Prepared for submission to JHEP

Null Surface Thermodynamics

H. Adamia,b , M.M. Sheikh-Jabbaric , V. Taghilooc,d , H. Yavartanoob

a Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, Chinab Beijing Institute of Mathematical Sciences and Applications (BIMSA), Huairou District, Beijing 101408, P. R.Chinac School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O.Box 19395-5531, Tehran, Irand Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), P.O. Box 45137-66731,Zanjan, Iran

E-mail: hamed.adami@bimsa.cn, jabbari@theory.ipm.ac.ir, v.taghiloo@iasbs.ac.ir,yavar@bimsa.cn

Abstract: We establish that boundary degrees of freedom associated with a generic co-dimension one nullsurface in D dimensional pure Einstein gravity naturally admit a thermodynamical description. The nullsurface thermodynamics which we expect to universally hold is a result of the diffeomorphism invarianceof the theory and do not rely on other special features of the null surface. Using standard surface chargeanalysis and covariant phase space method, we formulate laws of null surface thermodynamics which arelocal equations over an arbitrary null surface. This thermodynamical system is generally an open systemand can be closed only when there is no flux of gravitons through the null surface. Our analysis extendsthe usual black hole thermodynamics to a universal feature of any area element on a generic null surface.We discuss the relevance of our study for the membrane paradigm and black hole microstates.

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Contents

1 Introduction 1

2 Null Surface Solution Phase Space, A Review 2

3 Null Surface Thermodynamics 5

3.1 Null Boundary Thermodynamical Phase Space 6

3.2 Local First Law at Null Boundary 7

3.3 Local Extended Gibbs-Duhem Equation at Null Boundary 7

3.4 Local Zeroth Law 7

4 Outlook 9

A Null Boundary Focusing Theorem 10

1 Introduction

Despite apparent differences, there are various hints that gravity, as formulated by Einstein’s GeneralRelativity (GR), and thermodynamics are closely related to each other, both at conceptual and formulationlevels. Perhaps, the first hint was already in the statement of Einstein’s equivalence principle and theuniversality of GR and thermodynamics. In the context black hole physics the resemblance between lawsof black hole mechanics and laws of thermodynamics [1] was gradually completed into the equivalence ofthe two [2–5]; see e.g. [6–10] for some historical remarks and further references.

The connection between gravity and thermodynamics is not limited to black holes. In a seminal paper[11], Unruh showed that there is a non-zero temperature associated with a generic accelerated observer,as required by the equivalence principle. The next remarkable step was provided by Wald who showedthat black hole entropy is a conserved charge associated with bifurcate Killing horizons [12] and derivedthe first law of thermodynamics for generic probes around such black holes as a direct consequence ofdiffeomorphism invariance [13]. And finally, Jacobson derived Einstein’s field equations from the firstlaw of thermodynamics adapted around a null surface [14], see also [15, 16] and [17–19]. The connectionbetween gravity and thermodynamics was also reinforced through the holographic principle [20] and theAdS/CFT duality [21] and presented bluntly in [22].

Thermodynamical aspects of black holes and the possibility of black hole evaporation lead to theinformation problem, e.g. see [23–25] for recent reviews and insights. The ‘central dogma’ in resolving thisproblem has been that black hole thermodynamics and in particular, its entropy is a result of underlyingblack hole microstates governed by a unitary theory, like any ordinary thermodynamical system. Therehave been many ideas and proposals for identifying black hole microstates and/or counting them. Mostof them crucially rely on quantum gravity aspects and some, most notably the soft hair proposal by

– 1 –

Hawking, Perry and Strominger [26], rely only on (semi)classical aspects. Our analysis here is motivatedby the soft hair idea.

