1 Introduction↩︎

Black hole thermodynamics is one of the most profound developments in modern theoretical physics, as it unifies three originally independent frameworks: classical gravity, quantum theory, and statistical physics. Within this framework, a black hole behaves as a genuine thermodynamic system, equipped with temperature, entropy, and laws of evolution analogous to those of ordinary matter. This correspondence is made precise through the four laws of black hole mechanics, which establish a direct relation between horizon dynamics and thermodynamic behavior. Among these quantities, entropy plays a particularly central role: it not only governs the thermodynamic properties and stability of black holes, but also encodes microscopic information that is expected to shed light on the quantum structure of spacetime [1].

From a broader perspective, black hole entropy provides a unique window into several fundamental puzzles, including the statistical origin of gravitational degrees of freedom, the fate of information in gravitational collapse and evaporation, and the emergence of semiclassical geometry from an underlying quantum theory [2]–[4]. For this reason, understanding how entropy should be defined beyond strict stationarity is not merely a technical refinement, but a key step toward a deeper formulation of quantum gravity.

The earliest key insight into black hole entropy was provided by Bekenstein, who argued that the entropy of a black hole should be proportional to the area of its event horizon, \(S_{\mathrm{BH}} \propto A\) [5]. Subsequently, Bardeen, Carter, and Hawking formulated the four laws of black hole mechanics and established precise relations among mass, surface gravity, and horizon area [6]. Hawking later demonstrated that black holes emit thermal radiation with temperature \(T = \hbar \kappa/2\pi\) [7], leading to the celebrated entropy formula \(S = A/(4 G \hbar)\). These developments firmly established the thermodynamic interpretation of stationary black holes in general relativity.

Several decades later, Wald and collaborators developed the covariant phase space formalism in the context of gravity, in which black hole entropy is identified with a Noether charge associated with diffeomorphism invariance [8]–[11]. This approach yields a general definition of black hole entropy for arbitrary diffeomorphism-invariant theories of gravity. However, this construction is intrinsically tied to stationary backgrounds, where a preferred Killing horizon exists. Extending the notion of entropy to genuinely dynamical black holes remains a longstanding challenge.

A significant recent development was made by Wald and collaborators [12]. They uses the covariant phase space formalism to construct a dynamical extension of black hole entropy, which is also the approach we adopted. Their analysis shows that, in general, the entropy is not associated with the area of the cross-section of the event horizon, but rather with the area of the apparent horizon, suggesting that the latter provides a more local and physically meaningful notion of entropy in nonstationary settings. Another approach based on higher-curvature entropy functionals inspired by holographic entanglement entropy and the linearized second law, leading to the Dong-Wall entropy [13], [14]. Nevertheless, these results have been established mainly for first-order perturbations of stationary black holes [15]. Although there are some works on vacuum second-order perturbations [12], [16], an explicit expression for the second-order correction to the entropy remains unavailable, and no satisfactory conclusion has yet been reached for second-order perturbations incorporating matter fields.

Recently, Ashtekar et. al. developed a nonperturbative framework for black hole thermodynamics far from equilibrium, suggesting that the entropy is also proportional to the area of the marginally trapped surfaces associated with quasi-local horizons [17], [18]. Their approach is based on the quasi-local notations of horizons [19] and relies on the way identifying the thermodynamical qualities of the black holes. This is different from the approaches based on the covariant phase space analysis.

In this work, within covariant phase space formalism, we further investigate the structure of dynamical black hole entropy by considering second-order perturbations of spherically symmetric spacetimes. Working in Gaussian null coordinates, we perturbatively construct the apparent horizon and analyze the associated expansions and area element. By combining these geometric results with the covariant phase space formalism, we derive the modified canonical energy and relate it to the second-order variation of entropy. Our results show that, at second order, the entropy remains proportional to the area of the apparent horizon under the null energy condition, whereas additional conditions are required to establish the same proportionality without imposing the null energy condition. This result partially extends the proposal of Ref. [12] beyond linear order and supports the interpretation that the apparent horizon plays a distinguished role in characterizing dynamical black hole entropy.

The rest of this paper is organized as follows. In Section 2, we formulate the perturbed near-horizon geometry in Gaussian null coordinates, fix the gauge conditions, and construct the apparent horizon as a deformation of the event horizon. We also compute the corresponding null normals, expansions, and area element to the order needed for the entropy comparison. In Section 3, we review the covariant phase space formalism and express the second-order entropy variation in terms of the modified canonical energy and the constraint contribution on the horizon. In Section 4, we combine these ingredients with the Raychaudhuri equation, the shear term, the boundary condition, and the consequence of the null energy condition to show how the quadratic contributions cancel, thereby relating the entropy to the apparent-horizon area. We conclude with a summary of the main result and its implications in Section 5.

2 Black Hole Geometry and Apparent Horizon↩︎

In this section, we set up the near-horizon geometry in Gaussian null coordinates and fix the gauge conditions used in the perturbative analysis. We then construct the apparent horizon as a deformation of the event horizon, calculate the relevant null expansions, and derive the second-order expansion of its area element. These geometric results provide the ingredients needed to compare the apparent-horizon area with the entropy in later sections.

2.1 Geometric Setup↩︎

We consider a dynamical black hole perturbed from a stationary black hole background \((M,g)\) in \(n\)-dimensional spacetime. The event horizon of the stationary black hole is taken to be a bifurcate Killing horizon \(\mathcal{H}\), where we denote the future horizon as \(\mathcal{H}^+\), the past horizon as \(\mathcal{H}^-\), and the codimension-\(2\) bifurcation surface as \(\mathcal{B}\). The background spacetime is assumed to be stationary and spherically symmetric, possessing a Killing vector field \(\xi^a\) which generates the Killing horizons [20].

In the stationary case, near \(\mathcal{H}^+\), the spacetime metric in affinely parametrized Gaussian null coordinates \(\{u,v,x\}\) with \(x=(x^1,x^2,\ldots,x^{n-2})\) is given by \[\begin{align} g_{ab}(u,v,x) =& -(du)_a (dv)_b -(du)_b (dv)_a - u^2 \alpha(u,v) (dv)_a (dv)_b - u \beta_i(u,v) (dv)_a (dx^i)_b \\ &- u \beta_i(u,v) (dv)_b (dx^i)_a + \gamma_{ij}(u,v,x)\,(dx^i)_a (dx^j)_b \, , \end{align}\] where the \(u=0\) hypersurface corresponds to \(\mathcal{H}^+\), and \(v\) is chosen as an affine parameter along the null generators, the functions \(\alpha(u,v)\), \(\beta_i(u,v)\) and \(\gamma_{ij}(u,v,x)\) are assumed to be finite at \(u=0\) [21]. Under spherical symmetry assumption, \(\beta_i(u, v)\) vanishes and the transverse spatial metric takes the form \[\gamma_{ij}(u,v,x) = [r(u,v)]^2 \, \Omega_{ij}(x) \, ,\] where \(r(u,v)\) is the areal radius and \(\Omega_{ij}(x)\) is the metric on the unit \((n-2)\)-sphere. The inverse metric is \[\begin{align} g^{ab}(u,v,x) &=& u^2[\alpha(u,v) + \beta^2](\partial_u)^a(\partial_u)^b -(\partial_u)^a(\partial_v)^b-(\partial_u)^b(\partial_v)^a-u\beta^i(u,v)(\partial_i)^a(\partial_u)^b\nonumber\\ && -u\beta^i(u,v)(\partial_i)^b(\partial_u)^a +\gamma^{ij}(u,v,x)(\partial_i)^a(\partial_j)^b \, , \end{align}\] where \(\gamma^{ij}\) denotes the inverse of \(\gamma_{ij}\), \(\beta^{i}=\gamma^{ij}\beta_j\), and \(\beta^2=\beta_i\beta^i\). The metric can be decomposed as \[g_{ab} {=} -2 k_{(a} l_{b)} + \gamma_{ab} \, ,\] and the transverse metric can be expressed as \[\gamma_{ab} = \gamma_{ij}\,(dx^i)_a (dx^j)_b = \gamma_{ij}\,m^i_a m^j_b \, ,\] where \[k^a := (\partial_v)^a\,,\quad l^a := (\partial_u)^a\,,\quad m_i^a := (\partial_i)^a\,,\] and \[m^i_a:= \gamma^{ij}(u,v,x)m_j^b g_{ab} = \gamma^{ij}(u,v,x)[- u\beta_j(u,v)(dv)_a + \gamma_{jk}(u,v,x)(dx^k)_a] = (dx^i)_a \,.\] It is easy to check that they satisfy that \[k_al^a = -1\,,\quad k_am_i^a=0\,,\quad l_am_i^a=0\,.\] Please note that \[k_a = g_{ab}k^b = -(du)_a - u^2\alpha(u,v)\,(dv)_a - u\beta_i(u, v)\,(dx^i)_a \overset{\mathcal{H^+}}{=} -(du)_a\] is a null normal covector to \(\mathcal{H}^+\), but it is not necessarily null away from the horizon. Here, \(\overset{\mathcal{H^+}}{=}\) denotes equality evaluated on \(\mathcal{H^+}\).

Below we demonstrate how to decompose \(\xi^a\) in terms of the null basis \((k^a,l^a)\). Since \(\xi^a\) is normal to \(\mathcal{H}\), it is tangent to the null geodesic generators of \(\mathcal{H}\). Along \(\mathcal{H}^+\) it is therefore proportional to \(k^a\), whereas along \(\mathcal{H}^-\), it is proportional to \(-l^a\), because \(\xi^a\) is past directed on \(\mathcal{H}^-\) whereas \(l^a\) is future directed. On \(\mathcal{H}^+\), the Killing vector \(\xi^a\) obeys \(\xi^b\nabla_b\xi^a \overset{\mathcal{H}^+}{=} \kappa\xi^a\) and the outgoing null normal is affinely parameterized, \(k^b\nabla_b k^a \overset{\mathcal{H}^+}{=} 0\). Correspondingly, on \(\mathcal{H}^-\), one similarly has \(\xi^a\nabla_a\xi^b \overset{\mathcal{H}^-}{=} -\kappa\xi^b\) and \(l^a\nabla_a l^b=0\). Therefore, one can choose a combination of \(k^a\) and \(l^a\) such that [21] \[\xi^a = \kappa \left( v k^a - u l^a \right) \, ,\] where \(\kappa\) is the surface gravity. This ensures that \(\xi^a\) acts like a local Lorentz boost near the horizon and respects the fact that \(\xi^a=0\) at \(\mathcal{B}\), where \(u = v = 0\). Therefore, the isometry generated by \(\xi^a\) restricts the form of the metric functions \(\alpha\), \(\beta_i\) and \(\gamma_{ij}\) so that the dependence on \(u\) and \(v\) occurs only through the combination \(\kappa u v\) [22], [23]: \[\alpha = \alpha (\kappa u v) , \qquad \beta_i = \beta_i(\kappa u v) , \qquad \gamma_{ij} = \gamma_{ij}(\kappa u v, x) \, .\]

We now proceed with the perturbative analysis and first give the definition of the metric perturbation. We introduce a one-parameter family of metrics \(g_{ab}(s)\) labeled by \(s\). \[g_{ab}(s) = g_{ab} + s \delta g_{ab} + \frac{1}{2} s^2 \delta^2 g_{ab} + \mathcal{O}(s^3) \, .\] The validity of the approximation, upon setting \(s = 1\), depends on the smallness of the perturbation \(\delta g_{ab}\) and \(\delta^2 g_{ab}\), i.e., \[\delta g_{ab} := \left.\frac{d}{ds} g_{ab}(s)\right|_{s=0} \, , \qquad \delta^2 g_{ab} := \left.\frac{d^2}{ds^2} g_{ab}(s)\right|_{s=0} \, .\] To proceed with the perturbative analysis, we use the diffeomorphism gauge freedom to fix the spacetime point identification between the unperturbed and perturbed configurations. This leads us to introduce the following gauge conditions to simplify the perturbative analysis [21].

