surface of the horizon terpreted as the entropy of the horizon. The variation of
so that the expansion θ of the horizon generators at the this entropy is then considered, and a formula is written
equilibriumpointhasthenonzerovalueθ =−f˙/f. The in terms of an integral of a covariant directional deriva-
p
Clausiusrelationinvolvesaθ2termviatheRaychaudhuri tive of part of the integrand of the previously mentioned
p
equation, and one ends up with a field equation that is integral. Itisuncleartouswhythisexpressionshouldbe
inconsistent with matter energy conservation unless this interpreted as the change of entropy. A similar formula
term is identified as internal entropy production due to is written for the boost energy flux across the horizon.
a bulk viscosity proportional to f(R) and added to the Also, the binormal on horizon cross sections is identified
entropy balance law. Hence, instead of equilibrium ther- with the covariant derivative of the approximate Killing
modynamics,anon-equilibriumapproachisneeded. The vector. Whereas this can certainly be satisfied to lowest
resulting field equation is the one that follows from the order,giventhesmallchangesinvolvedintheClausiusre-
Lagrangian L, where f = dL/dR. That is, L is the La- lation it would be necessary to check whether neglecting
grangian for which f(R) is the Wald entropy [3]. (The the differences at subleading order is justified. Another
properties of Wald entropy will be briefly reviewed in issue is that the approximate Killing vector ξa in these
section VII.) calculationsistreatedasifitsatisfiestheKillingidentity
InRef.[13],thisentropyproductiontermisinterpreted ∇a∇bξc =Rdabcξd exactly. Although this is not possible
as a separate contribution to the heat flux from the ad- in general, we will show in the present paper that the
ditionalscalardegreeoffreedompresentinL(R)gravity. identity can be satisfied to the required order, provided
Then it is possible to derive the field equations using one restricts to a narrow neighborhood of a particular
only reversible thermodynamics as in the case of general horizon generator.
relativity. Although such an interpretation may offer a In Ref. [16] the proposal is to define the entropy of
possible understanding of entropy production terms for horizon slices as the integral of a Noether potential
the specific case of L(R) gravity, it is unclear how to formallyidenticaltoapotentialintroducedin[3], except
generalize that for a broader class of theories. that in the former case no matter fields are included in
In Ref. [14] it was proposed that the need for internal the Lagrangian with which the potential is associated.
entropy production could be eliminated by adopting Using Stokes’ theorem, the entropy change between two
the instantaneous boost invariant (IBI) prescription slices of the horizon is then expressed as an integral
for dynamical horizon entropy proposed in [3]. This of the correspondig Noether current over the enclosed
4
horizon patch. However, since slices of a LCH have The Clausius relation asserts that the entropy change
boundaries, the entropy change also involves a surface δS is equal to δQ/T, where T = (cid:126)/2π is the boost tem-
integral over the null outer boundary, unless the two perature,δQistheheatflux(4)throughthehorizonand
sliceshavetheirboundaryincommon. Thiscontribution ξa is the approximate Killing vector which vanishes at
was missed in Ref. [16]. In addition, as in Ref. [15], the final equilibrium point p. Since the heat integrand
the Killing identity was used without justification at vanishes at p, the Clausius relation can hold only if the
an order it cannot be expected to hold in a generic δS integrand also vanishes there. This requires that the
spacetime. Thus this derivation too is not complete. expansion at p is nonzero, to wit
Thepresentpaperstartsfromasimilarbutmoregeneral
(cid:12)
fmoremnt fboyr stthuedyeinntgrosplyic,esantdhafitlldsothhaevsee gthaepisr ibnoutnhdeaarryguin- θp =−1s ddλs(cid:12)(cid:12)(cid:12) . (8)
p
commonandthusencloseacompactpatchofthehorizon.
InthecaseofGRsisaconstant,sothisconditionstates
thatthehorizonmusthavevanishingexpansionatp. The
Finally we note that Refs. [19, 20] (see also [21] and
Raychaudhuri equation can be used to find θ(λ). If the
references therein) discuss a thermodynamic interpreta-
horizon shear does not vanish at p we must add internal
tion of gravitational field equations beyond general rel-
entropyproductionduetoshearviscositytotheClausius
ativity in terms of a local entropy balance law. A mat-
relation [12]. Assuming the shear at p vanishes, we have
ter entropy flux across the horizon is associated with a
θ =θ +λ(−θ2/(D−2)−R kakb)+O(λ2), where ka is
smallspatialvolumeandisrelatedtotheboostenergyby p p ab
the affine horizon tangent vector (d/dλ)a, and λ =0.
