Dynamical Black Hole Entropy
Robert M. Wald

S. Hollands, R.M. Wald, and V.G. Zhang, arXiv:2402.00818.

Black Hole Entropy
As Iyer and I showed in 1994, in an arbitrary theory of gravity
obtained from a diffeomorphism covariant Lagrangian,stationary black holes satisfy a “first law of black holemechanics,” thereby enabling one to identify a quantity
representing black hole entropy. However, our derivation
required evaluation of the entropy on the bifurcation surface, B,of the black hole, thus restricting the validity of our formula forentropy to stationary black holes and their linear perturbations,evaluated at the “time” represented by B. It is of considerable
interest to obtain an expression for the entropy of anon-stationary black hole in a general theory of gravity at a“time” represented by an arbitrary cross-section C, since thiswould allow one to investigate whether, classically, black holeentropy satisfies a second law (i.e., whether it is non-decreasing
with time) and whether, semiclassically, black hole entropy
satisfies a generalized second law (i.e., whether the sum of blackhole entropy and a matter contribution to entropy isnon-decreasing).

Previous Work on Dynamical Black Hole Entropy

Iyer and I proposed a formula for dynamical black hole entropy,
but our formula was not field redefinition invariant, and weretracted our proposal in a “note in proof” in the publishedversion of their paper. In 2013, Dong proposed a formula for
dynamical black hole entropy in theories whose Lagrangian is
an arbitrary function of curvature (but not derivatives ofcurvature) based on holographic entanglement arguments. In2015, Wall proposed a general prescription for dynamical blackhole entropy for perturbations of a stationary black hole. Whenhe evaluated this in the case of a Lagrangian that is a functionof curvature and obtained agreement with Dong’s formula.

Our Approach and Results
We apply a new strategy to the definition of dynamical blackhole entropy—intended to be applied only at leading nontrivial
order in perturbation theory about a stationary black
hole—based upon the validity of a local, “physical processversion” of the first law of black hole mechanics. In the case ofgeneral relativity, we obtain a nontrivial dynamical correction
to the Bekenstein-Hawking entropy formula, namely we obtainZA[C] 1−VϑSC =44 C
where A[C] is the area of the cross-section C (so A[C]/4 is theusual Bekenstein-Hawking entropy), V is an affine parameter ofthe null generators of the horizon (with V = 0 corresponding tothe bifurcation surface B), and ϑ is the expansion of thesegenerators. For general theories, our formula differs from theDong-Wall formula, but there is a close relation between them.

Lagrangian FormalismLagrangian:
L = L(gab , Rabcd , ∇a1 Rbcde , · · · , ∇(a1 · · · ∇am ) Rbcde ; ψ)Symplectic potential:
δL = Eδφ + dθ
Symplectic current:ω(φ, δ1 φ, δ2 φ) = δ1 θ(φ, δ2 φ) − δ2 θ(φ, δ1 φ)
Noether current associated to vector field χa :J[χ] = θ(φ, Lχ φ) − χ · L

Lagrangian Formalism (Continued)
Noether Charge:J[χ] = dQ[χ] + χa C a“Fundamental Identity”:
ω(φ, δφ, Lχ φ) = χ · (Eδφ) + χa δC a + d[δQ − χ · θ(φ, δφ)]If χa is a Killing field and the equations of motion are hold in
the background spacetime (E = 0), then the fundamental
identity reduces tod[δQ − χ · θ(φ, δφ)] = −χa δC a

First Law of Black Hole Mechanics
Consider a source-free perturbation δC a = 0 of a stationary,non-extremal black hole with horizon Killing field ξ a . Integrated[δQ[ξ] − ξ · θ(φ, δφ)] = 0
over a hypersurface extending from the bifurcation surface B tospatial infinity. ObtainκδS = δM − ΩH δJ2π
where2πS≡κZQ[ξ]BEvaluation at B (where ξ a = 0) was needed to get rid of theξ · θ that otherwise would have appeared in the horizon
boundary term. This restricts the validity of the formula to“time” B.

Entropy on an Arbitrary Cross-Section CWe would like to define SC on a cross-section C of the event
horizon H so thatZ2π
δSC =[δQ[ξ] − ξ · θ(φ, δφ)]κ CIf, on H, the symplectic potential θ were of the form θ = δB H
for some quantity B H defined on H, then the desired quantity
SC could be defined byZ2πSC =[Q[ξ] − ξ · B H ]κ CHowever, it is not possible for θ to be of the form of a totalvariation δB H on H in general because such a form would
imply the vanishing of the symplectic current flux through thehorizon (since the symplectic current ω is the antisymmetrized
second variation of θ).

Analogous Example: ADM Mass
Want to define ADM mass, M , at spatial infinity with respect
to an asymptotic time translation ta so that it is theHamiltonian conjugate to ta , i.e., so that it generates
asymptotic time translations on phase space. This requires thatZ
δM =[δQ[t] − t · θ(φ, δφ)]∞However, the symplectic current goes to zero sufficiently
rapidly that the symplectic current flux vanishes at spatial
infinity, so there is no obstruction to finding a quantity B ∞such that to leading asymptotic orderθ = δB ∞One may then define the ADM mass asZ
M=[Q[t] − t · B ∞ ]∞and there is no difficulty in obtaining an M that satisfies thedesired relation.