1. The event horizon of the perturbed black hole is identified with the Killing horizon of the unperturbed background. Moreover, \(\mathcal{H}^+\) and \(\mathcal{H}^-\) remain at \(u = 0\) and \(v = 0\), respectively, after the perturbation.

2. The null vectors \(k^a\) and \(l^a\) are held fixed under the perturbation, \[\delta^m k^a = 0 \, , \qquad \delta^m l^a = 0 \, , \qquad (m=1,2)\,.\] Additionally, \(k^a\) remains null only on \(\mathcal{H}^+\) and normal to \(\mathcal{H}^+\), while \(l^a\) stays null everywhere under the perturbation. Combined with the requirement \(\delta^m(k^a l_a)=0\), this implies the following conditions \[k^a \delta^m g_{ab} \overset{\mathcal{H}^+}{=} 0 \, , \qquad l^a \delta^m g_{ab} = 0 \, , \qquad (m=1,2)\,. \label{gauge32condition32k44l}\tag{1}\] We further require that \(k^a\) remains affinely parametrized on \(\mathcal{H}^+\) and \(l^a\) remains affinely parametrized throughout the entire spacetime under the perturbation, \[\delta^m \!\left( k^b \nabla_b k^a \right) \overset{\mathcal{H}^+}{=} 0 \, , \qquad \delta^m \!\left( l^b \nabla_b l^a \right) = 0 \, , \qquad (m=1,2)\,.\]

3. The Killing field \(\xi^a\) continues to be null and tangent to the geodesic generators of the perturbed horizon. \[\delta^m(g_{ab}\xi^a\xi^b)\overset{\mathcal{H}}{=}0 \, , \qquad \zeta_a\delta^m\xi^a\overset{\mathcal{H}}{=}0 \, , \qquad (m=1,2)\,,\] where \(\zeta^a\) is an arbitrary spacelike vector orthogonal to both \(k^a\) and \(l^a\). Together with condition 1 , \(\xi^a\delta^m g_{ab}\overset{\mathcal{H}}{=}0\), this means on the future horizon \(\delta^m\xi^a\) is proportional to \(k^a\) and on the past horizon to \(l^a\).

From the above gauge conditions, it follows that the affine parameters \(v\) and \(u\) are fixed under the perturbation. We will also fix the spatial coordinates \(x^i\). Therefore, the perturbation of a stationary background affects only the metric functions, which we expand as \[\begin{align} \alpha(s;u,v,x) &= \alpha(\kappa u v) + s\delta\alpha(u,v,x) + \frac{1}{2} s^2\delta^2\alpha(u,v,x) \, , \\ \beta_i(s;u, v, x) &= \beta_i(\kappa u v) + s\delta \beta_i(u,v,x) + \frac{1}{2} s^2\delta^2\beta_i(u,v,x) \, , \\ \gamma_{ij}(s;u,v,x) &= \gamma_{ij}(\kappa u v,x) + s \delta\gamma_{ij}(u,v,x) + \frac{1}{2} s^2 \delta^2\gamma_{ij}(u,v,x) \, . \end{align}\] Thus, after introducing the perturbation, the metric takes the following form \[\begin{align} g_{ab}(s;u,v,x) =& -(du)_a (dv)_b -(du)_b (dv)_a - u^2 \alpha(s;u,v,x) (dv)_a (dv)_b - u \beta_i(s;u,v,x) (dv)_a (dx^i)_b \\ &- u \beta_i(s;u,v,x) (dv)_b (dx^i)_a + \gamma_{ij}(s;u,v,x)\,(dx^i)_a (dx^j)_b \, , \end{align} \label{eq:metric}\tag{2}\] which is the foundation of all the following discussions.

2.2 Geometric Quantities on the Apparent Horizon↩︎

The apparent horizon is defined as a marginally trapped surface on which the expansion of outgoing null congruence vanishes, while the expansion for ingoing ones remains negative. Because the expansion of null congruences on the apparent horizon is defined locally, the apparent horizon provides a more local notion of the black-hole boundary than the event horizon, and overcoming the teleological deficiency of the event horizons in the discussion of local physics. Therefore, below we study its relation to dynamical black-hole entropy by analyzing the area of the apparent horizon.

In the stationary case, the apparent horizon coincides with the event horizon. The outgoing null expansion is then given by \(\theta_v = \nabla_a k^a\) on \(\mathcal{H}^+\), which describes the rate of change of the area element \(dA\) along the affine parameter \(v\): \[\theta_v \, dA = \frac{d}{dv}(dA) \, . \label{eq:theta95v95area95element}\tag{3}\] Here \(dA = \sqrt{\det \gamma}\, d^{n-2}x\) is the area element on a \((n-2)\)-dimensional horizon cross section. From Eq. 3 , we obtain \[\theta_v = \frac{1}{2}\, \gamma^{ab}\, \partial_v \gamma_{ab} = \partial_v \ln \sqrt{\det \gamma} \, .\] The position of the apparent horizon \(\mathcal{A}\) is denoted by [21] \[u=U(s;v,x)\geqslant 0\, .\] Here \(U(s;v,x)\) describes the affine null distance away from the event horizon and admits the following expansion under perturbations: \[U(s;v,x) = s \delta U(v,x) + \frac{1}{2} s^2 \delta^2 U(v,x) + \mathcal{O}(s^3) \, .\] The prosition of the apparent horizon is illustrated in the Fig. 1. For an arbitrary constant affine parameter \(v\), the cross section of the apparent horizon \(\mathcal{A}\) is denoted by \(\mathcal{T}(v)\), which is a future marginally outer trapped surface. The ingoing null normal covectors on \(\mathcal{T}(v)\) is given by \[\tilde{l}_a \overset{\mathcal{T}}{=} l_a = - (dv)_a \, ,\] whose dual vector is \[\tilde{l}^a \overset{\mathcal{T}}{=} g^{ab} \tilde{l}_b = (\partial_u)^a = l^a\, .\] The outgoing null normal covector \(\tilde{k}_a\) is constructed from two linearly independent normal covectors on the codimension-two surface \(\mathcal{T}(v)\), which are given by \[(df_1)_a = (du)_a - \partial_vU(dv)_a - \partial_iU(dx^i)_a\, ,\] and \[(df_2)_a = (dv)_a \, ,\] where \(f_1 = u - U(s;v,x)\) and \(f_2 = v\). After that, we construct \(\tilde{k}_a\) as a linear combination of \((df_1)_a\) and \((df_2)_a\), with the coefficients fixed by imposing the normalization condition \(\tilde{k}^a \tilde{l}_a=-1\) and the null condition \(\tilde{k}^a\tilde{k}_a=\mathcal{O}(s^3)\). This gives, on \(\mathcal{T}(v)\), \[\tilde{k}_a \overset{\mathcal{T}}{=} -(du)_a - \frac{1}{2}s^2\left[\alpha(\delta U)^2 + (D\delta U)^2\right](dv)_a + \left(sD_i\delta U + \frac{1}{2}s^2D_i\delta^2U\right)(dx^i)_a + \mathcal{O}(s^3)\, ,\] whose dual vector is \[\tilde{k}^a \overset{\mathcal{T}}{=} g^{ab} \tilde{k}_b = k^a + \frac{1}{2}s^2\left[(D\delta U)^2 - \alpha(\delta U)^2\right]l^a +\left(sD^i\delta U + \frac{1}{2}s^2D^i\delta^2 U\right)m^a_i + \mathcal{O}(s^3)\, .\] Here, \(D_i\) represents the codimension-2 intrinsic covariant derivative, \(D^i = \gamma^{ij}D_j\), and \((D\delta U)^2=(D_i\delta U)( D^i \delta U)\). The metric admits the following decomposition on \(\mathcal{T}(v)\), valid to second order in \(s\): \[g_{ab}(s) \overset{\mathcal{T}}{=} -2 \tilde{k}_{(a} \tilde{l}_{b)} + \gamma_{ab}(s) \, .\] Using the projection operator \[\gamma^{a}{}_{b} \overset{\mathcal{T}}{=} \delta^{a}{}_{b} + \tilde{k}^{a} \tilde{l}_{b} + \tilde{l}^{a} \tilde{k}_{b} \,,\] we express the expansion \(\tilde{\theta}_{\tilde{k}}\) on \(\mathcal{T}(v)\) as \[\begin{align} \tilde{\theta}_{\tilde{k}} &= \gamma^a{}_b \nabla_a \tilde{k}^b \\ &\overset{\mathcal{T}}{=} \gamma^a{}_b \nabla_a k^b + \frac{1}{2}s^2\left[(D\delta U)^2 - \alpha(\delta U)^2\right] \gamma^a{}_b \nabla_a l^b + \gamma^a{}_b \nabla_a \;\left\{(sD^i\delta U + \frac{1}{2}s^2D^i\delta^2 U)m^a_i\right\} \\ &= \theta_v + \frac{1}{2}s^2\left[(D\delta U)^2 - \alpha(\delta U)^2\right] \theta_u + D_i\left(sD^i\delta U + \frac{1}{2}s^2D^i\delta^2 U\right) + \mathcal{O}(s^3)\, . \end{align}\] Here \(\theta_u\) denotes the ingoing null expansion in the stationary case, which can be written as \(\theta_u = \nabla_a l^a\).

On the apparent horizon, an arbitrary function \(F(s;U(s;v,x),v,x)\) admits a second-order perturbative expansion about its unperturbed value on the event horizon: \[\begin{align} F(s;U(s;v,x),v,x) &= F(0,v,x) + s\delta F(0,v,x) + \frac{1}{2}s^2\delta^2 F(0,v,x) \\ &\quad+ s\delta U(v,x)\partial_u F(0,v,x) + \frac{1}{2}s^2[\delta U(v,x)]^2\partial_u^2 F(0,v,x) \\ &\quad+ s^2\delta U(v,x)\partial_u \delta F(0,v,x) + \frac{1}{2}s^2 \delta^2 U(v,x)\partial_u F(0,v,x) + \mathcal{O}(s^3)\, . \end{align} \label{eq:expansion95scheme}\tag{4}\] Here, \(\delta F(0,v,x)\) denotes \(\partial_s F(0;0,v,x)\), and similar notation is used for higher orders. Then, following the above expansion scheme, we expand \(\tilde{\theta}_{\tilde{k}}\) around the location of the event horizon, \[\begin{align} \tilde{\theta}_{\tilde{k}}(s;U(s;v,x),v,x) &= \theta_v(0,v) + s\delta \theta_v(0,v,x) + \frac{1}{2}s^2\delta^2 \theta_v(0,v,x) \\ &\quad + s\delta U(v,x)\partial_u \theta_v(0,v) + \frac{1}{2}s^2[\delta U(v,x)]^2\partial_u^2 \theta_v(0,v) \\ &\quad + s^2\delta U(v,x)\partial_u \delta \theta_v(0,v,x) + \frac{1}{2}s^2 \delta^2 U(v,x)\partial_u \theta_v(0,v) \\ &\quad + \frac{1}{2}s^2\{[D\delta U(v,x)]^2 - \alpha[\delta U(v,x)]^2\} \theta_u(0,v) \\ &\quad + D_i[sD^i\delta U(v,x) + \frac{1}{2}s^2D^i\delta^2 U(v,x)] + \mathcal{O}(s^3)\, . \end{align}\] Next, we impose the defining condition for an apparent horizon, namely that the expansion \(\tilde{\theta}_{\tilde{k}}\) must vanish which implies \[\gathered[b] \theta_v = 0 \, , \\ \delta \theta_v + \delta U \partial_u \theta_v + D^2\delta U = 0 \, , \\ \delta^2 \theta_v + (\delta U)^2(\partial_u^2 \theta_v - \alpha\theta_u) + 2\delta U \partial_u \delta \theta_v + \delta^2 U \partial_u \theta_v + D^2\delta^2U + (D\delta U)^2\theta_u = 0 \, . \end{gather} \label{eq:apparent95horizon95condition}\tag{5}\] Here, for convenience, we have suppressed the explicit dependence of all quantities on the coordinates in the calculation.