the Clausius relation. This is then equated to a gravita- p
Now we require that the Clausius relation hold for all
tionalentropychangeinthevolume,whichisconstructed
local horizons through p, i.e. for all ka. If the entropy
from the Noether current associated with a Lagrangian
density is independent of ka, as would be the case if it is
L[g ,Ra ]. This balance law at a point is then shown
ab bcd aspacetimescalar,thenweobtainthetensorialequation
to imply that the gravitational field equations hold, thus
giving a thermodynamical interpretation of those equa- D−1 2π
tions. The approach we will pursue in this paper uses sRab−∇a∇bs+ D−2s−1s,as,b+Ψgab = η Tab, (9)
similar ingredients, but we have different objectives. We
startbyassigningahorizonentropyfunctionaltoasmall whereΨissomescalarfunction. Thisfunctioncanbede-
but finite sized horizon slice, and investigate what prop- termined by imposing energy conservation, i.e. ∇aTab =
erties it must have if it is to satisfy the Clausius relation 0, which yields the condition
with the matter boost energy flux.
1 (cid:18)D−1 (cid:19);b
Ψ, =− sR, +∂ (cid:3)s− s−1s, s, . (10)
a 2 a a D−2 a b
IV. IMPOSSIBILITY OF GENERALIZING THE
SinceΨ isthegradientofascalar,asolutionexistsonly
NON-EQUILIBRIUM APPROACH ,a
if the right hand side is also the gradient of a scalar.
InthecaseofGRsisaconstant,soΨ=−sR/2isthe
A natural question to ask is whether the non-
unique solution up to a constant (the cosmological con-
equilibrium approach of Ref. [12] can be extended be-
stant). If s=s(R) is a function only of the Ricci scalar,
yond theories for which the horizon entropy density s is
thenthefirsttermontherighthandsideof(10)isagra-
a function only of the Ricci scalar. In this section it will
dient, but the last term is not. In the non-equilibrium
be shown that the non-equilibrium approach of Ref. [12]
approach of Ref. [12] the Clausius relation is replaced by
cannot be extended beyond the results found there. For
an entropy balance relation δS =δQ/T +δS , where δS
this discussion, we adopt the geometric set up of [1, 12] i i
isaninternalentropyproductiontermduetobulkviscos-
andweassumethattheentropyassociatedwiththeLCH
ity, which is proportional to the square of the expansion
takes the form
and which cancels the last term in (10). This yields an
(cid:90)
equation of state that coincides with the field equations
S = s dA, (6)
for L(R) gravity. For more general spacetime scalar en-
tropy densities, e.g. s(R ), even the first term is not
abcd
wheresissomearbitraryscalarfunctionandtheintegra-
a gradient, so there is apparently no way to satisfy an
tion is over a (D−2)-dimensional spacelike slice of the
entropy balance law, even allowing for internal entropy
horizon with ‘area’ element dA. The change in entropy
production. Hence, the methods devolved in [1] and [12]
from one horizon slice to another is
can not be generalized beyond L(R).
(cid:90) (cid:18)ds (cid:19) The situation is even more problematic if the entropy
δS = +θs dλ dA, (7) density is not a spacetime scalar but, for example, is
dλ
constructed from the intrinsic curvature of the (D−2)-
where λ is an affine parameter on the local horizon gen- dimensional horizon slice, as in Lovelock gravities [22].
erators and θ =d(lndA)/dλ is the expansion of the gen- In that case, we do not even have a tensorial relation-
erators. ship like Eq.(9). The Raychaudhuri equation then does
5
by this choice. The role in the new approach will be
p p explained in Sec. VIII.
B The role of the approximate Killing field ξa in the ap-
proach of Ref. [1] is to define the heat in the Clausius
relation (4). The bifurcation surface of this Killing field
was previously taken to coincide with the future bound-
B ary through p, whereas in the first law of global horizon
p0 p0 mechanics the bifurcation surface lies to the past of the
Γ Γ perturbation. Moreover, the approximate Killing field
was spacelike rather than timelike in the region outside
the horizon, so it was necessary to think of the heat as
Figure1: Causalspacetimediagramofthelocalcausalhori- going into the reservoir behind the horizon (see Fig. 1).3
zon (LCH) and Killing field in the old (left) and new (right)
However, this reservoir is not observable on the outside,
setups. Eachpointinthediagramrepresentsapatchofspace-
so should play no role in the outside thermodynamics.
