Dynamical black hole entropy beyond general relativity from the Einstein frame

The discovery of the laws of black hole thermodynamics is one of the great achievements of fundamental physicsBardeen:1973gs ; Bekenstein:1973ur ; Hawking:1975vcx . These laws combine gravity, quantum theory and thermodynamics within one stunning framework. Moreover, they provide some of the deepest insights on the fundamental nature of the quantum theory of gravity. For general relativity (GR), the black hole entropy $S$ is given by the Bekenstein-Hawking formulaBekenstein:1973ur ; Hawking:1975vcx

where $A$ is the area of the event horizon and $G$ is Newton’s constant. While for an arbitrary theory of gravity obtained from a diffeomorphism covariant Lagrangian, the entropy of a stationary black hole at the bifurcation surface $\mathcal{B}$ is given by the Wald entropy, which is defined as the Noether charge and for the $f(\text{Riemann})$ theories of gravity it is written asWald:1993nt ; Iyer:1994ys

where $v$ is the future-directed affine parameter of the null generator of the future horizon, $u$ denotes the affine null distance away from the horizon and is also future-directed. Furthermore, Iyer and Wald proposed a formula $S_{\text{Iyer-Wald}}$ for dynamical black hole entropy on an arbitrary cross-section $\mathcal{C}$ of the horizon by expanding the Wald entropy in fields and their derivatives, and keeping the terms that depend only on boost-invariant fieldsIyer:1994ys . However, the second law of black hole mechanics does not seem to hold for $S_{\text{Iyer-Wald}}$. And that proposed dynamical entropy formula is not field redefinition invariant. Moreover, the Noether charge method used to derive the black hole entropy is subjected to a number of ambiguities for non-stationary black holes, identified by Jacobson, Kang and Myers (JKM)Jacobson:1993vj . Wall derived a second law for higher curvature gravity in a perturbative context, and resolved some ambiguities in Wald’s Noether charge methodWall:2015raa . For $f(\text{Riemann})$ gravity the Wall entropy is given by

Where $K_{ij(a)}$ is the extrinsic curvature of the horizon in the $a$ direction. At the bifurcation surface both $S_{\text{Iyer-Wald}}$ and $S_{\text{Wall}}$ reduce to $S_{\text{Wald}}$, and they are equal to $S_{\text{BH}}$ for general relativity.

Although the second law for $S_{\text{BH}}$ and the linearized second law for $S_{\text{Wall}}$ hold for non-stationary perturbations, this is not the case for the first law. The first law often does not hold for non-stationary perturbations of a stationary black hole, and if it does, the black hole entropy cannot be evaluated at an arbitrary cross-section of the event horizon of the perturbed non-stationary black holeVisser:2024pwz . Recently, Hollands, Wald and Zhang proposed a strategy to overcome these two limitationsHollands:2024vbe . They derived the first law by applying the Noether charge method to non-stationary perturbations of a stationary black hole background, and introduced a new program to the definition of dynamical black hole entropy valid to leading order, on the basis of the validity of a local, “physical process version” of the first law of black hole mechanics. Visser and Yan generalized and improved their work in a number of ways, and gave a more pedagogical proof of the physical process first law for black holes in general relativityVisser:2024pwz .

Next let us review the key results of Hollands:2024vbe for the dynamical black hole entropy and the non-stationary first law. For a non-stationary dynamical black hole, the entropy $S$ is no longer equal to the Bekenstein-Hawking entropy for GR or to the Wall entropy for higher curvature gravity. The formula for dynamical black holes differs from the usual Noether charge formula by a non-trivial dynamical correction term. For general relativity, the dynamical black hole entropy was defined as

where $v$ is the future-directed affine null parameter along the future horizon, and it is equal to 0 at the bifurcation surface $\mathcal{B}$. Notably, the derivative term is invariant under the scaling transformation of the affine parameter $v\to a(x^{i})v$, where $x^{i}$ are codimension-2 spatial coordinates on the horizon. It was also shown that, to leading order in perturbation theory, the formula of dynamical black hole entropy for general relativity is equal to the Bekenstein-Hawking entropy of the apparent horizon. Furthermore, the dynamical black hole entropy was generalized to higher curvature gravity asHollands:2024vbe