Below, we introduce two relations to simplify the second and third equalities in Eq. 5 . The first is, at zeroth order, the two null expansions satisfy \(\partial_u \theta_v = \partial_v \theta_u\) [21]. The second condition is \[\theta_u(0,v)=-cv\, ,\] where \(c>0\) is a constant. This follows from Appendix B of [20], together with spherical symmetry, which implies that \(\theta_u(0,v)\) is independent of \(x\). Moreover, in deriving the second equality in Eq. 6 , we also used the first equality in Eq. 6 . \[\gathered[b] \delta \theta_v = (-D^2 + c)\delta U \, ,\\ \delta^2 \theta_v = 2\frac{\delta \theta_u}{\theta_u}\delta\theta_v + 2\left(\frac{\delta\theta_u}{\theta_u} + \frac{\alpha v^2\delta\theta_v}{\theta_u}\right)D^2\delta U - \frac{\alpha v}{c}\left[(\delta\theta_v)^2 + (D^2\delta U)^2\right] + (- D^2 + c)\delta^2U - (D\delta U)^2\theta_u \, . \end{gather} \label{eq:theta95relations}\tag{6}\] Finally, we expand the apparent-horizon area element \(dA_\mathcal{T}\) at affine parameter \(v\) around the event-horizon cross section at the same \(v\), following the same procedure as in Eq. 4 : \[\begin{align} dA_\mathcal{T}(s;U(s;v,x),v,x) &= dA_{\epsilon}(0,v,x) + s\delta dA_{\epsilon}(0,v,x) + \frac{1}{2}s^2\delta^2 dA_{\epsilon}(0,v,x) \\ &\quad+ s\delta U(v,x)\partial_u dA_{\epsilon}(0,v,x) + \frac{1}{2}s^2[\delta U(v,x)]^2\partial_u^2 dA_{\epsilon}(0,v,x) \\ &\quad + s^2\delta U(v,x)\partial_u \delta dA_{\epsilon}(0,v,x) + \frac{1}{2}s^2 \delta^2 U(v,x)\partial_u dA_{\epsilon}(0,v,x) + \mathcal{O}(s^3)\, , \end{align} \label{eq:apparent95horizon95area95expansion95second95order1}\tag{7}\] where \(dA_{\epsilon}(0,v,x)\) denotes the area element of the cross section \(\mathcal{C}(v)\) of the event horizon. Following the facts that, on the event horizon, the ingoing null expansion satisfies \(\partial_u dA_{\epsilon} = \theta_u dA_{\epsilon}\), together with \(\partial_u \theta_v = \partial_v \theta_u\) and \(\theta_u(0,v) = -cv\) as given above, and by using Eq. 6 to perform variable substitutions, we have \[\begin{align} dA_\mathcal{T}(s;U(s;v,x),v,x) &= dA_{\epsilon} + s\left[\delta dA_{\epsilon} - v\delta\theta_v dA_{\epsilon} - v(D^2\delta U)dA_{\epsilon}\right] + s^2\Bigg{\{}\frac{1}{2}\delta^2dA_{\epsilon} - v\delta\theta_v\delta dA_{\epsilon} - \frac{1}{2}v\delta^2\theta_vdA_{\epsilon} \\ &\quad + \frac{1}{2}\left(1-\frac{\alpha}{c}\right)v^2\left[(\delta\theta_v)^2 + (D^2\delta U)^2\right]dA_{\epsilon} - \frac{1}{2}v(D\delta^2 U)^2\theta_u dA_{\epsilon} - \frac{1}{2}(D^2\delta^2 U)dA_{\epsilon}\\ &\quad + (D^2\delta U)\left[\left(1-\frac{\alpha}{c}\right)v^2\delta\theta_vdA_{\epsilon} - v\delta dA_{\epsilon}\right]\Bigg{\}} + \mathcal{O}(s^3)\, . \end{align} \label{eq:apparent95horizon95area95expansion95second95order2}\tag{8}\] Similarly to the above calculation, we again suppress the explicit dependence of all quantities on the coordinates. Considering the result obtained in the first relation in Eq. 5 , we get \[\begin{align} dA_\mathcal{T}(s;U(s;v,x),v,x) &= dA_{\epsilon} + s\delta\left[\left(1 - v\theta_v\right)dA_{\epsilon}\right] + \frac{1}{2}s^2\delta^2\left[\left(1 -v\theta_v\right)dA_{\epsilon}\right] + s^2\Bigg{\{}\frac{1}{2}\left(1-\frac{\alpha}{c}\right)v^2\left(\delta\theta_v + D^2\delta U\right)^2 \\ &\quad - \frac{1}{2}v\left[(D\delta U)^2\theta_u + (D^2\delta U)\gamma^{ab}\delta\gamma_{ab}\right]\Bigg{\}}dA_{\epsilon} - vD^2\left(s\delta U + \frac{1}{2}s^2\delta^2 U\right)dA_{\epsilon} + \mathcal{O}(s^3)\, . \end{align} \label{eq:apparent95horizon95area95expansion95second95order}\tag{9}\]

In the following, we expand the apparent-horizon area in the perturbation parameter \[A_\mathcal{T}(s;v) = A_\mathcal{T}(v) + s\delta A_\mathcal{T}(v) + \frac{1}{2}s^2 \delta^2 A_\mathcal{T}(v) + \mathcal{O}(s^3)\, ,\] and retain only the second-order perturbative contribution. We then integrate both sides of the equality in Eq. 9 over the codimension-2 horizon cross section at fixed \(v\). At leading order, the area of \(\mathcal{T}(v)\) is the same as the one of \(\mathcal{C}(v)\). It is also easy to find that at firs order we have \[\delta A_{\mathcal{T}}(v)=\delta\int_{\mathcal{C}} dA_{\epsilon} \left(1 - v\theta_v\right)\, .\] Since the last second-order term on the right-hand side of Eq. 9 is a total derivative, its integral vanishes. Thus, we obtain \[\begin{align} \delta^2 A_\mathcal{T}(v) &= \delta^2 \int_{\mathcal{C}} dA_{\epsilon} \left(1 - v\theta_v\right) + \int_{\mathcal{C}}dA_{\epsilon} \Bigg{\{}\left(1 + \frac{\alpha v}{\theta_u}\right)(\theta_u\delta U)^2 - v\left[(D\delta U)^2\theta_u + (D^2\delta U)\gamma^{ab}\delta\gamma_{ab}\right]\Bigg{\}} \, . \label{eq:apparent95horizon95area95final} \end{align}\tag{10}\] The last term in the right hand of the above equation is non-vanishing in general.

3 Modified Canonical Energy↩︎

In this section, we give a brief review on the covariant phase space formalism and introduce a modified canonical energy which will be used in the second-order analysis in following sections. We include the effect of varying the vector field \(\xi^a\), identify the relevant constraint and boundary terms, and evaluate the corresponding fluxes through \(\mathcal{H}^+\) and \(\mathcal{I}^+\). This yields the relation between the modified canonical energy and the second-order variation of the entropy.

3.1 Covariant Phase Space Formalism↩︎

First, we briefly review the covariant phase space formalism following the strategy of  [8], [11], [24]. Consider a diffeomorphism-covariant theory in \(n\) dimensions with Lagrangian \(n\)-form \(\boldsymbol{L}(\phi)\) whose variation takes the standard form \[\delta \boldsymbol{L}(\phi) = \boldsymbol{E}(\phi)\delta\phi + d\,\boldsymbol{\Theta}(\phi,\delta\phi) \, . \label{eq:lagrangian95variation}\tag{11}\] Here \(\phi\) denotes the full set of dynamical fields including the metric and matter fields, \(\boldsymbol{E}(\phi)=0\) gives the equations of motion, and \(\boldsymbol{\Theta}(\phi,\delta\phi)\) is the symplectic potential \((n-1)\)-form, which is linear in \(\delta\phi\). The associated symplectic current \((n-1)\)-form is \[\boldsymbol{\omega}(\phi;\delta_1\phi,\delta_2\phi) = \delta_1\boldsymbol{\Theta}(\phi,\delta_2\phi) - \delta_2\boldsymbol{\Theta}(\phi,\delta_1\phi) \, .\] Now let \(\xi^a\) be an arbitrary smooth vector field. Replacing \(\delta\phi\) in Eq. 11 by \(\mathcal{L}_\xi\phi\), namely the variation induced by the diffeomorphism generated by \(\xi^a\), we can get \[\mathcal{L}_\xi\boldsymbol{L}(\phi) = d[\xi\cdot\boldsymbol{L}(\phi)] = \boldsymbol{E}(\phi)\mathcal{L}_\xi\phi + d\boldsymbol{\Theta}(\phi,\mathcal{L}_\xi\phi) \, . \label{eq:diffeomorphism95covariant95lagrangian95variation}\tag{12}\] The first equality follows from the Cartan magic formula \[\mathcal{L}_\xi\boldsymbol{L}(\phi) = d[\xi\cdot\boldsymbol{L}(\phi)] + \xi\cdot d\boldsymbol{L}(\phi) \, , \label{CK}\tag{13}\] where \(``\cdot"\) denotes the contraction of a vector field with the first index of a differential form. When the fields are on-shell, i.e., \(\boldsymbol{E}(\phi)=0\), Eq. 12 implies that there exists a Noether current \((n-1)\)-form \[\boldsymbol{J}_\xi(\phi) = \boldsymbol{\Theta}(\phi,\mathcal{L}_\xi\phi) - \xi\cdot\boldsymbol{L}(\phi) \, , \label{eq:noether95current95definition}\tag{14}\] which is closed. The Noether current also satisfies an off-shell identity and can equivalently be written as [25] \[\boldsymbol{J}_\xi(\phi) = d\boldsymbol{Q}_\xi(\phi) + \xi^a\boldsymbol{C}_a(\phi) \, , \label{eq:noether95current95decomposition}\tag{15}\] where \(\boldsymbol{Q}_\xi(\phi)\) is the Noether-charge \((n-2)\)-form, and \(\boldsymbol{C}_a(\phi)\) is a dual vector-valued \((n-1)\)-form that vanishes on shell [12] \[\begin{align} \boldsymbol{C}_a(\phi) &=&\boldsymbol{\epsilon}_c^{(n)} \Bigg[ 2(E_G)^{c}{}_{a} - \sum_A A^{d_1\cdots d_k}{}_{b_1\cdots a \cdots b_l}\, (E_M)_{d_1\cdots d_k}{}^{b_1\cdots c \cdots b_l} \nonumber\\ &&+ \sum_A A^{d_1\cdots c \cdots d_k}{}_{b_1\cdots b_l}\, (E_M)_{d_1\cdots a \cdots d_k}{}^{b_1\cdots b_l} \Bigg] \, . \label{eq:full95Ca95expression} \end{align}\tag{16}\] Here, \(\boldsymbol{\epsilon}_c^{(n)}=\epsilon_{c a_1\cdots a_{n-1}}\) is the spacetime volume form, \((E_G)_{ab}=0\) are the equations of motion for \(g_{ab}\), and \((E_M)_{a_1\cdots a_k}{}^{b_1\cdots b_l}=0\) are the equations of motion for the matter field \(A^{a_1\cdots a_k}{}_{b_1\cdots b_l}\).