like 2-surface. The boundary of the past of a patch of 2-
It seems much more satisfactory to place the bifurcation
surface through p defines the LCH. The arrows indicate the
flow lines of the local Killing field ξa, which is of third order surface to the past, so that the Killing field is timelike
onthebifurcationsurfaceB,andwhichvanishesatp(left)or outside the horizon, and the reservoir can be thought of
p (right). The central horizon generator Γ runs from p to indirectanalogywiththethermalatmosphereofablack
0 0
p. The“heat” istheboost energyfluxacross thehorizon. In hole, below the stretched horizon.4
thenewsetuptheKillingvectoristimelikewithinthehorizon It appears at first that modifying the location of the
and spacelike beyond, while the old setup is the opposite. bifurcation point of the Killing vector will change the
heat flux and, in the original approach of Ref. [1], ruin
thederivationoftheEinsteinequationfromtheClausius
not seem to be of any help at all, so a very different ap- relation. However, that is not what happens. The old
proach is needed. In this paper we will present such an Killing vector on the generator through p (see Fig. 1)
alternative approach, which involves an entropy density was ξold = −λk , where λ is the affine parameter that
b b
that depends explicitly on the Killing vector, a different vanishesatp. ThenewKillingvectorvanishesinsteadat
choice of horizon slices, and the application of Stokes’ p and is given by ξ =(λ−λ )k , where λ is the value
0 b 0 b 0
theorem. of λ at p . The corresponding boost energy currents are
0
thus
−Tabξold = λTabk (11)
V. NEW CHOICE OF APPROXIMATE b b
KILLING VECTOR −Tabξb = (λ0−λ)Tabkb (12)
Althoughthesediffer,theirintegralsfromthebifurcation
It turns out that the new approach to the thermody-
point to p are the same, since
namic derivation of field equations, even in the case of
general relativity, calls for a small but essential adjust-
(cid:90) 0 (cid:90) 0
ment in the choice of approximate Killing vector. This λdλ=−λ2/2= (λ −λ)dλ. (13)
adjustment amounts to locating the bifurcation surface2 λ0 0 λ0 0
atatimeearlierthantheterminalpoint,ratherthanbe-
ing coincident with the terminal point as originally for- With the heat defined using ξa, applying the Clausius
mulated in Ref. [1]. The change also makes the applica- relationtothehorizoninterval[λ0,0]inthelimitλ0 →0
tion of the Clausius relation more closely analogous to thus yields the field equations as described above. This
the first law of black hole mechanics [23] and resolves correspondstoatransitionbetweenastationarystateat
an uncomfortable aspect of the earlier derivation. In λ=λ0 (where ξa vanishes) and one at λ=0 (where the
this section we explain the thermodynamic motivation expansion and shear vanish). In analogy to the physical
for the new choice of approximate Killing vector, and process version of the first law of black hole mechanics,
show how the original derivation of Ref. [1] is modified the Clausius relation holds only when applied to this en-
tire interval.
2 In the current context, the term “bifurcation surface” is used
somewhat loosely, mainly to fix attention to the analogous role 3 We use the term “outside” to refer to the region accessible to
playedbythetruebifurcationsurfaceofaquasi-stationaryblack observerstothepastofp,liketheobserversoutsideablackhole
hole in the physical process version of the first law [23]. What horizon. For a cosmological horizon, the standard terminology
wemeanisasliceonwhichthecomponentsoftheapproximate would be opposite: outside a cosmological horizon is the region
Kiling vector are third order in an appropriate coordinate sys- thatcannotbeseen.
tem(seeSec.VID).Accordingly,theapproximateKillingvector 4 This same point was made recently in Ref. [24], with reference
vanishesatthecenterofthebifurcationsurface(seeFig. 1). tothechoiceofthe“observer”whoisdefiningtheheatflux.
6
Note that in the approach of Ref. [1], with the new of p lies on a unique geodesic orthogonal to Σ at some
p
choice of Killing vector, one must still impose the condi- point q. Let the tangent to that geodesic at q be given
tionthattheexpansionθvanishesatp,becausethatcon- by Vka+Ula when r lies at unit affine parameter from
dition is used in applying the Raychaudhuri equation to q. The NNCs of r are then defined by5
obtaintheareachange. Bycontrast,intheNoetheresque
approachofthepresentpapertheexpansionatpwillturn xαr =(U,V,xAq). (14)
out to play no role whatsoever.
These coordinates are defined uniquely up to a rotation
of the (D −2)-frame at p and a q-dependent rescaling
of ka, with inverse rescaling of la. The horizon H is
VI. PROPERTIES OF THE LOCAL HORIZON
the surface U = 0, restricted to V ≤ 0. In particular,
AND KILLING VECTOR
the central horizon generator Γ is the coordinate curve
U = 0, xA = 0, V ≤ 0, with bifurcation point p ∈ Γ
0
In this section we spell out the detailed geometric
at V = V < 0. Note that on H the coordinate V is an
0
construction and properties of the local causal horizon
affineparameteralongthenullgenerators. Thedetailsof
(LCH) and Killing vector.
theconstruction,includingthederivationofmetriccoeffi-
cientsandChristoffelsymbols,arepresentedinRef.[25].