By applying the Noether charge method to non-stationary perturbations, Hollands, Wald and Zhang have shown that their formula for dynamical black hole entropy satisfies both the non-stationary comparison first law and the non-stationary physical process first law. The non-stationary comparison version of the first law for an arbitrary horizon cross-section compares two vacuum black hole geometries

where $M$ and $J$ are the mass and angular momentum of the black hole, respectively. $\Omega_{\mathcal{H}}$ is the angular velocity of the horizon. And $\kappa$ denotes the surface gravity of the black hole. For vacuum perturbations of a stationary black hole, the dynamical entropy is “time independent” to first order, i.e., $\delta S_{\text{dyn}}[\mathcal{C}(v)]=\delta S_{\text{dyn}}[\mathcal{B}]$, where $\mathcal{C}(v)$ is an arbitrary cross-section. In order to study the non-trivial time variation of black hole entropy, we may unseal an external stress-energy, $\delta T_{ab}$, in the first-order perturbation. For perturbations sourced by external matter fields, the non-stationary physical process first law reads¹¹1Note that in the traditional treatments of the physical process first law, the black hole starts and ends in a stationary stateGao:2001ut ; Poisson:2009pwt .

where $\Delta$ denotes the difference between two cross-sections $\mathcal{C}(v_{1})$ and $\mathcal{C}(v_{2})$, and $\delta$ stands for the first-order perturbation around the stationary background. As a corollary of the physical process first law, the second law holds for the dynamical black hole entropy to first order as long as the stress-energy tensor satisfies the null energy condition $\delta T_{vv}\geq 0$. While for vacuum perturbations we have to pay attention to second order in perturbation theory to study the leading-order change of the dynamical black hole entropy. And for vacuum perturbations the second law is obeyed at second order in general relativity since the “modified canonical energy flux” is positive for GR. However, in more general theories of gravity the “modified canonical energy flux” is not necessarily positive, and the second law presumably would not hold in more general theories of gravity for vacuum perturbations.

In this paper we present an alternative, pedagogical proof of both the non-stationary physical process first law and the non-stationary comparison first law for the $f(R)$ gravity as well as the canonical scalar-tensor theory via conformal transformations. To begin with, we generalize Visser and Yan’s pedagogical derivation of the physical process first law for GR to black holes that may not be in vacuum. We also present a pedagogical deviation of the second-order behavior of $S_{\text{dyn}}$ for vacuum perturbations in GR by studying the second variation of the Raychaudhuri equation. Since conformal transformations preserve the causal structures of spacetimes, and the conformal factors that put theories into the Einstein frames are independent of the Killing time $\tau$ in the stationary spacetimes, the spacetimes given by conformal transformations are stationary black hole solutions to the Einstein equation. We are capable of applying the proof of the first law for GR to theories in the Einstein frames, and then recast them in their original frames. As the perturbations are non-stationary, the gauge conditions for perturbations that fix the affine parameter $v^{E}$ in the Einstein frame will not necessarily fix the affine parameter $v$ in the original frame. However, those gauge conditions do not restrict the way we perturb the stationary black hole background. Instead, they tell us how to identify spacetime points in the two slightly different spacetimes. We show that the switching from $v^{E}-$identification to $v-$identification leads to second-order corrections, and the first laws are invariant to first order under the re-identification of spacetime points. For vacuum perturbations we further study the second-order behavior of the dynamical black hole entropy, and observe that the second law holds at second order both in the Einstein frames and in the original frames. Using the Einstein frame we also determine the relationship between the dynamical black hole entropy of the cross-section and the Wald entropy of the generalized apparent horizon in the original frame, which was not given before.

The rest of the paper is organized as follows. In section 2 we introduce the geometric setup and impose gauge conditions on perturbations. In Section 3 we generalize Visser and Yan’s pedagogical proof of the physical process first law for GR to non-vacuum solutions, and study the second-order behavior of the dynamical black hole entropy for vacuum perturbations. In Sections 4 and 5 we present the proof of both the physical process first law and the comparison first law for those theories, study the second-order behavior of the dynamical black hole entropy for vacuum perturbations, and determine the relationship between the dynamical black hole entropy and the Wald entropy of the generalized apparent horizon in the original frame. Section 6 is devoted to the summary of the paper and some discussions. In Appendix A we calculate the modified canonical energy flux for $f(R)$ gravity via the covariant phase space formalism, and give a non-trivial check of the second-order behavior of $S_{\text{dyn}}$ for vacuum perturbations.

We will mainly follow the notation and conventions of Wald:1984rg . Throughout this paper, we set $c=\hbar=k_{B}=1$ while keep Newton’s constant $G_{D}$ explicit.

In this section we introduce the geometry of the stationary black hole background and impose gauge conditions on non-stationary perturbations used to study the first law. Our geometric setup mainly follows that of Visser:2024pwz .