In the following, we consider a variation of the Noether current \(\boldsymbol{J}_\xi(\phi)\) in which both the dynamical fields \(\phi\) and the vector field \(\xi^a\) are varied. Varying Eq. 14 , we obtain \[\begin{align} \delta\boldsymbol{J}_\xi(\phi) &= \delta_\phi\boldsymbol{\Theta}(\phi,\mathcal{L}_\xi\phi) + \boldsymbol{\Theta}(\phi,\mathcal{L}_{\delta\xi}\phi) - \delta\xi\cdot\boldsymbol{L}(\phi) - \xi\cdot\delta\boldsymbol{L}(\phi) \\ &= \boldsymbol{J}_{\delta\xi}(\phi) -\xi\cdot[\boldsymbol{E}(\phi)\delta\phi] + \boldsymbol{\omega}(\phi;\delta\phi,\mathcal{L}_\xi\phi) + d[\xi\cdot\boldsymbol{\Theta}(\phi,\delta\phi)] \, , \end{align} \label{eq:noether95current95variation}\tag{17}\] where \[\delta_\phi\boldsymbol{\Theta}(\phi,\mathcal{L}_\xi\phi) = \delta\boldsymbol{\Theta}(\phi,\mathcal{L}_\xi\phi) -\boldsymbol{\Theta}(\phi,\mathcal{L}_{\delta\xi}\phi) \, .\] Substituting the variation of Eq. 15 into Eq. 17 , we obtain the so-called “fundamental identity" [24] \[\boldsymbol{\omega}(\phi;\delta\phi,\mathcal{L}_\xi\phi) = \xi\cdot[\boldsymbol{E}(\phi)\delta\phi] + \xi^a\delta\boldsymbol{C}_a(\phi) + d[\delta_\phi\boldsymbol{Q}_\xi(\phi) - \xi\cdot\boldsymbol{\Theta}(\phi,\delta\phi)] \, , \label{eq:varied95symplectic95current95identity}\tag{18}\] where \[\delta_{\phi}\boldsymbol{Q}_\xi(\phi) = \delta\boldsymbol{Q}_\xi(\phi) - \boldsymbol{Q}_{\delta\xi}(\phi) \, .\] In the following, we take \(\xi^a\) to be the Killing vector field of the stationary background, i.e., \(\mathcal{L}_\xi\phi=0\). Using this condition and varying Eq. 18 once more, we obtain \[\begin{align} \boldsymbol{\omega}(\phi;\delta\phi,\mathcal{L}_\xi\delta\phi) + \boldsymbol{\omega}(\phi;\delta\phi,\mathcal{L}_{\delta\xi}\phi) =& \xi\cdot[\delta\boldsymbol{E}(\phi)\delta\phi] + \xi\cdot[\boldsymbol{E}(\phi)\delta^2\phi] + \xi^a\delta^2\boldsymbol{C}_a(\phi) + d[\delta_{\phi}^2\boldsymbol{Q}_\xi(\phi) - \xi\cdot\delta\boldsymbol{\Theta}(\phi, \delta\phi)] \\ &+ \delta\xi\cdot[\boldsymbol{E}(\phi)\delta\phi] + \delta\xi^a\delta\boldsymbol{C}_a(\phi) + d[\delta_\phi\boldsymbol{Q}_{\delta\xi}(\phi) - \delta\xi\cdot\boldsymbol{\Theta}(\phi,\delta\phi)] \, . \end{align}\] Tracing the dependence on \(\delta\xi^a\) shows that the first term on the left-hand side corresponds to the first line on the right-hand side, whereas the second term corresponds to the second line, all of whose terms are linear in \(\delta\xi^a\), therefore, we obtain \[\boldsymbol{\omega}(\phi;\delta\phi,\mathcal{L}_\xi\delta\phi) = \xi\cdot[\delta\boldsymbol{E}(\phi)\delta\phi] + \xi\cdot[\boldsymbol{E}(\phi)\delta^2\phi] + \xi^a\delta^2\boldsymbol{C}_a(\phi) + d[\delta^2_{\phi}\boldsymbol{Q}_\xi(\phi) - \xi\cdot\delta\boldsymbol{\Theta}(\phi, \delta\phi)]\, . \label{eq:second95varied95symplectic95current95identity}\tag{19}\] So far, we have gathered enough formulas on covariant phase space formalism which is crucial to study the canonical energy for the perturbation fields.

When external matter fields are completely absent, taking a variation of Eq. 11 shows that \(d\boldsymbol{\omega}=0\) provided the perturbations satisfy the linearized equations of motion and \(\xi^a\) is fixed. In this case, the canonical energy of \(\delta\phi\) can be defined as the integral of \(\boldsymbol{\omega}(\phi,\delta\phi,\mathcal{L}_\xi\delta\phi)\) over a Cauchy surface [24]. In the next section, we introduce a modified canonical energy which incorporates external matter fields.

3.2 Construction of the Modified Canonical Energy↩︎

Start from this section, we restrict attention to the case in which matter fields are merely absent in the background, so \(\phi\) is identified with the metric field \(g\) only. In order to define a conserved canonical energy, we introduce a \((n-1)\)-form \(\boldsymbol{\omega}'(g;\delta g,\mathcal{L}_{\xi}g)\) as follows \[\begin{align} \boldsymbol{\omega}'(g;\delta g,\mathcal{L}_\xi\delta g) &= \boldsymbol{\omega}(g;\delta g,\mathcal{L}_\xi\delta g) - \xi \cdot [\delta\boldsymbol{E}(g)\delta g] - \xi \cdot [\boldsymbol{E}(g)\delta^2 g] - \xi^a\delta^2\boldsymbol{C}_a(g) \, , \end{align} \label{eq:modified95symplectic95current95definition}\tag{20}\] where \(\xi^a\) is the Killing vector field of the stationary background and \[d\boldsymbol{\omega}'(g;\delta g,\mathcal{L}_\xi\delta g)=0\] by considering Eq.(19 ). We refer to this \((n-1)\)-form as the “canonical energy form". The canonical energy of the metric perturbation on a Cauchy surface \(\Sigma(t)\) is defined as \[\mathcal{E}[\Sigma(t)] = \int_{\Sigma(t)} \boldsymbol{\omega}'(g;\delta g,\mathcal{L}_\xi\delta g) .\] Here, the canonical-energy \((n-1)\)-form is defined differently from the one in [24] to include the contribution associated with the external matter fields. This definition reduces to the one used in [24] only when external matter fields are completely absent. Therefore, by Stokes’ theorem, in a compact region bounded by the Cauchy surfaces \(\Sigma(t_2)\) and \(\Sigma(t_1)\) with \(t_2>t_1\), \(\mathcal{H}^+_{12}\), and \(\mathcal{I}^+_{12}\), we have \[\int_{\Sigma(t_1)}\boldsymbol{\omega}' = \int_{\Sigma(t_2)}\boldsymbol{\omega}' + \int_{\mathcal{H}^+_{12}}\boldsymbol{\omega}' + \int_{\mathcal{I}^+_{12}}\hat{\boldsymbol{\omega}}' \, . \label{eq:stokes95modified95symplectic95current}\tag{21}\] The hypersurfaces \(\Sigma(t_2)\), \(\Sigma(t_1)\), \(\mathcal{H}_{12}^+\), and \(\mathcal{I}_{12}^{+}\) in the above formula are illustrated schematically in Fig.2. ¹

Here we have suppressed the dependence of \(\boldsymbol{\omega}'\) on its arguments. The signs in this formula are determined by the boundary orientation. A hatted quantity denotes the corresponding unhatted quantity expressed in the unphysical spacetime and evaluated at conformal infinity \(\mathcal{I}^+\). The region \(\mathcal{H}^+_{12}\) denotes the portion of \(\mathcal{H}^+\) bounded by the two cross sections specified by the affine parameters \(v_1\) and \(v_2\). Similarly, \(\mathcal{I}^+_{12}\) denotes the portion of \(\mathcal{I}^+\) bounded by the two cross sections specified by the affine parameters \(\hat{u}_1\) and \(\hat{u}_2\). It follows that we can calculate the change of the canonical energy \(\mathcal{E}[\Sigma(t)]\) between the two Cauchy surfaces \(\Sigma(t_2)\) and \(\Sigma(t_1)\) as \[{\mathcal{E}}[\Sigma(t_2)] - {\mathcal{E}}[\Sigma(t_1)] = - \int_{\mathcal{H}^+_{12}}\boldsymbol{\omega}' - \int_{\mathcal{I}^+_{12}}\hat{\boldsymbol{\omega}}' \, . \label{eq:modified95canonical95energy95change}\tag{22}\] In order to calculate the contribution of \(\boldsymbol{\omega}'(g;\delta g,\mathcal{L}_\xi\delta g)\) on \(\mathcal{I}^+_{12}\), we need to specify the geometric structure near \(\mathcal{I}^+_{12}\). For this purpose, we choose the Bondi gauge for the physical background metric \(g_{ab}\). The corresponding unphysical background metric \(\hat{g}_{ab}=\Omega^2 g_{ab}\), where the conformal factor is \(\Omega=1/r\), then admits, in a neighborhood of \(\mathcal{I}^+\), the asymptotic form \[\hat{g}_{ab} = (d\hat{\Omega})_a (d\hat{u})_b + (d\hat{\Omega})_b (d\hat{u})_a + \hat{\gamma}_{ij} (dx^i)_a (dx^j)_b + \mathcal{O}(\Omega) \, . \label{eq:bondi95gauge95unphysical95metric}\tag{23}\] Here \(\hat{u}\) is a future directed affine parameter on the null geodesic generators of \(\mathcal{I}^+\). On \(\mathcal{I}^+\), the generator is given by \(\hat{n}^a\equiv(\partial/\partial\hat{u})^a\), while \(\hat{\gamma}_{ab}\) denotes the unit round metric on the \(S^{n-2}\) cross sections of \(\mathcal{I}^+\) and obeys \(\hat{\gamma}_{ab}\hat{n}^a=0\) and \(\hat{\gamma}_{ab}(\partial/\partial\Omega)^a=0\). The Killing field \(\xi^a\) can be continued smoothly to \(\mathcal{I}^+\) as a vector field \(\hat{\xi}^a\) parallel to \(\hat{n}^a\); equivalently, \[\hat{\xi}^a = (\hat{\xi}^b \hat{\nabla}_b \hat{u})\hat{n}^a \, , \label{eq:timelike95killing95field95at95null95infinity}\tag{24}\] where \(\hat{\xi}^b\hat{\nabla}_b\hat{u}\) is positive and constant on \(\mathcal{I}^+\) [24].