It is also shown there that the ambiguity in the choice of
A. Local causal horizon ka can be exploited to make the coordinates locally in-
ertial at p and to further specialize the properties of the
WedefinealocalcausalhorizonH asfollows. Consider metric components. A preferred choice is made by start-
anyspacetimepointpinaD-dimensionalspacetime,and ingwiththeD-dimensionalRiemannnormalcoordinates
let Σp be any small patch of spacelike (D −2)-surface at p and adjusting them. This also induces a particular
through p. The boundary of the past of Σp in the neigh- choice of affine parameter V.
borhoodofphastwocomponents,eachofwhichisanull
surface generated by a congruence of null geodesics or-
thogonal to Σp. The LCH H is defined as one of these C. Horizon slices
components.
As mentioned in the previous section, it will not mat-
TheClausiusrelationappliedtoahorizonreferstothe
ter for the present approach whether the congruence is
entropy change δS between two times. Those times are
stationaryatp. However,ifpistobeastationarypoint,
spacelike hypersurfaces, one to the future of the other,
the expansion and shear of this congruence must vanish
which intersect the horizon in two slices. For a black
there. That is, the extrinsic curvatureof Σ must vanish
p hole with a compact horizon, the two slices may bound
at p, which is equivalent to saying that Σ is generated
p a cylindrical region of the horizon.
bygeodesicsatp. Forconcretenesswewillgofurtherand
Foralocalcausalhorizon,theconsideredprocessmust
assumethatΣ isfullygeneratedbygeodesicsemanating
p belocal,sincetheLCHisnotevenwelldefinedexceptin
fromp. Thiswillbeconvenientforstudyingconcreteex-
a small neighborhood of the terminal point p. To local-
amples of the approximate Killing vector and of entropy
ize the process we can restrict attention to cases where,
values.
rather than pushing the spacelike hypersurface forward
in time everywhere, it is deformed to the future only in
a small neighborhood of the bifurcation point p .6 Then
0
B. Null normal coordinates the two corresponding horizon slices also coincide every-
where except in a small region. Parts of two such slices,
To establish the existence of an approximate Killing Σ and Σ, are depicted in Fig. 2. Σ corresponds to the
0 0
vector with the required properties, we will employ a V = V surface and Σ lies to the future. If we truncate
0
“null normal coordinate” (NNC) system [25] adapted to the horizon slices outside the region where they differ,
the LCH. This is an explicit realisation of the coordi- their union Σ ∪Σ forms the closed boundary of a patch
0
nate systems first introduced in [3, 26] and is defined of the horizon. This will allow the difference of the en-
as follows. At a spacetime point p an orthonormal set of tropies on the two slices to be computed using Stokes’
(D−2)spacelikevectors{ea},A=1,2,..,(D−2),ischo- theorem.
A
sen, and a (D−2)-surface Σ is generated by geodesics
p
with tangent vectors at p in the space spanned by {ea}.
A
The point reached on such a geodesic at unit affine pa-
rameterisassignedthecoordinatesxA,whenxAea isthe 5 We use greek indices, and upper case latin indices, for coordi-
A
tangentvectorinthatparametrisationatp. Thisdefines nates and coordinate components in the NNC coordinate sys-
standard Riemann normal coordinates on Σ based at p. tem. Lowercaselatinindicesarereservedforabstractindiceson
p
Apairoffuturepointingnullvectorfields(ka,la)normal tensors.
6 Hypersurface deformations of this sort were considered in
to Σ is chosen on Σ , normalized such that k la = −1.
p p a Ref. [27] in the context of proving a local form of the Gener-
Each point r in a small enough spacetime neighborhood alizedSecondLaw.
7
p
V =0 to refer to components in a particular coordinate sys-
Σ
p
tem, reserving lower case latin indices for abstract ten-
sor indices.) An approximate boost generator ξa can be
defined in terms of local inertial coordinates by the for-
mula for an exact flat spacetime boost generator, e.g.
(x,t,0,0)inMinkowskicoordinates. Thisvectorsatisfies
the Killing equation
ka
∇ ξ +∇ ξ =0 (15)
Σ a b b a
Γ
exactly at p, and to O(x) near p, but in general ∇ ξ
(a b)
willhaveO(x2)terms. Thisisadequateforourpurposes.