The thermodynamical aspects and black hole microstates are generically attributed to the horizon,typically a null surface which is the boundary of the spacetime as viewed by non-free fall observers outsidethe hole. Recalling the equivalence principle, which is manifested in diffeomorphism invariance of thetheory and the usual no-hair theorems for black hole solutions, there remains no room for seeking blackhole microstates within usual GR. Nonetheless, one should note that the equivalence principle needsamendment in presence of spacetime boundaries [27]: There are ‘boundary degrees of freedom’ whichreside only at the (timelike or null) boundaries and interact with bulk (graviton) modes. The presence ofboundary degrees of freedom opens the way to attribute presumed black hole microstates to the horizon.The soft hair proposal may be viewed as a statement of this idea.

To formulate the idea outlined above, one should study gravity theory on spacetimes with a nullboundary. This boundary can be an arbitrary one in spacetime. This has been the research programpursued in some recent works [28–31] and in particular in [32], see also [33–45] for related recent literature.It has been established that the most general solution phase space of D dimensional pure Einstein gravitytheory around a given null surface besides the bulk graviton modes involves boundary modes parametrizedby D arbitrary functions over the D− 1 dimensional null boundary. In this work, building upon analysisand results of [32], we show that the null boundary dynamics may be concisely recast through equationswhich are local versions of usual thermodynamical equations.

In what follows we present a quick review of the results of [32], for more detailed analysis the reader isreferred to that work and references therein. Section 3 contains our main results and we close with furtherdiscussions and outlook in section 4. In the appendix, we discuss the null boundary focusing theorem.

2 Null Surface Solution Phase Space, A Review

We start with a D dimensional (D ≥ 3) generic metric adopting v, r, xA coordinates,

ds2 = −V dv2 + 2η dv dr + gAB

(dxA + UA dv

) (dxB + UB dv

), (2.1)

such that r = 0 is the null surface N and metric coefficients admit the near N expansion,

V = −η(

Γ− 2

D − 2

DvΩΩ

+Dvηη

)r +O(r2)

UA = UA − r ηΩJ A +O(r2)

gAB = ΩAB +O(r)

(2.2)

where all the fields are functions of v, xA and

ΩAB = Ω2/(D−2)γAB, Ω :=√

det ΩAB, det γAB = 1. (2.3)

We use the definitionDv := ∂v − LU , (2.4)

where LU is the Lie derivative along UA direction.Let Θ be the expansion of vector field generating thenull surface N and NAB be the news tensor associated with flux of gravitons through N :

Θ := Dv ln Ω, NAB :=1

2Ω2/(D−2)DvγAB (2.5)

– 2 –

We use ΩAB and ΩAB respectively for raising and lowering capital Latin indices. Note that NAB as definedabove is a symmetric-traceless tensor.

The coefficients appearing in the metric are subject to Einstein field equations. In particular, thereare Raychaudhuri and Damour equations which play a crucial role in our analysis, see [32] for a moredetailed treatment. These two equations respectively are

DvΘ +1

2

(Γ− Dvη

η

)Θ +NABN

AB = 0, (2.6a)

DvJA + ΩΘ∂Aη

η+ Ω∂A (Γ− 2Θ) + 2Ω∇BNAB = 0. (2.6b)

Here ∇A denotes the covariant derivative with respect to the metric ΩAB. The new variable P defined as

P := lnη

Θ2, (2.7)

may substitute η and in terms of which (2.6) simplify to

DvΩ = ΩΘ, (2.8a)

DvP = Γ +2

ΘNABN

AB, (2.8b)

DvJA + ΩΘ∂AP + Ω∂AΓ + 2Ω∇BNAB = 0. (2.8c)

Off-shell pre-symplectic form. Starting from the Einstein-Hilbert action,

SEH =1

16πG

∫dr dv dD−2x η

√det gAB LEH, LEH = R− 2Λ, (2.9)

one can compute the usual Lee-Wald pre-symplectic form [46] over the set of geometries (2.1), yielding

ΩLW =1

16πG

∫N

dv dD−2x[δUA ∧ δJA − δΓ ∧ δΩ + δ(ΩΘ) ∧ δP + δΩAB ∧ δ(ΩNAB)

]. (2.10)

Note that the pre-symplectic form (2.10) involves off-shell quantities. While the above expressionclearly shows which variables are canonical conjugate of each other, the functions appearing there aresubject to equations (2.8) and not all of them are independent. In other words, the solution phase spaceis obtained after imposing the constraints (2.8) upon the parameter space and the symplectic form, e.g.using the Dirac bracket method or going to the reduced phase space.