In the following, we carry out the calculation by taking Einstein gravity as an example. We first evaluate the contribution from the horizon portion \(\mathcal{H}^+_{12}\) in Eq. 22 . Using Eq. (21) of [24], we find that \(\boldsymbol{\omega}(g,\delta g,\mathcal{L}_\xi \delta g)\) is given by \[\boldsymbol{\omega}(g,\delta g,\mathcal{L}_\xi \delta g) = \frac{1}{16\pi G} \, w^a \, \boldsymbol{\epsilon}_a^{(n)} \, ,\] where \[\begin{align} w^a &= P^{abcdef}\Bigl( \mathcal{L}_\xi \delta g_{bc}\,\nabla_d \delta g_{ef} - \delta g_{bc}\,\nabla_d \mathcal{L}_\xi \delta g_{ef} \Bigr) \, , \end{align}\] and \[P^{abcdef} = g^{ae}g^{bf}g^{cd} - \frac{1}{2} g^{ad}g^{be}g^{cf} - \frac{1}{2} g^{ab}g^{cd}g^{ef} - \frac{1}{2} g^{bc}g^{ae}g^{df} + \frac{1}{2} g^{bc}g^{ad}g^{ef} \, .\] Using the stationarity condition on the background together with the perturbed metric in Eq. 2 , one can show that \[\boldsymbol{\omega}(g,\delta g,\mathcal{L}_\xi \delta g) \overset{\mathcal{H}^+}{=} \frac{\kappa}{32\pi G} \, \boldsymbol{\epsilon}^{(n-1)} \left\{(\delta \gamma^{ab} + \gamma^{ab}\gamma^{cd}\delta \gamma_{cd})\partial_v(v\partial_v \delta \gamma_{ab}) - v\left[\partial_v\delta\gamma^{ab}\partial_v\delta\gamma_{ab} + 4(\delta\theta_v)^2\right]\right\} \, . \label{eq:omega95on95horizon}\tag{25}\] Here \(\boldsymbol{\epsilon}^{(n-1)}\) is the volume \((n-1)\)-form on \(\mathcal{H}^+\). Finally, for the last term in Eq. 20 , using 16 , its second-order perturbative expansion simplifies to \[\xi^a\delta^2\boldsymbol{C}_a(g) \overset{\mathcal{H}^+}{=} \kappa v[\delta \boldsymbol{\epsilon}^{(n-1)} 2 k^a k^b\delta T_{ab} + \boldsymbol{\epsilon}^{(n-1)}k^a k^b\delta^2T_{ab}] \, . \label{eq:constraint95term95on95horizon}\tag{26}\] Here \(\delta T_{ab}\) and \(\delta^2 T_{ab}\) the external infalling matter fields. They are determined by the first- and second-order perturbations of the gravitational equations of motion, \[2\delta (E_G)_{ab}=\delta T_{ab} \, , \qquad 2\delta^2 (E_G)_{ab}=\delta^2 T_{ab} \, .\] In order to evaluate the horizon contribution appearing in Eq. 25 , it is necessary to analyze the perturbations of the geometric quantities on the event horizon. First, we need to calculate the contribution from the shear tensor. The shear tensor is defined as \[\sigma_{ab} = \frac{1}{2} \mathcal{L}_k \gamma_{ab} - \frac{1}{n-2} \gamma_{ab} \theta_v \, .\] Expanding the Lie derivative along the \(k\) direction and using \(g_{ab} \overset{\mathcal{H}^+}{=}-2k_{(a}l_{b)} + \gamma_{ab}\), one obtains \[\sigma_{ab} = \frac{1}{2} \partial_v \gamma_{ab} - \frac{1}{n-2} \gamma_{ab} \theta_v \, .\] Here and in the following calculations, \(\partial_v\) denotes \(k^a\partial_a\), while \(\partial_u\) denotes \(l^a\partial_a\). After first-order perturbation and using the zeroth-order \(\theta_v = 0\), one can find that \[\delta \sigma_{ab} = \frac{1}{2} \partial_v \delta \gamma_{ab} - \frac{1}{n-2} \gamma_{ab} \delta \theta_v \, , \qquad \delta \sigma^{ab} = - \frac{1}{2} \partial_v \delta \gamma^{ab} - \frac{1}{n-2} \gamma^{ab} \delta \theta_v \, .\] Using \(\delta \theta_v = \frac{1}{2} \gamma^{ab} \partial_v \delta \gamma_{ab}\), the contraction of the first-order shear perturbation can be simplified as \[\delta \sigma_{ab} \, \delta \sigma^{ab} \overset{\mathcal{H}^+}{=} - \frac{1}{4} \partial_v \delta \gamma^{ab} \partial_v \delta \gamma_{ab} - \frac{( \delta \theta_v )^2}{ n-2 } \, . \label{eq:first95order95shear95contraction}\tag{27}\] Second, we need to calculate the contribution from the twist tensor. The twist tensor is defined as \[\omega_{ab} = \gamma_a{}^c\gamma_b{}^d \nabla_{[c}k_{d]} \, ,\] where the projection operator \(\gamma_a{}^b \overset{\mathcal{H}^+}{=} \delta_a{}^b + k_al^b + l_ak^b\). In the stationary case, we have \[\omega_{ab} = \gamma_a{}^c\gamma_b{}^d \partial_{[c} k_{d]} \overset{\mathcal{H}^+}{=} 0 \, , \qquad \omega^{ab} = g^{ac}g^{bd}\omega_{cd} \overset{\mathcal{H}^+}{=} 0 \, .\] Using \(\partial_{[c} k_{d]} = 0\), the variation of the twist tensor is \[\delta\omega_{ab} = \gamma_a{}^c \gamma_b{}^d \delta\!\left(\partial_{[c} k_{d]}\right) \overset{\mathcal{H}^+}{=} 0 \, , \qquad \delta\omega^{ab} = g^{ac}g^{bd}\delta\omega_{cd} \overset{\mathcal{H}^+}{=} 0 \, .\] Finally, to obtain the relation between the relevant geometric quantities and the expansion, we introduce the Raychaudhuri equation for the null geodesic congruence generated by \(k^a\): \[\frac{d\theta_v}{dv} = -\frac{1}{n-2}\,\theta_v^2 - \sigma^{ab}\sigma_{ab} + \omega^{ab}\omega_{ab} - R_{ab}k^a k^b \, . \label{eq:raychaudhuri95horizon}\tag{28}\] Here \(\sigma_{ab}\) and \(\omega_{ab}\) are the shear and twist of the null congruence. Notice that \(\omega_{ab}\overset{\mathcal{H}^+}{=}0\) since \(k^a\) is hypersurface orthogonal on \(\mathcal{H}^+\), and according to our calculation of the twist tensor, we also have \(\delta\omega_{ab}\overset{\mathcal{H}^+}{=}0\). Moreover, \(\theta_v,\sigma_{ab}\) vanishes in the background. Furthermore, according to the gauge conditions, the perturbation of the Einstein equation satisfies \(\delta R_{ab} k^a k^b = 8\pi G\, \delta T'_{ab} k^a k^b\) and \(\delta^2 R_{ab} k^a k^b = 8\pi G\, \delta^2 T'_{ab} k^a k^b\). Here \(T'_{ab}\) denotes the total stress-energy tensor, including the contributions from all matter fields. Thus, the first-order perturbation and the second-order perturbation of the Raychaudhuri equation give \[\frac{d\delta\theta_v}{dv} \overset{\mathcal{H}^+}{=} - 8\pi G\, \delta T'_{ab} k^a k^b \, , \qquad \frac{d\delta^2\theta_v}{dv} \overset{\mathcal{H}^+}{=} \frac{1}{2} \partial_v \delta \gamma^{ab} \partial_v \delta \gamma_{ab} - 8\pi G \delta^2 T'_{ab} k^a k^b \, . \label{eq:first95second95order95raychaudhuri}\tag{29}\] Since we only consider contributions from external matter fields, \(T'_{ab}\) is equivalent to \(T_{ab}\) in the following analysis. Combining Eqs. 25 , 26 , and 27 , then integrating over \(\mathcal{H}^+_{12}\), we obtain the contribution of \(\boldsymbol{\omega}'\) on \(\mathcal{H}^+\). \[\begin{align} \int_{\mathcal{H}^+_{12}}\boldsymbol{\omega}' =& -\frac{1}{16\pi G}\int_{\mathcal{H}^+_{12}}\boldsymbol{\epsilon}^{(n-1)}(\xi^c\nabla_c v) \left[\delta\sigma^{ab}\delta\sigma_{ab} - \frac{n-2}{n-3}(\delta\theta_v)^2\right] \\ &+ \frac{1}{32\pi G}\Delta \int_{\mathcal{C}}\boldsymbol{\epsilon}^{(n-2)}(\xi^c\nabla_c v)(\delta \gamma^{ab}\partial_v \delta \gamma_{ab} + 2\delta \theta_v \gamma^{ab}\delta \gamma_{ab}) \\ &+ \int_{\mathcal{H}^+_{12}}(\xi^c\nabla_c v)[\delta \boldsymbol{\epsilon}^{(n-1)} 2 k^a k^b\delta T_{ab} + \boldsymbol{\epsilon}^{(n-1)}k^a k^b\delta^2 T_{ab}] \, . \end{align} \label{fluxathorizon1}\tag{30}\] The symbol \(\Delta\) denotes the difference of a quantity evaluated between the two cross-sections \(\mathcal{H}^+_1\) and \(\mathcal{H}^+_2\).