Closely Analogous Example: Bondi MassWould like to define Bondi mass, MB , so that on a cross-section
C of null infinity we haveZ
δMB = [δQ[t] − t · θ(φ, δφ)]CHowever, this is not possible, since the symplectic flux does notvanish at null infinity. Nevertheless, Zoupas and I found that
one could define a symplectic potential θ 0 (φ, δφ) at I + suchthat θ 0 = 0 in a stationary background φ. We then defined
B I + byθ − θ 0 = δB I +and we defined the Bondi mass relative to an asymptotic timetranslation ta on a cross-section, C, of I + byZ
MB = [Q[t] − t · B I + ]CDynamical Black Hole Entropy
We follow the same strategy that Zoupas and I used at null
infinity. We prove that for first order perturbations of astationary black hole, there exists a quantity B H defined on theblack hole horizon H that satisfiesθ = δB H(This is equivalent to the requirement of the existence of analternative symplectic potential θ 0 that vanishes for a stationary
background, since for a non-stationary background, one could
define θ 0 = θ − δB H .) We then define the entropy on anarbitrary cross-section C of H byZ2π[Q[ξ] − ξ · B H ]SC =κ CPhysical Process Version of the First Law
For vacuum perturbations of a stationary black hole, δSC = δS,i.e., the entropy is “time independent” to first order. In order toobtain nontrivial time variation of black hole entropy, we must
either work to second order in perturbation theory in thevacuum case or allow an external stress-energy, δTab , to bepresent in the first order perturbation. In the presence of δTab ,the fundamental identity on the black hole horizon becomesd[δQ[ξ] − ξ · θ(φ, δφ)] = −ξ a δCa
Integration over the region of the horizon bounded bycross-sections C1 and C2 yields
ZZ√
κ[δSC2 − δSC1 ] = − ξ a δCa = δTab ξ a k b hdV dn−2 x2π
where k a is the tangent to the affinely parametrized generatorsof the horizon. This has the form of the “physical processversion” of the first law
κ∆δS = ∆δE2π

Second Law
For first order perturbations with an external stress-energy
δTab , that satisfies the null energy condition, we have ∆δE ≥ 0.
Thus, the physical process version of the first law then
immediately implies that ∆δS ≥ 0, i.e., the second law holds tofirst order for matter satisfying the null energy condition. Forvacuum perturbations, we have ∆δS = 0, so we have to work tosecond order to see the change in entropy. We obtain,Zκ∆δS = ω(δg, Lξ δg) − d[ξ · (δθ − δ 2 BH )]2π
The integrand on the right side is what Hollands and I called
the “modified canonical energy” so the second law holds at
leading order for vacuum perturbations if and only if themodified canonical energy is positive. This holds in generalrelativity but would not be expected to hold in more general
theories of gravity.

Entropy of a Dynamical Black Hole in General
Relativity
For the case of general relativity, the additional term −ξ · B H
in our entropy formula gives rise to a“dynamical correctionterm” to the Bekenstein-Hawking entropy, namely,ZA[C] 1−VϑSC =44 C
Consequently, SC is smaller than the Bekenstein-Hawking
entropy. From the local form of the physical process first law, it
follows that, to first order, our entropy increases only whenmatter crosses the horizon. This contrasts with theBekenstein-Hawking entropy, which increases before matter isthrown into the black, since the event horizon moves outward in
anticipation of matter being thrown in at a later time. We haveshown that, to first order, our entropy is, in fact, the area of theapparent horizon corresponding to “time” C, and thus can be
determined locally, without knowledge of the future behavior ofthe spacetime.

Relationship to the Dong-Wall Entropy

Consider the entropy S[V ] evaluated on a cross-section ofconstant affine time V . Then we have
S[V ] = SDW [V ] − V∂
SDW [V ]∂VIn particular, we have∂∂2
S = −VSDW∂V∂V 2Generalized Second Law
Quantum null energy condition (QNEC) applied to Killing
horizon:Z√∂ 2 SvN≤ 2πhTab ik a k b hdn−2 x2∂VV =constwhereSvN = −trρ ln ρis the von Neuman entropy of the matter outside the blackhole. Since
1 aka =ξκV
this is equivalent toZ√ n−2∂ 2 SvN2π∂Sa bV≤hdx=hTiξ
kab∂V 2κ V =const∂VGeneralized Second Law (continued)
where the semiclassical version of the local, physical process
first law was used. Define
Sdyn matter = SvN − V∂SvN∂VThen∂2∂
Sdyn matter = −VSvN∂V∂V 2
so QNEC takes the form of the generalized second law∂[S + Sdyn matter ] ≥ 0∂V

Generalized Second Law (continued)
Since we have∂S∂ 2 SDW= −V∂V∂V 2we may also express QNEC asV∂ 2 SDW∂ 2 SvN≤−V∂V 2∂V 2Dividing by V , integrating from V to ∞, and setting(∂SvN /∂V )|∞ = (∂SDW /∂V )|∞ = 0, we obtain∂[SDW + SvN ] ≥ 0∂VThus, the generalized second law using S and Sdyn matter isequivalent to QNEC, whereas the generalized second law using
SDW and SvN is equivalent to an integrated form of QNEC.

Conclusions
We have proposed a new definition of the entropy of dynamical
black holes, intended to be valid to leading nontrivial order for
perturbations of stationary black holes.
I For general relativity, it gives a correction to theBekenstein-Hawking entropy that yields the area of theapparent horizon rather than the area of the event horizon.
I The general definition satisfies a local version of thephysical process version of the first law.
I It satisfies the classical second law for first order for

perturbations with matter satisfying the null energy
condition, and it satisfies the second law for vacuum
perturbations if and only if the modified canonical energy
flux is positive.
I It satisfies a version of the generalized second law that isequivalent to QNEC.
This should provide plenty of food for thought!