V=V0 Σ However, our application of the Clausius relation is
P0(ξa=0) 0 also sensitive to the O(x) part of the Killing identity
Figure2: Thelocalcausalhorizonispartoftheboundaryof
the past of Σp, a spacelike D−2 dimensional surface. The ∇a∇bξc =Rdabcξd. (16)
vertical lines are horizon generators with affine parameter V
and tangent vector ka, and Γ is the central generator. Σ is
p For a true Killing field, this identity follows from the
geodesic at p, so the expansion θ and shear vanish at p. The
Killing equation (15). Conversely, the Killing identity
horizontallinesrepresentconstantV slices. ThelocalKilling
implies the Killing equation if the latter holds at one
vector vanishes at the bifurcation point p . The entropy is
0
compared on two slices, Σ and Σ, which differ only in a point, because of the antisymmetry of the Riemann ten-
0
compact region. sor in the last index pair. Our computations will rely on
this identity being satisfied at O(x), but for the approxi-
mateKillingfielddefinedaboveitwillgenerallynothold
In fact, we shall need to further restrict the choice of atthatorder. Wecantrytomodifythedefinitionofthis
horizon cuts so that they bound a narrow region of the vectorfieldsoastosatisfyboththeKillingequationand
horizon,becauseonlythencantheKillingidentitybesat- the Killing identity at O(x), but in fact this is not possi-
isfied to sufficient accuracy for the approximations made ble. However, narrowing our sights, ξb can be chosen so
inthederivationoftheequationofstatetobevalid7. By thattheKillingidentityholdstothisorder(infactitcan
“narrow”, we mean that the ratio of the width in xA to bechosentoholdexactly)onthesinglehorizongenerator
thelengthinV goestozerointhelimitasp approaches Γ that ends on the terminal point p, and this turns out
0
p. to be good enough. In effect, the local Killing symmetry
can be extended away from p to a better approximation
along a single null generator Γ than across the whole
LCH. This calls to mind null Fermi coordinates[28], but
D. Local Killing vector
we have used the NNCs to describe the situation since
they are better adapted to the LCH.
NextwedefinepreciselytheapproximateKillingvector
Given that we confine the horizon to a narrow region
ξa thatplaysacentralroleindefiningboththeheatflux
surroundingthecentralhorizongeneratorΓ,theintegrals
and, in the Noetheresque approach of Sec. VIIIB, the
appearing in the Clausius relation will be dominated by
entropy density. We will refer to this vector field as the
their integrands evaluated on Γ. As such, any conditions
“local Killing vector”. that ξa may need to satisfy in order for the Clausius
Of course a general curved spacetime has no Killing relation to lead to a consistent equation of state will be
vectors. Nevertheless, in a small enough neighborhood conditions imposed on Γ.
of any point, any spacetime is approximately flat. In
As motivated in Sec. V, the local Killing vector ξa is
particular, in local inertial coordinates at p, the metric
taken to have a bifurcation surface Σ —or at least a bi-
components take the form g =η +O(x2/L2), where 0
αβ αβ furcation point—at p to the past of p, where it vanishes
η is the Minkowski metric, x denotes the coordinates, 0
αβ andwhereitscovariantderivativegeneratesboostsinthe
and L characterizes the shortest radius of curvature of
plane orthogonal to the bifurcation surface. The NNCs
thespacetimeatp. (Hereandbelowweusegreekindices
of p are (U,V,xA) = (0,V ,0). It will be convenient to
0 0
shift the affine parameter such that its origin coincides
with the bifurcation point, i.e. we define V˜ ≡V −V , so
0
p lies at V˜ = 0. We also want ξa to be approximately
7 An alternative option would be to restrict to a horizon patch 0
tangent to the generator Γ that connects p to p, so that
that is symmetric in the transverse coordinates, to cancel con- 0
the evolution of the LCH can be interpreted as a small
tributions from terms linear in those coordinates. However, we
considerthissymmetryrestrictiontobeartificial. perturbation of a stationary background, which justifies
8
the use of the Clausius relation.8 More specifically, the out to be consistent with the Killing identity on Γ. In
equation of state derivation will require that, at least to NNCs,theapproximateKillingequationimpliesthatthe
lowestorder,ξa coincideswithV˜ka,aswouldbethecase components of the local Killing vector are of the form
on a Killing horizon. And, finally, the derivation will re- ξα =V˜δα −Uδα +O(x3).