Similarly, in close analogy with the calculation on \(\mathcal{H}^+\), we evaluate the contribution from the null infinity portion \(\mathcal{I}^+_{12}\) to the variation of \(\mathcal{E}[\Sigma(t)]\). At this stage, the corresponding quantities in Eq. 20 are replaced by their counterparts in the conformally transformed unphysical spacetime. For the term \(\hat{\boldsymbol{\omega}}(\hat{g},\delta\hat{g},\mathcal{L}_{\hat{\xi}}\delta\hat{g})\), its contribution, after pullback to \(\mathcal{I}^+\), is determined by the symplectic current [27] \[\hat{\boldsymbol{\omega}}(\hat{g},\delta\hat{g},\mathcal{L}_{\hat{\xi}}\delta\hat{g}) \overset{\mathcal{I}^+}{=} \delta \hat{\boldsymbol{\Theta}}'(\hat{g},\mathcal{L}_{\hat{\xi}}\delta\hat{g}) - \mathcal{L}_{\hat{\xi}}\delta \hat{\boldsymbol{\Theta}}'(\hat{g},\delta\hat{g}) \, , \label{eq:symplectic95current95definition95null95infinity}\tag{31}\] where \[\hat{\boldsymbol{\Theta}}'(\hat{g},\delta\hat{g}) = -\frac{1}{32\pi G}\hat{N}^{ab}\delta\hat{g}_{ab}\hat{\boldsymbol{\epsilon}}^{(n-1)} \, . \label{eq:theta95prime95null95infinity}\tag{32}\] Here \(\hat{N}_{ab}\) is the Bondi news tensor, defined by \[\begin{align} \hat{N}_{ab} ={}& \hat{\gamma}_a{}^{c}\hat{\gamma}_b{}^{d}\,\Omega^{-\frac{n-4}{2}} \left[ \frac{2}{n-2}\hat{R}_{cd} - \frac{1}{(n-1)(n-2)}\hat{R}\,\hat{g}_{cd} \right] \\ &- \frac{1}{n-2}\hat{\gamma}_{ab}\hat{g}^{ef}\hat{\gamma}_e{}^{c}\hat{\gamma}_f{}^{d}\, \Omega^{-\frac{n-4}{2}} \left[ \frac{2}{n-2}\hat{R}_{cd} - \frac{1}{(n-1)(n-2)}\hat{R}\,\hat{g}_{cd} \right] \, . \end{align} \label{eq:bondi95news95tensor95null95infinity}\tag{33}\] Here, \(\hat{R}_{ab}\) and \(\hat{R}\) are the Ricci tensor and Ricci scalar associated with the unphysical metric \(\hat{g}_{ab}\), respectively. Substituting Eqs. 32 and 33 into Eq. 31 , together with the fact that \(\hat{N}_{ab}=0\) on the stationary background and \[\delta\hat{N}_{ab}=-\mathcal{L}_{\hat{n}}\delta\hat{g}_{ab}+O(\Omega)\, ,\] we obtain \[\hat{\boldsymbol{\omega}}(\hat{g},\delta \hat{g},\mathcal{L}_{\hat{\xi}}\delta \hat{g}) \overset{\mathcal{I}^+}{=}\frac{1}{32\pi G}\hat{\boldsymbol{\epsilon}}^{(n-1)}\left(\mathcal{L}_{\hat{\xi}}\delta \hat{g}_{ab}\,\mathcal{L}_{\hat{n}}\delta \hat{g}^{ab} - \delta \hat{g}_{ab}\,\mathcal{L}_{\hat{\xi}} \mathcal{L}_{\hat{n}}\delta \hat{g}_{ab}\right) \, . \label{eq:null95infinity95symplectic95current95news}\tag{34}\] Here \(\hat{\boldsymbol{\epsilon}}^{(n-1)}\) is the volume \((n-1)\)-form on \(\mathcal{I}^+\). Similarly, as on \(\mathcal{H}^+\), the second and the third term in Eq. 20 vanish when pulled back to \(\mathcal{I}^+\). Finally, we consider the contribution from the last term of Eq. 20 . We now explain why the corresponding integral over future null infinity does not contribute. In the present setup, the perturbation is produced by external matter fields falling into the black hole through the future event horizon \(\mathcal{H}^+\). Consequently, the corresponding stress-energy perturbations are localized near the black hole and do not give rise to outgoing matter flux through future null infinity \(\mathcal{I}^+\). Under the standard asymptotic flatness conditions, the matter stress tensor decays sufficiently rapidly toward null infinity, so that \(\delta\hat{T}_{ab}\to0\) and \(\delta^2\hat{T}_{ab}\to0\) at \(\mathcal{I}^+\). As a result, the pullback of the last term of Eq. 20 to \(\mathcal{I}^+\) vanishes. Thus, the last term of Eq. 20 contributes only on the horizon \(\mathcal{H}^+\), while the flux through \(\mathcal{I}^+\) is entirely determined by the radiative gravitational and matter degrees of freedom appearing in the symplectic current. According to Eq. 34 and integrating over \(\mathcal{I}^+_{12}\), we obtain the contribution of \(\boldsymbol{\omega}'\) on \(\mathcal{I}^+\). \[\int_{\mathcal{I}^+_{12}}\hat{\boldsymbol{\omega}}' = \frac{1}{16\pi G}\int_{\mathcal{I}^+_{12}}\hat{\boldsymbol{\epsilon}}^{(n-1)}(\hat{\xi}^c \hat{\nabla}_c \hat{u}) \delta \hat{N}^{ab}\delta \hat{N}_{ab} - \frac{1}{32\pi G}\Delta\int_{\mathcal{\hat{C}}}\hat{\boldsymbol{\epsilon}}^{(n-2)}(\hat{\xi}^c\hat{\nabla}_c\hat{u})\delta\hat{\gamma}^{ab}\delta\hat{N}_{ab} \, . \label{fluxatinfinity}\tag{35}\] The symbol \(\Delta\) denotes the difference of a quantity evaluated between the two cross-sections \(\mathcal{I}^+_1\) and \(\mathcal{I}^+_2\).

From the calculations above, we see that the change of \(\mathcal{E}[\Sigma(t)]\) not only depends on the fluxes through \(\mathcal{H}^+_{12}\) and \(\mathcal{I}^+_{12}\), but also on two boundary terms. This motivates us to subtract these boundary terms and define a modified canonical energy \(\mathcal{E}'[\Sigma(t)]\). \[\mathcal{E}'[\Sigma(t)] \equiv \mathcal{E}[\Sigma(t)] + \frac{1}{32\pi G}\left[\int_{\mathcal{C}}\boldsymbol{\epsilon}^{(n-2)}(\xi^c\nabla_c v)(\delta \gamma^{ab}\partial_v \delta \gamma_{ab} + 2\delta \theta_v \gamma^{ab}\delta \gamma_{ab}) - \int_{\mathcal{\hat{C}}}\hat{\boldsymbol{\epsilon}}^{(n-2)}(\hat{\xi}^c\hat{\nabla}_c\hat{u})\delta\hat{\gamma}^{ab}\delta\hat{N}_{ab}\right] \, . \label{eq:modified95canonical95energy95prime95definition}\tag{36}\] Consequently, the change of the modified canonical energy \(\mathcal{E}'[\Sigma(t)]\) between two Cauchy surfaces \(\Sigma(t_2)\) and \(\Sigma(t_1)\) is given by \[\begin{align} {\mathcal{E}'}[\Sigma(t_2)] - {\mathcal{E}'}[\Sigma(t_1)] ={}& -\frac{1}{16\pi G}\int_{\mathcal{H}^+_{12}}\boldsymbol{\epsilon}^{(n-1)}(\xi^c\nabla_c v)\left[\delta\sigma^{ab}\delta\sigma_{ab} - \frac{n-2}{n-3}(\delta\theta_v)^2\right] \\ &- \int_{\mathcal{H}^+_{12}}(\xi^c\nabla_c v)[\delta \boldsymbol{\epsilon}^{(n-1)} 2 k^a k^b\delta T_{ab} + \boldsymbol{\epsilon}^{(n-1)}k^a k^b\delta^2T_{ab}] \\ &- \frac{1}{16\pi G}\int_{\mathcal{I}^+_{12}}\hat{\boldsymbol{\epsilon}}^{(n-1)}(\hat{\xi}^c \hat{\nabla}_c \hat{u}) \delta \hat{N}^{ab}\delta \hat{N}_{ab} \, . \label{eq:canonical95energy95entropy95relation} \end{align}\tag{37}\] It follows that the change of the modified canonical energy \(\mathcal{E}'[\Sigma(t)]\) is totally determined by the fluxes on the horizon and null infinity, which is in sharp constract to the canonical energy \(\mathcal{E}[\Sigma(t)]\).

4 Dynamical black hole entropy↩︎

In this section, we combine the geometric results with the modified canonical energy to compare the entropy with the apparent-horizon area at second order. After evaluating the relevant boundary and flux terms, we use the Raychaudhuri equation and the shear contribution to simplify the entropy variation.

Our result shows that, at second order, the entropy remains proportional to the area of the apparent horizon under the null energy condition, whereas additional conditions are required to establish the same proportionality without the null energy condition.

4.1 Derivation of the Balance Law↩︎

In the following, we show that the two boundary terms in Eq.(36 ) can be canceled by the integrals of two exact forms, i.e., \[d\!\left[\xi \cdot \delta \boldsymbol{\Theta}(g,\delta g) - \xi \cdot \delta^2 \boldsymbol{B}_{\mathcal{H}^+}(g)\right]\, , \label{exactformhorizon}\tag{38}\] and \[d\!\left[\hat{\xi} \cdot \delta \hat{\boldsymbol{\Theta}}'(\hat{g},\delta\hat{g}) - \hat{\xi} \cdot \delta^2 \hat{\boldsymbol{B}}_{\mathcal{I}^+}(\hat{g})\right] \label{exactforminfinity}\tag{39}\] on \(\mathcal{H}^+_{12}\) and \(\mathcal{I}^+_{12}\), respectively. Here \(\boldsymbol{B}_{\mathcal{H}^+}(g)\) is defined by \[\underline{\boldsymbol{\Theta}}(g,\delta g)\overset{\mathcal{H}^+}{=}\delta\boldsymbol{B}_{\mathcal{H}^+}(g) \, \label{eq:horizon95counterterm95definition}\tag{40}\] where \(\underline{\boldsymbol{\Theta}}(g,\delta g)\) denotes the pullback of \(\boldsymbol{\Theta}(g,\delta g)\) to the event horizon. This equation shows that, for a stationary background, there exists a quantity \(\boldsymbol{B}_{\mathcal{H}^+}(g)\) on \(\mathcal{H}^+\) such that \(\boldsymbol{\Theta}(g,\delta g)\) is exact in field space when pulled back to \(\mathcal{H}^+\), with \(\boldsymbol{B}_{\mathcal{H}^+}(g)\overset{\mathcal{H}^+}{=}0\) on the stationary background (see Sec. 4.7 of Ref. [12]). Moreover, \(\hat{\boldsymbol{B}}_{\mathcal{I}^+}(\hat{g})\) and \(\hat{\boldsymbol{\Theta}}^{\prime}(\hat{g},\delta\hat{g})\) are given by [27] \[\delta \hat{\boldsymbol{B}}_{\mathcal{I}^+}(\hat{g}) \overset{\mathcal{I}^+}{=} \hat{\boldsymbol{\Theta}}(\hat{g},\delta\hat{g}) - \hat{\boldsymbol{\Theta}}'(\hat{g},\delta\hat{g}) \, . \label{eq:null95infinity95counterterm95definition}\tag{41}\] Here \(\hat{\boldsymbol{\Theta}}'(\hat{g},\delta\hat{g})\) denotes the symplectic potential associated with the pullback of \(\hat{\boldsymbol{\omega}}(\hat{g},\delta\hat{g},\mathcal{L}_{\hat{\xi}}\delta\hat{g})\) to \(\mathcal{I}^+\).