Consider a stationary $D-$dimensional stationary black hole background geometry $(M,g_{ab})$. We assume that the black hole spacetime is asymptotic flat and electrically neutral. The event horizon of this black hole coincides with the bifurcate Killing Horizon $\mathcal{H}$. We label the future horizon by $\mathcal{H}^{+}$, the past horizon by $\mathcal{H}^{-}$, and the bifurcation surface by $\mathcal{B}$. The Killing field normal to the Killing horizon is denoted by $\xi^{a}$, and it is a Killing symmetry of the metric $g_{ab}$ as well as the matter fields $\phi$. i.e.,

We are mainly interested in the part of $\mathcal{H}^{+}$ that lies to the future of $\mathcal{B}$. We can always erect a set of null zweibein bases $(k^{a},l^{a})$ on $\mathcal{H}^{+}$, where $k^{a}$ is the future directed null normal to $\mathcal{H}^{+}$ and it is affinely parameterized as $k^{a}\overset{\mathcal{H}^{+}}{=}(\partial_{v})^{a}$ with $v$ being set to $0$ on $\mathcal{B}$, $l^{a}$ is an (future-directed) ingoing auxiliary null vector field and $k^{a}l_{a}=-1$ holds on the horizon. We can also extend $l^{a}$ off the horizon by solving $l^{b}\nabla_{b}l^{a}=0$ and denote the affine null distance away from $\mathcal{H}^{+}$ by $u$ to identify $l^{a}=(\partial_{u})^{a}$. Finally, we can extend $k^{a}$ off the horizon such that it commutes with $l^{a}$, i.e., $\left[k,l\right]^{a}=0$. Since $\xi^{b}\nabla_{b}\xi^{a}\overset{\mathcal{H}^{+}}{=}\kappa\xi^{a}$ and $k^{b}\nabla_{b}k^{a}\overset{\mathcal{H}^{+}}{=}0$, we can scale $k^{a}$ properly such that

where $C$ is a constant. We can also decompose the metric on $\mathcal{H}^{+}$ as

where $\gamma_{ab}$ is the intrinsic codimension-2 spatial metric of the cross-section $\mathcal{C}(v)$ satisfying $\gamma_{ab}=\gamma_{(ab)}$ and $\gamma_{ab}k^{a}\overset{\mathcal{H}^{+}}{=}\gamma_{ab}l^{a}\overset{\mathcal{H}^{+}}{=}0$.

We would like to perturb the stationary black hole background $g_{ab}\to g_{ab}+\delta g_{ab}$ and the matter fields $\phi\to\phi+\delta\phi$ to study the first law for non-stationary variations, where $\delta g_{ab},\ \delta\phi\sim\mathcal{O}(\epsilon)$ are of first order in the perturbation. Since the stationary black hole spacetime and the perturbed spacetime are different, there exists certain gauge freedoms about how we should establish the coordinate systems in the perturbed geometry and how to identify spacetime points in the two slightly different spacetimes. In order to simplify the deviation of the first law, we impose the following gauge conditions on perturbationsVisser:2024pwz :

1. 1.

   The event horizon of the perturbed black hole is identified with the Killing horizon of the background geometry. And $\mathcal{H}^{+}$ is still described by $u=0$ and $\mathcal{H}^{-}$ described by $v=0$ after the perturbation.

2. 2.

   $k^{a}$ and $l^{a}$ are fixed under the perturbation

   $$\delta k^{a}=0,\quad\delta l^{a}=0.$$

   (11)

   and $k^{a}$ remains null normal to $\mathcal{H}^{+}$ and $l^{a}$ remains null everywhere under the perturbation, which together with the condition $\delta(k^{a}l_{a})=0$ yields the following conditions

   $$k^{a}\delta g_{ab}\overset{\mathcal{H}^{+}}{=}0,\quad l^{a}\delta g_{ab}=0.$$

   (12)

   We further require that $k^{a}$ is still affinely parameterized on $\mathcal{H}^{+}$ and $l^{a}$ affinely parameterized everywhere after the perturbation

   $$\delta(k^{b}\nabla_{b}k^{a})\overset{\mathcal{H}^{+}}{=}0,\quad\delta(l^{b}\nabla_{b}l^{a})=0.$$

   (13)

3. 3.

   The Killing vector field $\xi^{a}$ remains null and tangent to the geodesic generators of the perturbed black hole. Combining with $\xi^{a}\delta g_{ab}\overset{\mathcal{H}}{=}0$ means that $\delta\xi^{a}$ is proportional to $k^{a}$ on $\mathcal{H}^{+}$ and to $l^{a}$ on $\mathcal{H}^{-}$.