V U
quiretheKillingequationandtheKillingidentitytohold In summary, the main properties exhibited by our
at O(V˜). Expressed in NNC components, the full set of choice of local Killing vector are given by (17) together
requirements that must be imposed on ξα is: with
∇ ξ | = (k l −l k )| , (17) ξα| = V˜δα, (22)
α β p0 α β α β p0 Γ V
ξα| = V˜δα +O(V˜2), (18) ∇(αξβ) = O(x2), (23)
Γ V
∇(αξβ)(cid:12)(cid:12)Γ = O(V˜2), (19) ∇α∇βξγ|Γ = (cid:0)Rδαβγξδ(cid:1)(cid:12)(cid:12)Γ. (24)
∇α∇βξγ|Γ = (cid:0)Rδαβγξδ(cid:1)(cid:12)(cid:12)Γ+O(V˜2). (20) To conclude this section, we turn to the perturbative
argument that the Killing identity (16) can be satisfied
As alluded to above, ξa may be subjected to rather exactly on Γ. We specify ξa in a neighborhood of Γ by
stronger conditions under which it approximates a true its (covariant) components in NNCs as the Taylor series
Killingvectormoreclosely. Inparticular,wemaychoose
ξa such that the Killing identity (16) holds exactly on ξ =UδV −V˜δU +C x˜βx˜γ +D x˜βx˜γx˜δ+...(25)
α α α βγα βγδα
Γ. This can be demonstrated by a perturbative argu-
ment which is given at the end of this section (see also wherex˜α =xα−V δα andinparticularV˜ =V−V . Sim-
0 V 0
[25] for further details). The identity then implies that ilarly, on Γ the deviation away from the Killing identity
the Killing equation also is exact on Γ, provided it holds iswrittenasapowerseriesintheaffineparameterV˜. As
exactly at one point. Another consequence is that ξa shown in [25], the latter series may be set to zero order
becomes exactly tangential to Γ, as will be shown next. by order through an appropriate choice of the expansion
Hence our claim is that all terms of quadratic or higher coefficients occurring in (25).
order in V˜ in (18) - (20) may actually be set to zero. More specifically, at the linear order (20) needed for
If a vector field ζa satisfies the Killing identity along a the equation of state derivation, one finds
geodesic with affine tangent vector va, it follows that
C ∼ V ×(Riemann components at p), (26)
αβγ 0
va∇a(cid:0)vb∇bζc(cid:1)=Rdabcζdvavb. (21) DVαβγ ∼ (Riemann components at p). (27)
Eqn.(21) is an ordinary differential equation of second Thus the terms that are quadratic and cubic in x˜
order and has a unique solution along the geodesic, once in the expansion (25) both contribute to ξ only at
α
ζa and vb∇bζa are given at an initial point. It is easily O(x3), counting V0 as O(x). This is consistent with the
verified that sva is a solution to (21), where s is any Killing equation (23) at O(x), from which one finds that
affine parameter on the geodesic, and vb∇b(sva) = s˙va. quadratic order terms must be absent in (25).
Hence if ζa = sva and vb∇bζa = va at one point, it For the purpose of the general derivation of field
follows that ζa = sva everywhere on the geodesic. Now equations all we need to know is that the conditions
recall that ka is an affine tangent to Γ, and the local (17)-(20) can be met. For the purpose of computing the
Killing vector is chosen to vanish at p0 and to satisfy actual entropy (which will have a dependence on the
∇vba∇ξbb|ξpa0(cid:12)(cid:12)p=0 =(kavlab|p−0,laskob)w|pe0.mTayakcionngclvuade=tkhaatthξaus=yiV˜elkdas lthohicgeahleaKrreidalleirninigvavttheivceteoGgr,RrasveceiatySs,eecto.hreVwIdIeiIttBhai)loeatdnhdeforcromemxpoprfaerstishnieognlwoscitianhl
everywhere on Γ.9
Killing vector is generally needed. For more explicit
As regards the Killing equation, we recall that the
expressions of the higher order coefficients in (25) we
NNC system is local inertial at p. As discussed at the
refer to [25].
beginning of the current section, it is then possible for
the Killing equation (15) to hold at O(x), and this turns
VII. BLACK HOLE ENTROPY AS NOETHER
CHARGE
8 Moreover,ifweweretoletthelocalKillingvectorandthetan-
genttothegeneratorvaryindependently,itappearstheClausius
relationwouldbeoverlyrestrictive. In this section we review Wald’s expression for black
9 ItwillgenerallybeinconsistentwiththeKillingidentityonΓto hole entropy in terms of the Noether charge[2, 3]. This
chooseξatobetangentialtoallthegeneratorsofasmallhorizon expression will motivate the form of LCH entropy that
patch. Moreover, if the horizon expansion does not vanish at
we adopt.