We first evaluate the contribution from the term (38 ). Here the symplectic potential is given by [12] \[\boldsymbol{\Theta}(g,\delta g) = \frac{1}{16\pi G} \, \boldsymbol{\epsilon}_a^{(n)} g^{ab} g^{cd} \left(\nabla_c \delta g_{bd} - \nabla_b \delta g_{cd}\right) ,\] and \[\boldsymbol{B}_{\mathcal{H}^+}(g) = \frac{1}{16\pi G} \, l^a \boldsymbol{\epsilon}_{a}^{(n)} g^{bc} \mathcal{L}_k g_{bc} = \frac{1}{8\pi G} \, \boldsymbol{\epsilon}^{(n-1)}\, \theta_v \, . \label{eq:BH95boundary95term95definition}\tag{42}\] Using the gauge conditions together with the stationary condition on the background, one finds that \[\delta\boldsymbol{\Theta}(g,\delta g) = \frac{1}{16\pi G} \, \boldsymbol{\epsilon}^{(n-1)}(2\delta^2 \theta_v + \gamma^{ab} \delta\gamma_{ab} \delta\theta_v - \delta^2\gamma^{ab} \partial_v\gamma_{ab} - \frac{1}{2}\delta\gamma^{ab} \partial_v\delta\gamma_{ab}) \, , \label{eq:delta95theta95on95horizon}\tag{43}\] and \[\delta^2\boldsymbol{B}_{\mathcal{H}^+}(g) = \frac{1}{8\pi G} \, \boldsymbol{\epsilon}^{(n-1)}(\delta^2\theta_v + \gamma^{ab} \delta\gamma_{ab} \delta\theta_v) \, . \label{eq:delta295BH95on95horizon}\tag{44}\] Notice that for an \((n-1)\)-form \({\boldsymbol{p}}\), the pullback of \(\xi \cdot d{\boldsymbol{p}}\) to the horizon vanishes. Using Eq. 13 to simplify the expression, we obtain the following result. \[\begin{align} d\!\left[\xi \cdot \delta \boldsymbol{\Theta}(g,\delta g) - \xi \cdot \delta^2 \boldsymbol{B}_{\mathcal{H}^+}(g)\right] \overset{\mathcal{H}^+}{=}& - \frac{\kappa}{32\pi G}\boldsymbol{\epsilon}^{(n-1)}(2\gamma^{ab}\delta \gamma_{ab}\delta \theta_v + \delta \gamma^{ab}\partial_v \delta \gamma_{ab}) \\ &- \frac{\kappa}{32\pi G}\boldsymbol{\epsilon}^{(n-1)} v\partial_v(2\gamma^{ab}\delta \gamma_{ab}\delta \theta_v + 2\delta^2 \gamma^{ab}\partial_v \gamma_{ab} + \delta \gamma^{ab}\partial_v \delta \gamma_{ab}) \, . \end{align} \label{eq:boundary95term95on95horizon}\tag{45}\] Integrating this expression over \(\mathcal{H}^+_{12}\) gives the corresponding boundary contribution on the horizon. \[\int_{\mathcal{H}^+_{12}}d\!\left[\xi \cdot \delta \boldsymbol{\Theta}(g,\delta g) - \xi \cdot \delta^2 \boldsymbol{B}_{\mathcal{H}^+}(g)\right] = -\frac{1}{32\pi G}\Delta \int_{\mathcal{C}}\boldsymbol{\epsilon}^{(n-2)}(\xi^c\nabla_c v)(\delta \gamma^{ab}\partial_v \delta \gamma_{ab} + 2\delta \theta_v \gamma^{ab}\delta \gamma_{ab}) \, . \label{eq:horizon95boundary95integral}\tag{46}\] We then calculate the contribution from the term in Eq.(39 ). Substituting Eq. 41 into this term and noting that, for an \((n-1)\)-form \(\hat{\boldsymbol{p}}\), the pullback of \(\hat{\xi}\cdot d\hat{\boldsymbol{p}}\) to \(\mathcal{I}^+\) vanishes, together with Eq. 13 , one obtains \[d\!\left[\hat{\xi} \cdot \delta \hat{\boldsymbol{\Theta}}(\hat{g},\delta \hat{g}) - \hat{\xi} \cdot \delta^2 \hat{\boldsymbol{B}}_{\mathcal{I}^+}(\hat{g})\right] \overset{\mathcal{I}^+}{=}-\frac{1}{32\pi G}\hat{\boldsymbol{\epsilon}}^{(n-1)}\hat{\xi}^c \partial_c(\delta \hat{g}_{ab}\delta \hat{N}^{ab}) \, . \label{eq:boundary95term95null95infinity}\tag{47}\] Integrating this expression over \(\mathcal{I}^+_{12}\) gives the corresponding boundary contribution at future null infinity. \[\int_{\mathcal{I}^+_{12}}d\!\left[\hat{\xi} \cdot \delta \hat{\boldsymbol{\Theta}}(\hat{g},\delta \hat{g}) - \hat{\xi} \cdot \delta^2 \hat{\boldsymbol{B}}_{\mathcal{I}^+}(\hat{g})\right] = \frac{1}{32\pi G}\Delta\int_{\mathcal{C}}\hat{\boldsymbol{\epsilon}}^{(n-2)}(\hat{\xi}^c\hat{\nabla}_c\hat{u})\delta\hat{\gamma}^{ab}\delta\hat{N}_{ab} \, . \label{eq:null95infinity95boundary95integral}\tag{48}\] Comparing Eqs. 46 and 48 with Eq. 36 , we see that the boundary terms arising from difference of the last two terms in Eq. 36 between \(\Sigma(t_1)\) and \(\Sigma(t_2)\), except for opposite signs, are precisely given by the integrals of the exact forms (38 ) and (39 ) over \(\mathcal{H}^+_{12}\) and \(\mathcal{I}^+_{12}\), respectively.

Based on this and according to Eq. 20 , we can construct the corresponding “modified canonical energy \((n-1)\)-forms" \(\boldsymbol{e}_G\) on \(\mathcal{H}^+\) \[\boldsymbol{e}_G \overset{\mathcal{H}^+}{=} \boldsymbol{\omega}'(g,\delta g,\mathcal{L}_\xi \delta g) + d\!\left[\xi \cdot \delta \boldsymbol{\Theta}(g,\delta g) - \xi \cdot \delta^2 \boldsymbol{B}_{\mathcal{H}^+}(g)\right] \, , \label{eq:modified95canonical95energy95definition}\tag{49}\] and \(\hat{\boldsymbol{e}}_G\) on \(\mathcal{I}^+\) \[\hat{\boldsymbol{e}}_G \overset{\mathcal{I}^+}{=} \hat{\boldsymbol{\omega}}'(\hat{g},\delta \hat{g},\mathcal{L}_{\hat{\xi}} \delta \hat{g}) + d\!\left[\hat{\xi} \cdot \delta \hat{\boldsymbol{\Theta}}'(\hat{g},\delta \hat{g}) - \hat{\xi} \cdot \delta^2 \hat{\boldsymbol{B}}_{\mathcal{I}^+}(\hat{g})\right] \, .\] So Eq.(37 ) can be written as \[\mathcal{E}'[\Sigma(t_2)]-\mathcal{E}'[\Sigma(t_1)]=-\int_{\mathcal{H}^+_{12}}\boldsymbol{e}_G - \int_{\mathcal{I}^+_{12}}\hat{\boldsymbol{e}}_G\, .\] Considering the fact that the second and third terms in Eq. 20 vanish on \(\mathcal{H}^+_{12}\) and \(\mathcal{I}^+_{12}\), and substituting the identity 19 , we find \[\boldsymbol{e}_G \overset{\mathcal{H}^+}{=} d\!\left[\delta_{\phi}^2 \boldsymbol{Q}_\xi(g) - \xi \cdot \delta^2 \boldsymbol{B}_{\mathcal{H}^+}(g)\right] \, , \label{eq:modified95canonical95energy95definition1}\tag{50}\] and \[\hat{\boldsymbol{e}}_G \overset{\mathcal{I}^+}{=} d\!\left[\delta_{\phi}^2 \hat{\boldsymbol{Q}}_{\hat{\xi}}(\hat{g}) - \hat{\xi} \cdot \delta^2 \hat{\boldsymbol{B}}_{\mathcal{H}^+}(\hat{g})\right] \, .\] Based on the first law of black hole mechanics and some innovative thermodynamic consideration, the authors in [12] introduce a entropy \((n-2)\)-form \[\boldsymbol{S} \equiv \frac{2\pi}{\kappa_3}\left[\boldsymbol{Q}_\xi(g) - \xi \cdot \boldsymbol{B}_{\mathcal{H}^+}(g)\right] \, ,\] where \(\kappa_3\) is the surface gravity defined as \[2\kappa_3^2=-(\nabla_{[a}\xi_{b]})(\nabla^{[a}\xi^{b]})\, .\] The integral of \(\boldsymbol{S}\) is nothing but the entropy of the system. Below, we will present the detailed relation between the modified canonical energy and the entropy.

To obtain the variation of entropy on \(\mathcal{H}^+\), we need the entropy \((n-1)\)-form on \(\mathcal{H}^+\), namely \[d\boldsymbol{S} = 2\pi d\left[\frac{\boldsymbol{Q}_\xi(g)}{\kappa_3} - \frac{\xi \cdot \boldsymbol{B}_{\mathcal{H}^+}(g)}{\kappa_3}\right] \, . \label{eq:entropy95n95minus95one95form}\tag{51}\] Now, we will show that, under second-order perturbations, the entropy depends only on the variation of the fields and not on the variation of \(\xi^a\). For general differeomorphsim covariance theories, the Noether charge \((n-2)\)-form \(\boldsymbol{Q}_{\xi}(g)\) has been thoroughly studied in [11]. However, for Einstein gravity, we simply have \[\boldsymbol{Q}_{\xi}(g)= -\frac{1}{16\pi G} \boldsymbol{\epsilon}^{ab}{}_{a_1\cdots a_{n-2}}\nabla_a\xi_b\, ,\] or \[\boldsymbol{Q}_{\xi}(g)= -\frac{1}{16\pi G} \boldsymbol{\epsilon}^{(n-2)}(\boldsymbol{\epsilon}^{ab}\nabla_a\xi_b)\, ,\] where \(\boldsymbol{\epsilon}_{ab}=k_al_b-l_ak_b\) is the so-called binormal when restricted on the event horizon. It is not hard to find  [11], [21] \[\boldsymbol{\epsilon}^{ab}\nabla_a\xi_b=(k^al^b-k^bl^a)\nabla_a\xi_b=k^a\nabla_a(l^b\xi_b) - l^a\nabla_a (k^b\xi_b)= -\kappa_2 - \kappa_1=-2\kappa_3\, ,\] here we have used the fact that \(l^a\) and \(k^a\) commute to each other, and the definitions of the surface gravities \(\kappa_1\), \(\kappa_2\) \[\nabla_a(\xi_b\xi^b)=-2\kappa_1\xi_a\, ,\qquad \xi^b\nabla_b\xi^a=\kappa_2\xi^a\,\] These can be equivalently expressed as \[\kappa_2=-k^a\nabla_a(l^b\xi_b)\, ,\qquad \kappa_1=l^a\nabla_a(k^b\xi_b)\, ,\] From the definition, it is easy to find \(\kappa_3=(\kappa_1+\kappa_2)/2\), and also on the stationary background, the surface gravities satisfies \(\kappa_2=\kappa_3=\kappa\). So, on the event horizon, we have \[\boldsymbol{Q}_{\xi}(g)= \frac{\kappa_3}{8\pi G} \boldsymbol{\epsilon}^{(n-2)}\, . \label{eq:noether95charge95horizon95slice}\tag{52}\] Moreover, on the horizon \(\mathcal{H}^+\), the variation of surface gravity \(\delta\kappa_2\) and \(\delta\kappa_3\) satisfy (see Appendix A of Ref. [21]) \[\delta\kappa_2\overset{\mathcal{H}^+}{=}-k^a\nabla_a(l_b\delta\xi^b) \, , \qquad \delta\kappa_3\overset{\mathcal{H}^+}{=}l_{[a}k_{b]}\nabla^a\delta\xi^b \, . \label{eq:delta95kappa295delta95kappa395definitions}\tag{53}\] From the gauge condition, one obtains \[\delta\xi^a\overset{\mathcal{H}^+}{=}\left(\int_0^v\delta\kappa_2\,dv'\right)k^a\, . \label{deltaxiingetral}\tag{54}\] Next, we derive the full second-order variation of the two terms in Eq. 51 . For the first term, we have \[\begin{align} \delta^2 \left[\frac{\boldsymbol{Q}_\xi(g)}{\kappa_3}\right] = \frac{1}{8\pi G}\delta_\phi^2\boldsymbol{\epsilon}^{(n-2)} = \frac{1}{\kappa}\delta^2_{\phi}\boldsymbol{Q}_\xi(g) \, . \end{align} \label{eq:second95variation95noether95charge95over95kappa}\tag{55}\] Similarly, for the second term, we have \[\delta^2\left[\frac{\xi\cdot\boldsymbol{B}_{\mathcal{H}^+}(g)}{\kappa_3}\right] \overset{\mathcal{H}^+}{=} \frac{1}{\kappa}\xi\cdot\delta^2_{\phi}\boldsymbol{B}_{\mathcal{H}^+}(g) + \frac{2}{\kappa}\left[\delta\xi\cdot\delta\boldsymbol{B}_{\mathcal{H}^+}(g) - \frac{\delta\kappa_3}{\kappa}\,\xi\cdot\delta\boldsymbol{B}_{\mathcal{H}^+}(g) \right] \, . \label{eq:second95variation95boundary95term95over95kappa}\tag{56}\] Combining Eqs. 51 , 55 , and 56 , we then find that \[\begin{align} \frac{\kappa}{2\pi}d\delta^2\boldsymbol{S}&\overset{\mathcal{H}^+}{=}& d\!\left[\delta_{\phi}^2 \boldsymbol{Q}_\xi(g) - \xi \cdot \delta^2 \boldsymbol{B}_{\mathcal{H}^+}(g)\right] + 2d\left[\delta\xi\cdot\delta\boldsymbol{B}_{\mathcal{H}^+}(g) - \frac{\delta\kappa_3}{\kappa}\xi\cdot\delta\boldsymbol{B}_{\mathcal{H}^+}(g)\right] \, . \label{eq:second95order95entropy95form95horizon} \end{align}\tag{57}\] When \(\delta\xi^a=0\) and only the variation of \(g\) is involved, combining Eqs. 37 , 49 , and 57 , then pulling the result back to \(\mathcal{H}^+\), we obtain \[\frac{\kappa}{2\pi} \Delta\delta_{\phi}^2S=\int_{\mathcal{H}^+_{12}}\boldsymbol{e}_G \, . \label{eq:canonical95energy95dynamical95entropy95relation95old}\tag{58}\] We see that, under the second-order perturbations, the decrease of the modified canonical energy on \(\mathcal{H}^+\) gives the variation of the entropy. We now consider the case \(\delta\xi^a\neq0\). In other words, the variation involves not only the metric field \(g\) but also \(\xi^a\). Therefore, the entropy formula should be corrected by the additional terms arising from the variation of \(\xi^a\), and the corresponding relation becomes \[\frac{\kappa}{2\pi} \, \Delta \delta^2 S=\int_{\mathcal{H}^+_{12}}\boldsymbol{e}_G - \int_{\mathcal{H}^+_{12}}2d\left[\delta\xi\cdot\delta\boldsymbol{B}_{\mathcal{H}^+}(g) - \frac{\delta\kappa_3}{\kappa}\xi\cdot\delta\boldsymbol{B}_{\mathcal{H}^+}(g)\right] \, . \label{eq:canonical95energy95dynamical95entropy95relation}\tag{59}\] Eqs.(58 ) and (59 ) are the so-called “balance law" that we expect to obtain.