The gauge condition 1 simply means that we compare the event horizon of the perturbed black hole and the Killing horizon of the stationary background. Condition 2 tells us how to establish the $u,v$ coordinate systems in the perturbed spacetimes. Moreover, we identify points with identical coordinates in the two slightly different spacetimes. These conditions do not mean that the Killing field should be fixed under the perturbation. If we allow the surface gravity $\kappa$ to vary, $\xi^{a}$ will change in the perturbed geometry.

In this section, we present the formula for dynamical black hole entropy based on the non-stationary physical process first law in GR. This derivation is completed by means of integrating the linearized Raychaudhuri equation on the horizon between two arbitrary cross-sections. A similar derivation was given in Visser:2024pwz for black hole solutions to the vacuum Einstein equation. While our deviation is less restrictive since we allow the metric to couple to matter fields via the Einstein equation in the stationary background. For vacuum perturbations we also provide a pedagogical derivation of the second-order behavior of the dynamical black hole entropy by considering the second-order variation of the Raychaudhuri equation. We also note that a similar deduction for vacuum perturbations was established in Rignon-Bret:2023fjq before us.

Let us consider the stationary black hole background and the perturbed geometry introduced in the last section. The Raychaudhuri equation for the congruence of null geodesics on the future horizon $\mathcal{H}^{+}$ reads

As the black hole background is stationary, the expansion $\theta$ and shear tensor $\sigma_{ab}$ vanish in the unperturbed spacetime geometry. What’s more, the rotation tensor $\omega_{ab}$ is equal to 0 since $k^{a}$ is hypersurface orthogonal. With the aid of the Einstein equation, the last term of (14) can be written as

Where $G_{D}$ is the $D-$dimensional Newton’s constant. As a corollary of the Raychaudhuri equation combined with the Einstein equation, $T_{ab}k^{a}k^{b}$ is equal to zero on the horizon in the stationary geometry. Otherwise the last term on the right hand side of (14) would not be zeroPoisson:2009pwt . Physically this condition can be interpreted as matter cannot be flowing across the event horizon. We multiply the Raychaudhuri equation on both sides by $\kappa v$, integrate it over the future horizon between cross-sections $\mathcal{C}(v_{1})$ and $\mathcal{C}(v_{2})$, and then vary this equation. Since $\frac{\mathrm{d}\theta}{\mathrm{d}v}$ and $T_{ab}k^{a}k^{b}$ vanish in the unperturbed geometry, $\delta$ acts only on $\theta$ on the left-hand side and only on $R_{ab}$ on the right-hand side, and the right-hand side is left with $8\pi G_{D}\kappa v\delta T_{ab}k^{a}k^{b}$ due to our gauge condition $\delta g_{ab}k^{a}k^{b}=0$. Recalling that $\xi^{a}=\kappa vk^{a}$ on the future horizon, to first order the result is

where $\mathrm{d}A=\mathrm{d}^{D-2}x\sqrt{\gamma(x,v)}$ is the area element of the cross-section $\mathcal{C}(v)$. Let $x=(x^{1},\cdots x^{D-2})$ be a coordinate system on $\mathcal{C}(v_{1})$, we can extend this coordinate system to an arbitrary cross-section by requiring that all points on each null geodesic share the same values of $(x^{1},\cdots x^{D-2})$. To exchange the order of integrals in (16), define the function $\mathcal{A}(x,v)$ on $\mathcal{H}^{+}$ asFlanagan:1999jp

The integral on the left hand side of (16) can then be split into

and it can be integrated by parts

where we have used the fact that $\mathcal{A}(x,v)=1$ in the stationary background. On the right-hand side of this equation, we may pull the variation $\delta$ to the front of the integrals as $\theta$ vanishes on the horizon of the unperturbed black hole background. And the second term on the right-hand side is

As a result, the left part of (16) reads

Thus we obtain the non-stationary physical process first law between two arbitrary cross-sections for GR

where $S_{\text{dyn}}$ is the dynamical black hole entropy of the cross-section of the horizon

The matter Killing energy flux between two cross-sections $\mathcal{C}(v_{1})$ and $\mathcal{C}(v_{2})$, relative to the Killing field $\xi^{a}$, is defined as