the bifurcation point, such a choice would also be inconsistent
Consider a diffeomorphism-invariant Lagrangian field
withtheKillingequationimposedatlinearorderinaspacetime
neighborhood. theory of gravity in arbitrary dimensions. A variation of
9
theactionisalwaysequaltoasumoftermsproportional If the theory admits stationary black hole solutions
to the field equations and a surface term. When the with a regular bifurcation surface, variations away from
variation is induced by a vector field ξa, this equality these solutions satisfy the first law (1), with the entropy
can be expressed as the statement that a certain current S defined by [3]
BH
ja is closed, i.e. ∇ ja = 0. The current is closed even
when evaluated off-ashell and takes the form S = 2π (cid:73) Qab[χˆ]N dA. (33)
BH (cid:126) ab
ja =θa−Lξa−(cid:0)2Eab−Tab(cid:1)ξ +..., (28) Σ
b
Here,Σdenotesanysliceofthestationaryhorizon,N =
ab
where the symplectic current θa stems from the sur- 2k l is its binormal (normalized as N Nab = −2),
[a b] ab
face term, L is the Lagrangian scalar in the action, dA is the (D −2) dimensional area element, and χˆa is
2Eab − Tab = 0 is the metric field equation and the the horizon-generating Killing vector normalized to unit
dots indicate off-shell terms corresponding to the mat- surface gravity. The integral is referred as the Noether
ter field equations. The Noether current Ja is defined as charge and is invariant under the three sources of am-
Ja = θa −Lξa, and on shell coincides with the closed biguity [3, 5]. Invoking the stationary symmetry, it was
current ja. Furthermore, since the on-shell Noether cur- further shown in these references that the entropy may
rent is closed for any possible vector field ξa, it can equally be expressed by substituting Qab[χˆ] in (33) with
be expressed as Ja = 2∇ Qab, where the antisymmet- XabcdN . Thus the W term does not contribute to the
b cd
ric tensor Qab is a local function of the fields and their black hole entropy. For general relativity, S becomes
BH
derivatives[29]. ThetensorQabisreferredtoasaNoether the familiar Bekenstein-Hawking entropy A/4(cid:126)G.
potential.10
Since the vector field ξa only enters the expression for
aNoetherpotentialwhentakingLiederivativesoftensor VIII. CLAUSIUS RELATION AND EQUATION
fields with respect to ξa, any Noether potential is of the OF STATE
form
In this section we adopt the assumption that LCH’s
Qab[ξ]=Wabcξc+Pabcd∇cξd, (29) have an entropy of Noetheresque form, i.e. the entropy
density depends on the local Killing vector in the same
apart from the possible addition of a total divergence.
waythattheNoetherpotential(29)dependsonthehori-
More specifically,
zon generating Killing field. With this entropy, we shall
find that the Clausius relation applied to all LCHs, to-
Qab[ξ]=Wabcξ +Xabcd∇ ξ +Yab+∇ Zabc, (30)
c [c d] c gether with the local conservation law for the matter
stress tensor, can be satisfied provided that i) the en-
wherethetensorsW,X,Y,Z arelocallyconstructedfrom
tropy density can be identified with (the gravitational
the dynamical fields, Y is linear in Lie derivatives with
respect to ξ, and Zabc is totally antisymmetric. The de- part of) a Noether potential of some Lagrangian, and ii)
the fields satisfy the metric field equation for that La-
composition of Q in terms of W,X,Y,Z is not unique,
grangian.11 This is the equation of state for the corre-
and in addition the Noether potential has three sources
sponding entropy function.
of ambiguity, coming from the freedom to add a total
divergence to the Lagrangian, the symplectic potential,
or the Noether potential itself. Using all this freedom Y
A. Clausius relation for a local causal horizon
and Z can be set to zero, X can be chosen [3] as
∂L This Clausius relation for a LCH is analogous to the
Xabcd =− +... (31)
∂R first law for stationary black holes. However, the latter
abcd
reliesonthestationarityoftheblackholehorizon,aswell
and W is then given by [30] as the fact that a horizon slice is a complete boundary
componentofaspacetimeslice,neitherofwhichholdfor
Wabc =2∇ Xabcd+matter terms+..., (32)
d a LCH. Nevertheless, it turns out that the construction
can be localized enough to make the Clausius relation at
wherethedotsindicatetermsthatstemfromderivatives
least well defined. First, as explained in Sec. VIC and
of the Riemann tensor in the Lagrangian.
Fig. 2, we compare the entropy on two LCH slices Σ and
Σ thatshareacommonboundary,sothattogetherthey
0
formtheboundaryofalocalpatchH ofthehorizon. The
10 Inthispaperweusetensornotation. TheformalisminRefs.[2,3]
makesuseofdifferentialforms,whicharedualtothesetensors.
The relation between a p-form F and the corresponding tensor
F˜isFa1...ap =F˜b1...bn−p(cid:15)b1...bn−pa1...ap,where(cid:15)isthevolume 11 Notethatfortheequationofstatetobewelldefinedthematter
element. ThetensorcorrespondingtotheexteriorderivativedF stress tensor need not be the functional derivative with respect
is(n−p)∇aF˜b1...bn−p−1a. tothemetricofanaction.