Null boundary symmetry generators. The vector field

ξ = T ∂v + r (DvT −W ) ∂r +(Y A − rη∂AT

)∂A +O(r2) , (2.11)

preserves the form of metric (2.1), keeps r = 0 a null surface and generates the following variations overthe solution phase space functions

δξη = T∂vη + 2ηDvT −Wη + Y A∂Aη , (2.12a)

δξΓ = −Dv(W − ΓT ) + (Y A + UAT )∂AΓ , (2.12b)

δξ(ΩΘ) = Dv(TΩΘ) + L(Y+TU)(ΩΘ) , (2.12c)

δξUA = Dv(Y A + TUA) , (2.12d)

δξΩAB = 2TNAB + L(Y+TU)ΩAB +2

D − 2TΘΩAB . (2.12e)

– 3 –

where LY denote the Lie derivative along Y A, and for associated conjugate charges (see below)

δξΩ = TΩΘ + L(Y+TU)Ω , (2.13a)

δξP = TDvP + L(Y+TU)P −W , (2.13b)

δξJA = TDvJA + L(Y+TU)JA + Ω[∂AW − Γ∂AT − 2NAB∂

BT], (2.13c)

δξNAB = Dv(TNAB) + L(Y+TU)NAB . (2.13d)

Note that Ω is a scalar density, hence ∇AΩ = 0 and L(Y+UT )Ω = Ω∇A(Y A + UAT ). Similarly, the factthat JA is a one-form density and NAB is a symmetric-traceless tensor should be considered in computingthe Lie-derivatives.

Surface charge variation. One may compute the charge variation associated with the boundary sym-metry generators using covariant phase space method [13, 46]. Detailed analysis yields [32]

/δQξ =1

16πG

∫Nv

dD−2x

[(W − ΓT ) δΩ + (Y A + UAT )δJA + TΩΘδP − TΩΩABδNAB

], (2.14)

where Nv is a constant v section on N . This charge variation is an integral over∑4

i=1 Gi δQi, where Qi

parameterize the solution phase space. Among the four families, NAB corresponds to the bulk degrees offreedom while three others Ω,JA,P parameterize boundary information. Γ,UA functions which appearin Gi are subject to field equations (2.8) and δQi subject to linearized equations of motion.

The Gi are ‘field dependent’ linear combinations of symmetry generators T,W, Y A, notably Gi dependon Γ,UA as well as ΩΘ and ΩAB and δξGi 6= 0. The charge variation /δQξ, as stressed in the notation/δ, is hence not integrable. Γ,UA may be respectively solved for in terms of the charges using (2.8b) and(2.8c) and therefore all these coefficients may be represented through the charges. Note also that thereare three symmetry generators and four towers of charges and these are functions over the null surfaceN . We crucially note that δΩ, δJA, δP, δNAB denote generic variations around solutions of equations ofmotion (EoM) and are subject to linearized field equations. These linearized equations may be viewedas equations for variations δΓ, δUA. The solution phase space is hence parametrized by the four tower ofcharges and their variations.

We close this part by giving expressions for three ‘zero mode’ charges, ξ = −r∂r, ξ = ∂A and ξ = ∂v.One may readily observe that the first two are integrable and the latter is not:

Q−r∂r :=S

4π=

1

16πG

∫Nv

dD−2x Ω,

Q∂A := JA =1

16πG

∫Nv

dD−2x JA,

/δQ∂v := /δH =1

16πG

∫Nv

dD−2x(−ΓδΩ + UAδJA + ΩΘδP − ΩΩABδNAB

).