And $\Delta$ denotes the difference between $\mathcal{C}(v_{1})$ and $\mathcal{C}(v_{2})$. By the argument similar to (16), if we vary $\Delta E$, $\delta$ acts only on $T_{ab}$. It was shown by Hawking that if a black hole is stationary, then it must be either static or axisymmetricHawking:1971vc . If the horizon Killing field is normalized as $\xi^{a}=\left(\partial_{t}\right)^{a}+\Omega_{\mathcal{H}}\left(\partial_{\theta}\right)^{a}$, where $\left(\partial_{t}\right)^{a}$ represents time translations at infinity, $\left(\partial_{\theta}\right)^{a}$ represents the rotational Killing vector and $\Omega_{\mathcal{H}}$ is the angular velocity of the black hole, then the mass and angular momentum transferred across the horizon arePoisson:2009pwt

The matter Killing energy flux is then related to the change in mass and angular momentum of the black hole as²²2We emphasize that the mass and the angular momentum of the black hole are not equal to those of the spacetime if the black hole is not in vacuum. The matter distribution outside the black hole also contributes to the total mass and total angular momentum of the spacetime.

Therefore, the physical process first law reads

As the first corollary of the physical process first law, if the perturbation of the stress-energy tensor satisfies the null energy condition $\delta T_{ab}k^{a}k^{b}\geq 0$, for first-order perturbations sourced by external matter fields, the “linearized” second law holds for the dynamical black hole entropy

And as the second corollary, for source-free perturbations of a stationary black hole, the comparison first law holds between an arbitrary cross-section $\mathcal{C}(v)$ and the spatial infinity $i_{0}$

where $M$ and $J$ are ADM definitions of the mass and angular momentum of the spacetime, respectively. This is because $\Delta\delta S_{\text{dyn}}=0$ for source-free perturbations $\delta T_{ab}=0$, $S_{\text{dyn}}$ equals the usual Bekenstein-Hawking entropy at the bifurcation surface, and it has been proven that Bekenstein-Hawking entropy satisfies the comparison first law at the bifurcation surface. It is noteworthy that this relation still holds if matter fields are present as long as the combined Einstein-matter system admits a Hamiltonian formulationWald:1995yp ; Sudarsky:1992ty .

In the case of vacuum perturbations there are no external matter fields and $\delta T_{ab}=0$. Then (22) indicates that the dynamical black hole entropy is a constant at first order in perturbation theory. So we have to keep track of the second order in perturbation theory to study the leading-order behavior of the dynamical black hole entropy. Hollands, Wald and Zhang obtained the leading-order behavior in that case based on the varied fundamental identity of the covariant phase space formalismHollands:2024vbe . In what follows we are going to present a different derivation of the second-order behavior of $S_{\text{dyn}}$ by considering the second variation of the Raychaudhuri equationRignon-Bret:2023fjq .

For vacuum perturbations, the first-order variation of (14) is

Thus the perturbed expansion $\delta\theta$ is a constant along the null generators of $\mathcal{H}^{+}$. Recalling that $\theta$ vanishes at future infinity in the perturbed spacetime due to the teleological definition of the event horizon, the perturbed expansion $\delta\theta$ must vanish on $\mathcal{H}^{+}$Hollands:2012sf . We then consider the second-order variation of (14) to obtain

We multiply this equation on both sides by $\kappa v$, integrate over the future horizon between $\mathcal{C}(v_{1})$ and $\mathcal{C}(v_{2})$, and integrate the left hand side by parts in the same way shown above

What’s more, on the left hand side of this equation we may pull $\delta^{2}$ to the front of the integrals as both $\theta$ and $\delta\theta$ vanish. Finally the result is

Our result agrees with that derived by varying the fundamental identity of the covariant phase space formalismHollands:2024vbe . The term $\delta\sigma_{ab}\delta\sigma^{ab}$ can be interpreted as the energy flux of the weak gravitational wavesAshtekar:2021kqj . Since $\delta\sigma_{ab}\delta\sigma^{ab}\geq 0$, the second law holds at second order for vacuum perturbations of GR.

In this section we derive both the non-stationary physical process first law and the non-stationary comparison first law for $f(R)$ gravity using the Einstein frame. Our results and and the dynamical black hole entropy formula for $f(R)$ gravity agree with those derived by means of the Noether charge methodHollands:2024vbe ; Visser:2024pwz . For vacuum perturbations we also study the second-order behavior of the dynamical black hole entropy, and observe that the second law is obeyed both in the Einstein frame and in the original frame. What’s more, we determine the relationship between the dynamical black hole entropy of the cross-section $\mathcal{C}$ and the Wald entropy of the generalized apparent horizon $\mathcal{T}$ in the original frame.