10
lackofstationaritylooksmoreproblematicbecause,fora changeintegrandof(35), multipliedbyT, mustbeequal
dynamicalblackholehorizoninageneraltheory,thereis totheO(x)heatfluxintegrand,uptoO(x2)terms. That
no well defined notion of entropy available12. The prob- is,
lemisthatthereisnoKillingvectorand,ifavectorfield
is somehow selected, the ambiguities in the definition of −((cid:126)/π)∇bsabka =Tabξbka+O(x2). (37)
theNoetherchargewillmaketheentropyambiguous,un-
like for a stationary black hole horizon.13 On the other This relation is imposed at all points p and for all null
hand, locallythereisalwaysanapproximateKillingvec- directions at p.
tor, the “local Killing vector” constructed in Sec. VID. Weemphasizethatthelimitp0 →pistakeninaformal
ThisturnsouttoactenoughlikeaKillingvectortomake sense only, in order to identify the leading order contri-
theClausiusrelationwelldefinedinacertainlocallimit, bution for a small region. From a physical point of view,
once a form for the entropy is adopted. it makes no sense to consider an arbitrarily small region,
Inthecontextofhorizonthermodynamicsitisnatural because quantum fluctuations of the metric presumably
toexpectthattheentropyisanextensivequantity. Thus invalidate our semiclassical considerations at sufficiently
we assume that it can be expressed as an integral of a short distances.
(D−2)-form over a spacelike slice Σ of the horizon. We
adopt the dual description, and express the entropy as
the integral B. Entropy density of Noetheresque form
(cid:90)
S = sabN dA, (34) Inspired by Refs. [16, 19], we propose a local entropy
ab
densityofthesameformastheNoetherpotentialofEqn.
Σ
(29), where ξa is the local Killing vector, but a priori no
where the entropy density sab is an antisymmetric ten- restrictionsareplacedonW andP,otherthanthatthey
sor, Nab is the binormal to the slice and dA is the area areconstructedlocallyfromthedynamicalfields. Weem-
elementontheslice. Thechangeofthisentropybetween phasize that, although we are using a similar notation,
the two horizon slices is we do not assume at this stage that the divergence of
the local entropy density is a Noether current associated
δS = S−S0 to a Lagrangian, in contrast to the entropy density of
(cid:73)
stationary black holes. This form of entropy density can
= sabN dA
ab be viewed as the most general one that depends linearly
Σ∪Σ0
(cid:90) onalocalKillingvectorthatsatisfiestheKillingidentity,
= 2 ∇bsabdHa andcontributestotheentropychangeatO(x2). Asmen-
H tioned in the introduction, since the local Killing vector
(cid:90)
= −2 ∇ sabk dVdA, (35) determines the local notion of equilibrium and heat, it
b a
H is not entirely unnatural that the entropy would depend
on the Killing vector. On the other hand, this entropy is
whereΣ istakenwiththeoppositeorientationtoΣ,and
0 not strictly intrinsic to the horizon, and has no immedi-
inthethirdlinewehaveusedStokes’theorem(fordetails
ate statistical interpretation that we are aware of.
of Stokes’ theorem on a null surface see Ref. [31]).
We thus assume the entropy density takes the form
On the right hand side of the Clausius relation, dS =
δQ/T, we have the heat flux 2π
sab = Qab, (38)
(cid:126)
(cid:90)
δQ= (−T )ξbdH (36)
ab a with
H
dividedbytheUnruhboosttemperatureT =(cid:126)/2π. The Qab = Wabcξc+Pabcd∇cξd
heat flux integrand is proportional to the local Killing = Wabcξ +(cid:0)Xabcd + Yabcd(cid:1)∇ ξ . (39)
c c d
vector which vanishes at p and is thus of O(x) in NNCs
0
in the neighborhood of p0. The Clausius relation is im- In the second line of (39), the tensor P is split into an
posed in the limit p0 →p, which means that the entropy antisymmetric part X and a symmetric part Y:
Xabcd =Xab[cd] and Yabcd =Yab(cd). (40)
12 Generalrelativityisanexception,wheretheareaofthehorizon We could of course absorb the factor of 2π/(cid:126) into the
slices is a good candidate for the dynamical entropy. This is definition of Qab. The only reason we factor it out here
motivatedbytheareatheorem,aswellasbytheformofentan- issothatthenotationwillmaketheanalogywiththefirst
glemententropy;seeSec. II.
law for stationary black hole horizons more transparent.
13 Although the proposed dynamical entropy of Ref. [3] (in terms
We now show that the symmetric part does not con-
oftheinstantaneousboostinvariantmetric)managestobypass
theambiguities,itmostlikelydoesnotsatisfyasecondlaw[32]. tribute to the Clausius relation. The contribution of