(2.15)

Surface charges and flux in thermodynamics slicing. As pointed out /δQξ in (2.14) is not in-tegrable. The charge variation may be split into Noether (integrable) part QN and the ‘flux’ part F :/δQξ = δQN

ξ +Fξ(δg; g). QN may be computed for the Einstein-Hilbert action using the standard Noetherprocedure, yielding

QNξ =

1

16πG

∫Nv

dD−2x[W Ω + Y A JA + T

(−ΓΩ + UAJA

)], (2.16)

– 4 –

and non-integrable flux part

Fξ(δg; g) =1

16πG

∫Nv

dD−2xT(ΩδΓ− JAδUA + ΩΘδP−ΩΩABδNAB

). (2.17)

Here we are assuming symmetry generators T,W, Y A to be field-independent, i.e. δT = δW = 0 = δY A.

For later use, we also present the expressions for the zero mode Noether charges,

QN−r∂r =

1

16πG

∫Nv

dD−2x Ω,

QN∂A

=1

16πG

∫Nv

dD−2x JA,

QN∂v := E =

1

16πG

∫Nv

dD−2x (−ΓΩ + UAJA) ,

(2.18)

Balance or ‘generalized charge conservation’ equation. In our general setup charges and the fluxare given by integrals over co-dimension two surface Nv. They are hence functions of ‘lightcone time’coordinate v and the charges are not conserved. From the expressions above one can deduce,

d

dvQNξ ≈ −F∂v(δξg; g), (2.19)

where ≈ denotes on-shell equality. Eq.(2.19) may be viewed as (1) manifestation of the boundary EoMwritten in terms of charges; (2) a ‘generalized charge conservation equation’ as it relates time dependence,or non-conservation, of the charge (as viewed by the null boundary observer) to the flux passing throughthe boundary; (3) how the passage of flux through the null boundary is ‘balanced’ by the rearrangementsin the charges. In this respect, it is very similar to the usual balance equation used at asymptotic nullsurfaces, which is now written for an arbitrary null surface in the bulk. Note also that the third viewpointyield null surface memory effects discussed in [32].

3 Null Surface Thermodynamics

Consider a usual thermodynamical system with chemical potentials µi(i = 1, 2, · · · , N) and temperatureT . This system is specified with charges Qi, the entropy S and the energy E; that is, there are N + 2

charges and N+1 chemical potentials. The distinction between charges and associated chemical potentialsis by convention and is specified with/specifies the ensemble. In microcanonical ensemble (which we havealready assumed), the first law takes the form

dE = T dS +

N∑i=1

µi dQi. (3.1)

This equation implies that the LHS is an exact one-form over the thermodynamic space. Moreover,chemical potentials and the charges are related by the Gibbs-Duhem relation

S dT +N∑i=1

Qi dµi = 0. (3.2)

Together with the first law (3.1) this yields E = TS +∑

i µiQi. This equation relates E to the othercharges and chemical potentials, e.g. E = E(S,Qi) (in microcanonical description) or E = E(T, µi) (in

– 5 –

grandcanonical description). Depending on the ensemble chosen, N + 1 number of chemical potentialsand/or charges may be taken to be ‘independent’ variables parameterizing the thermodynamical config-uration space and the rest of N + 1 of them as functions of the former N + 1 variables. In other words,the thermodynamic configuration space is (N + 1) dimensional and the change of ensemble is basicallya canonical transformation the generator of which is the difference between various ‘energy’ functionsassociated with each ensemble.

3.1 Null Boundary Thermodynamical Phase Space

Stirring at the expression of the charge variation (2.14), one can recognize that functions parameterizingthe solution space come in two categories: the bulk modes NAB (and its conjugate ‘chemical potential’determinant-free part of ΩAB, γAB) and the boundary modes. The latter may also be separated intothose whose variation appears Ω,P and JA, and those which appear only in the coefficients, in chemicalpotentials Γ,UA. There are hence D = 1+1+(D−2) charges and D−1 = 1+(D−2) chemical potentials.

We crucially note that if we treat Γ,UA and associated charges Ω,JA as independent variables, P isspecial as it does not appear in the integrable part of the charges (2.16) and only appears in the expressionfor the flux (2.17) through ΩΘδP term. Moreover, as already remarked (cf. discussions below (2.14)),the chemical potentials may be expressed in terms of the charges using field equations. Again we note atΘ = 0, P dependence completely drops out of the analysis. We will return to the special non-expandingcase in section 3.4.

Given all the above we are led to the following picture for the generic case.

I. Null boundary solution space relevant to the null boundary thermodynamics consists of three parts:

I.1) (D − 1) dimensional ‘thermodynamic sector’ parametrized by (Γ,UA) and conjugate charges(Ω,JA);

I.2) P, which only appears in the flux (2.17) and not in the Noether charge (2.16);

I.3) the bulk mode parameterized by determinant free part of ΩAB and its ‘conjugate charge’ NAB

which appear in the flux (2.17).

II. NAB parameterizes effects of the bulk and how they take the boundary system out-of-thermal-equilibrium (OTE) whereas P parameterizes OTE within the boundary dynamics. Put differently,OTE may come from inner boundary dynamics and/or from the gravity-waves passing through thenull boundary, parameterized by NAB.

III. Expansion parameter Θ is a measure of OTE, from both bulk and boundary viewpoints. WhenΘ = 0 the system is completely specified by the D − 1 dimensional thermodynamic phase space.

IV. The rest of the in-falling graviton modes parameterized through O(r) terms in gAB , do not enter inthe boundary/thermo dynamics, as of course expected from usual causality and that the boundaryis a null surface.

To establish the null surface thermodynamics depicted above, we need to discuss the laws of ther-modynamics. We start with the local first law, then local Gibbs-Duhem equation and come to localzeroth law, specifying the subsectors which can be brought to a (local) equilibrium. Before moving on,we introduce a piece of useful notation. By X we will denote the density of the quantity X,

X :=

∫Nv

dD−2x X . (3.3)

– 6 –

3.2 Local First Law at Null Boundary

Defining P := P/(16πG) and NAB := (16πG)−1NAB, (2.15) implies,

/δH = TN δS + UAδJ A + ΩΘδP − ΩΩABδNAB, TN := − Γ

4π(3.4)

The above is true at each v, xA over the null surface and represents the local null boundary first law.The LHS, unlike the usual first law (3.1), is not a complete variation, as the system is describing an openthermodynamic system due to the existence of the expansion and the flux. The above reduces to a usualfirst law for closed systems when NAB = 0 or in the non-expanding Θ = 0 case.

Note that Γ = −2κ+Dv ln(ηΩ2

D−2 ) where κ is the non-affinity parameter (surface gravity) associatedthe vector field generating the null surfaceN [32]. So, TN = κ

2π−1

4πDv ln(ηΩ2

D−2 ). For non-expanding Θ =

0 case where one may put η = 1 or when we have a Killing horizon, TN equals the usual Unruh/Hawkingtemperature, cf. section 3.4 for more discussions.

3.3 Local Extended Gibbs-Duhem Equation at Null Boundary

Gibbs-Duhem equation (3.2) is a relation among the thermodynamic charges (rather than charge varia-tions). Given the expressions for the zero mode charges (2.18) and for the densities in the same notationas in (3.3) we have1

E = TNS + UAJ A (3.5)

The above is an analogue of the Gibbs-Duhem equation if E is viewed as energy, S as entropy and J A

as other conserved charges and Γ,UA as the respective chemical potentials. This of course manifests thepicture we outlined in section 3.1. However, one should note that (3.5) is a local equation at the nullboundary, unlike its usual thermodynamic counterpart. This equation also holds for non-stationary/non-adiabatic cases when the system is out-of-thermal-equilibrium (OTE). So, we call (3.5) ‘local extendedGibbs-Duhem’ (LEGD) equation at the null boundary.

LEGD equation, like the local first law (3.4), is a manifestation of diffeomorphism invariance of thetheory. While the explicit expressions for the charges do depend on the theory, we expect (3.5) to beuniversally true for any diff-invariant theory of gravity in any dimension. This equation is on par withthe first law of thermodynamics but extends it in two important ways: it is a local equation in v, xA andholds also for OTE.

Since the integrable parts of the charge are (by definition) independent of the bulk flux NAB and alsoof P, the LEGD equation (3.5), also do not involve P and NAB. Nonetheless, the chemical potentials in(3.5), Γ and UA, implicitly depend on NAB and P through Raychaudhuri and Damour equations.

3.4 Local Zeroth Law

Zeroth law in the usual thermodynamics is a statement of thermal equilibrium: as a consequence ofthe zeroth law, two (sub)systems with the same temperature and chemical potentials are in thermalequilibrium. In the usual thermodynamics flow of charges is proportional to the gradient of associatedchemical potentials and hence the absence of such fluxes can be taken as a statement of the zeroth law.In our case, we are dealing with a system parameterized by chemical potentials Γ,UA and γAB which are

1See [36] for appearance of Gibbs-Duhem equation in the near horizon analysis in 3 dimensional cases.

– 7 –

functions of charges Qα ∈ Ω,P,JA, NAB. This system is not in general in equilibrium but there couldbe special subsectors which are. The zeroth law is to specify such subsectors.

Recalling (2.19), flow of charges vanishes on subsystems over which F∂v(δξg, g) vanishes. On a closelyrelated account, one can show that [47] this flux has the same expression as the on-shell variation of theaction. Nonetheless, while the charge variation (2.14) is invariant under the addition of a total derivativeterm to the Lagrangian, the Noether charge and hence the flux are not. In particular, upon addition of aboundary Lagrangian LB = ∂µBµ, the on-shell action variation and hence the flux F are shifted by δBr.For later convenience, let us call Br = G. This opens up the possibility of (partially) removing the fluxby an appropriate boundary term. The question is hence what are the subsectors in the solution phasespace for which flux can be removed by an appropriate boundary term.

So, we start with the variation of on-shell action. A direct computation leads to

δSEH|on-shell =1

16πG

∫N

dv dD−2x(ΩΘδP + ΩδΓ− JAδUA − ΩNABδΩAB

)=

∫dvF∂v(δg; g), (3.6)

where F∂v may be readily read from (2.17). Next, let us add a boundary term to the Lagrangian uponwhich δSEH|on-shell → δSEH|on-shell +

∫N δG. As the statement of the zeroth law we require there exists a

G = G(Ω,P,JA, NAB) such that,

δG = −S(δTN − 4GΘδP)−J AδUA + ΩNABδΩAB (3.7)

admits non-zero solutions. Integrability condition for the zeroth law (3.7) is δ(δG) = 0,2 which yields anequation like

∑α,β CαβδQα ∧ δQβ = 0, where Qα are generic charges and Cαβ is skew-symmetric. This

equation is satisfied only for Cαβ = 0. One can immediately see NAB = 0 = δNAB is a necessary (butnot sufficient) condition for (3.7) to have non-trivial solutions.

Before discussing the solutions in more detail, let us note that when (3.7) is fulfilled the charge H,which appears in the LHS of the local first law (3.4), becomes integrable and we obtain

H = G + TNS + UAJ A (3.8)

Besides NAB = 0, in terms of H = H(S,J A,P) local zeroth law implies,

TN =δHδS , UA =

δHδJ A

, DvS = SΘ =1

4G

δHδP (3.9)

where the last equation may be seen as the equation of state. For the special case of Θ = 0, one simplydeduces that H does not depend on P . In general, equations of motion together with (3.7) ensure thattotal energy and angular momentum are conserved, d

dvH = ddvJA = 0, where d

dvX :=∫Nv

dD−2x ∂vX .Total entropy, on the other hand, is not conserved as d

dvS =∫Nv

dD−2x ΘS; ddvS is zero only when

expansion vanishes, Θ = 0.

Generic Θ 6= 0 case. The zeroth law requires NAB = 0 for which (2.8) reduce to

TN = −4GDvP , Dv[J A + 4G∇A(SP)

]= 0. (3.10)

2δ(δG) is the pre-symplectic two-from (2.10), therefore (3.7) implies vanishing symplectic form over the null surface N .

– 8 –

The above imply that zeroth law (3.9) is satisfied for any H = H(S,P ,J A), when S,P and J A havethe following basic Poisson brackets [32]

S(x, v),P(y, v) =1

4GδD−2(x− y), S(x, v),S(y, v) = P(x, v),P(y, v) = 0,

S(x, v),J A(y, v) = S(y, v)∂

∂xAδD−2(x− y),

P(x, v),J A(y, v) =

(P(y, v)

∂

∂xA+ P(x, v)

∂

∂yA

)δD−2(x− y),

J A(x, v),J B(y, v) =1

16πG

(J A(y, v)

∂

∂xB−J B(x, v)

∂

∂yA

)δD−2(x− y)

(3.11)

and ∂vX = H,X. That is, H is the Hamiltonian over this phase space and (3.9) do not impose anyrestrictions on H which is a scalar over N .

Θ = 0 case. In this case trace of the extrinsic curvature of the null surface N vanishes, hence it is anextremal null surface. Vanishing of the expansion Θ has some important consequences. (1) Raychaudhuriequation implies NAB = 0. So, again we arrive at the vanishing flux; (2) η drops out from the chargevariation (2.14). (3) We lose one tower of the charge P, and the associated symmetry generator becomesa pure gauge. (4) We may fix the η = 1 gauge which yields W = 2DvT . We hence remain withT, Y A generators which form Diff(N ) symmetry algebra. (5) EoM (2.6) reduces to DvJ A = S∂ATN andDvS = 0, which may be viewed as equations for spatial derivatives of the chemical potentials.

Local zeroth law (3.9) is satisfied by any scalar Hamiltonian H = H(S,J A), together with basicPoisson brackets (3.11) but with P dropped [32] and again with ∂vX = H,X.

Closing remarks: (1) Zeroth law (3.9) is just defining the Poisson bracket structure over our chargespace and existence of Hamiltonian dynamics, but does not specify a Hamiltonian. (2) Choice of Hamil-tonian fixes a boundary Lagrangian and the boundary dynamical equations which in turn specifies localdynamics of charges on the null boundary N . (4) In analogy with isolated horizon [48] of black holes, ifthe zeroth law holds the null surface may be called an ‘isolated null surface’.3 (4) Our zeroth law is aweaker condition than stationarity as ∂v of the chemical potentials need not vanish. (5) The usual zerothlaw of black hole mechanics (for Killing horizons) that UA and Γ are constants over the horizon (our nullboundary N ) is a very special case which obeys our zeroth law. For the stationary asymptotic flat blackhole solutions to the vacuum Einstein gravity, i.e. the Myers-Perry solutions, we get E =

(D−3D−2

)H, and

we have the usual Smarr formula.

4 Outlook

Building upon the analyses of [28–31] and in particular [32], we established that the solution phase spacearound an arbitrary null surface in pure D dimensional Einstein gravity naturally admits a thermodynam-ical description and the charges and corresponding chemical potentials form a ‘thermodynamical phasespace’. The laws of thermodynamics are all local equations over the D − 1 dimensional null surface Nand our analysis does not fix the boundary dynamics, boundary Hamiltonian, which may still be chosen.

As discussed, the zeroth law necessitates vanishing of the flux of bulk gravitons through N andestablishes basic Poisson brackets on the thermodynamic phase space. The same condition, absence of

3One may show by direct computation that, upon zeroth law in the Θ = 0 sector, DvJ A = S∂ATN and DvS = 0 andalso DvE = −∂vG,DvH = −∂A(UAG).

– 9 –

NAB, has been discussed as the condition for the existence of a slicing in the solution phase space in whichthe charges become integrable [29, 31, 32]. The physics of change of phase space slicing and null surfacethermodynamics developed here is an interesting direction to study.

The second law of thermodynamics and how it can be realized in our setting is an important problemthat should be tackled next. As the first step in this direction, we have discussed the null surface focusingtheorem in appendix A.

Our analysis is based on covariant phase space formulation and hence readily generalizes to any dif-feomorphism invariant theory in any dimension. Given all the previous literature, especially [13], it isreasonably certain that the same ‘local’ thermodynamical description with exactly the same equationsshould also hold for this generic case. In other words, diffeomorphism invariance yields the local thermo-dynamical picture. One may do the reverse and show that the thermodynamical description results indiffeomorphism invariance. We should emphasize that this is already weaker than Einstein’s equivalenceprinciple and non-minimal coupling and generic modified gravity theories follow the same analysis. Theconnection between thermodynamics and gravity is nothing new, e.g. see [14]. However, our analysis hereand in particular the thermodynamical phase space structure provides a solid framework to push this tothe quantum level.

The local thermodynamical description in its basics and general ideas reminds one of the membraneparadigm [49–51]. It is interesting to relate the two more systematically. The first steps in this directionwere outlined in [52]. The interesting question is whether the membrane picture restricts the boundaryHamiltonian.

Among other things, our analysis here very clearly shows how the boundary and bulk degrees offreedom interact and that the boundary phase space admits the local thermodynamical description. Thisis expected to be very relevant for the black hole microstate problem in that the boundary d.o.f andquantization thereof can account for the microstates, whereas the interactions with bulk modes would berelevant for the information loss problem. Our analysis permits a semi-classical setting in which boundaryphase space is quantized while bulk modes are treated (semi)classically and hence potentially gives a betterhandle on both microstate and information loss problems.

Acknowledgement. We would like to especially thank Daniel Grumiller and Celine Zwikel for longterm collaborations and many fruitful discussions which were crucial in developing the ideas and analysisdiscussed here. We thank Glenn Barnich for correspondence and Mohammad Hassan Vahidinia for theuseful discussions. MMShJ acknowledges SarAmadan grant No. ISEF/M/400122. The work of VT ispartially supported by IPM funds.

A Null Boundary Focusing Theorem

From Raychaudhuri equation (2.6a) one learns that,

DvΘ− κΘ +1

D − 2Θ2 ≤ 0, κ := −Γ

2+

1

2Dv ln(ηΩ

2D−2 ), (A.1)

In terms of variableX(v) := exp

(∫v0

κ

), DvX − κX = 0, X(v0) = 1, (A.2)

assuming Θ(v) 6= 0, (A.1) implies

Θ ≤ Θ0X(v)

1 + Θ0

D−2

∫v0X(v)

, (A.3)

– 10 –

where Θ0 = Θ(v = v0) and without loss of generality we have taken v0 such that the equality is saturated.Since X(v) ≥ 0, then

∫v0X(v) is a growing function of v. If Θ0 < 0 (that is if we start in a contracting

phase) then there will always be a “trapping time” v1 > v0 where∫ v1v0X(v) = − 1

Θ0(D−2), and Θ(v1)→ −∞.

If at some v, v0, Θ = 0 then its derivative should be non-positive at that point DvΘ ≤ 0. For thecase of equality we have a non-expanding case and the NAB should also vanish and if DvΘ < 0 then atv0 + δv, Θ < 0 and again (A.3) implies existence of a trapping time.

In the absence of bulk modes, NAB = 0, the above inequality is replaced with equality. In this case(A.3) shows internal null boundary dynamics which is of course due to gravity effects. In other words,gravity dynamics is relating thermodynamical sector of the solution phase space to the other two parts,the η part and the bulk modes and this dynamics is featured in the focusing equation.

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