4.2 Relation to Apparent Horizon↩︎

Now, we will derive the explicit expression for the entropy by evaluating the modified canonical energy and the additional terms. According to Eqs. 37 , 49 , and 59 , it follows that \[\begin{align} \frac{\kappa}{2\pi} \, \Delta \delta^2 S =& \kappa\int_{v_1}^{v_2} dv \int_{\mathcal{C}}(\delta dA_{\epsilon}\, 2v k^a k^b\delta T_{ab} + dA_{\epsilon}\, v k^a k^b\delta^2T_{ab})\\ &- \frac{\kappa}{16\pi G}\int_{v_1}^{v_2} dv \int_{\mathcal{C}} dA_{\epsilon} v \left[\delta\sigma^{ab}\delta\sigma_{ab} - \frac{n-2}{n-3}(\delta\theta_v)^2\right] \\ &- \Delta\int_{\mathcal{C}}2 \left[\delta\xi\cdot\delta\boldsymbol{B}_{\mathcal{H}^+}(g) - \frac{\delta\kappa_3}{\kappa}\xi\cdot\delta\boldsymbol{B}_{\mathcal{H}^+}(g)\right] \, . \end{align}\] Substituting the explicit form of \(\delta \sigma_{ab} \delta \sigma^{ab}\) and Eqs. 27 , 29 and 42 , we find \[\begin{align} \frac{\kappa}{2\pi} \, \Delta \delta^2 S =& - \frac{\kappa}{4\pi G} \int_{v_1}^{v_2} dv \int_{\mathcal{C}} \delta dA_{\epsilon}\, v \frac{d \delta \theta_v}{dv} \\ & - \frac{\kappa}{4\pi G} \int_{v_1}^{v_2} dv \int_{\mathcal{C}} dA_{\epsilon}\left[\frac{1}{2}v \frac{d \delta^2 \theta_v}{dv} +v (\delta \theta_v)^2\right] \\ & + \frac{1}{4\pi G}\Delta\int_{\mathcal{C}}dA_{\epsilon} \left(v\delta\kappa_3 + \delta\xi\cdot l\right)\delta\theta_v \, . \end{align}\] After applying \(\partial_v\delta dA_{\epsilon} = \delta\theta_vdA_{\epsilon}\), performing integration by parts with respect to \(v\), and using the fact that, in general relativity, in the absence of matter fields, we obtain \[\begin{align} \frac{\kappa}{2\pi} \, \Delta \delta^2 S =& - \frac{\kappa}{4\pi G} \,\Delta \int_{\mathcal{C}} \delta dA_{\epsilon}\, v\, \delta \theta_v + \frac{\kappa}{4\pi G} \int_{v_1}^{v_2} dv \int_{\mathcal{C}} \delta dA_{\epsilon}\, \delta \theta_v \\ & - \frac{\kappa}{8\pi G} \,\Delta \int_{\mathcal{C}} dA_{\epsilon}\, v\, \delta^2{\theta}_v + \frac{\kappa}{8\pi G} \int_{v_1}^{v_2} dv \int_{\mathcal{C}} dA_{\epsilon}\, \delta^2 \theta_v \\ & + \frac{1}{4\pi G}\Delta\int_{\mathcal{C}}dA_{\epsilon}[\left(v\delta\kappa_3 - \delta h\right)\delta\theta_v] \,. \end{align}\label{Deltadelta2S}\tag{60}\] Integrating the first equation of Eq. 53 on \(\mathcal{H}^+\), one obtains Eq.(54 ). We then define \[\delta h \equiv \int_0^v \delta\kappa_2\,dv'\, .\] Together with \(\theta_vdA_{\epsilon} = \partial_vdA_{\epsilon}\) and \(\partial_v\delta dA_{\epsilon} = \delta\theta_vdA_{\epsilon}\), by exchanging the order of variation and integration, 60 reduces to \[\frac{\kappa}{2\pi} \, \Delta \delta^2 S = \frac{\kappa}{8 \pi G} \Delta \delta^2 \int_{\mathcal{C}} dA_{\epsilon}(1 - v\theta_v) + \frac{1}{4\pi G}\Delta\int_{\mathcal{C}}dA_{\epsilon}[\left(v\delta\kappa_3 - \delta h\right)\delta\theta_v] \,. \label{eq:canonical95energy95simplified}\tag{61}\] Replacing the modified canonical energy by the entropy through Eq. 59 gives \[\delta^2 S = \frac{1}{4G} \delta^2 \int_{\mathcal{C}} dA_{\epsilon}(1 - v\theta_v) + \frac{1}{2G\kappa}\int_{\mathcal{C}}dA_{\epsilon}[\left(v\delta\kappa_3 - \delta h\right)\delta\theta_v] \, . \label{eq:entropy95from95canonical95energy}\tag{62}\] We can now discuss Eq. 62 in two cases.

4.2.1 Case I↩︎

In this case, we assume that the null energy condition holds and show that \(\delta\theta_v=0\). In the following, we prove this statement using the method of  [28], together with the boundary conditions and a consequence of the null energy condition.

First, we assume that \(\delta\theta_v \to 0\) at late times along the event horizon, which means that the perturbed black hole eventually settles into a new stationary state. Since \(\theta_v=0\) for stationary configurations, the perturbation of the expansion must decay asymptotically; otherwise, a persistent nonzero expansion would lead to a continual growth or focusing of \(k^a\), preventing the spacetime from approaching equilibrium.

Second, consider a one-parameter family of perturbed solutions \(\{g_{ab}(s),\phi(s)\}\), with \(s=0\) corresponding to the stationary background. On the horizon, for each \(s\), the null energy condition requires \[f(s)\equiv T_{ab}(s)k^a k^b \geqslant 0 \,.\] For the stationary background, Eq. 28 , together with the Einstein equations, gives \(f(0)=T_{ab}(0)k^a k^b=0\). Since \(f(s)\geqslant 0\) in a neighborhood of \(s=0\), \(f(0)=0\) is a local minimum value of \(f\). Assuming differentiability at \(s=0\), Fermat’s theorem (interior extremum theorem) then implies \[\frac{d}{ds}f(s)=\delta T_{ab}k^a k^b = 0 . \label{eq:deltaT95integral95zero}\tag{63}\] This conclusion is also manifest in other matter models, such as Klein–Gordon and Maxwell fields. Alternatively, one may argue as follows. For a typical field theory, the stress-energy tensor is quadratic in the field \(\psi\). Hence, if \(\psi\) vanishes on the background, then the first non-vanishing contribution to \(T_{ab}(s)\) appears only at second order. Therefore \(\delta T_{ab}=0\), and the above equation holds. Using Eq. 63 and integrating the first equation of Eq. 29 over the interval from an arbitrary \(v'\) to \(+\infty\), one finds that \(\delta \theta_v\) is a constant for arbitrary \(v'\).

Then, considering the boundary conditions together with the above consequence of the null energy condition, we conclude that [29] \[\delta \theta_v = 0 \, . \label{eq:delta95theta95v95zero}\tag{64}\] Substituting Eqs. 27 , 63 , and 64 into Eq. 37 , we obtain the following relation: \[\begin{align} {\mathcal{E}'}[\Sigma(t_2)] - {\mathcal{E}'}[\Sigma(t_1)] ={}& -\frac{1}{4\pi G}\int_{\mathcal{H}^+_{12}}\boldsymbol{\epsilon}^{(n-1)}(\xi^c\nabla_c v)\delta\sigma_{ab}\delta\sigma^{ab} \\ &- \int_{\mathcal{H}^+_{12}}\boldsymbol{\epsilon}^{(n-1)}(\xi^c\nabla_c v)k^a k^b\delta^2T_{ab} \\ &- \frac{1}{16\pi G}\int_{\mathcal{I}^+_{12}}\tilde{\boldsymbol{\epsilon}}^{(n-1)}(\tilde{\xi}^c \tilde{\nabla}_c \tilde{u}) \delta \tilde{N}^{ab}\delta \tilde{N}_{ab} \, \\ &\leqslant 0 \, . \label{eq:reduced95canonical95energy95entropy95relation} \end{align}\tag{65}\] According to Eq. (106) of Ref. [24], the sum of the first and third terms is nonpositive. Moreover, using Eq. 64 and taking into account the stationary background condition, one obtains \(\delta^2 T_{ab}k^a k^b\geqslant 0\), and therefore the second term is also nonpositive. It follows that the difference of the modified canonical energy between \(\Sigma(t_2)\) and \(\Sigma(t_1)\) is nonpositive. Substituting Eq. 64 into the first equation of Eq. 6 , we obtain \[(-D^2+c)\delta U= 0 \, .\] However, \(\delta U\) can be expanded in eigenfunctions of the Laplacian. Since the eigenvalues of the Laplacian on a compact manifold (the codimension-2 horizon cross section at fixed \(v\)) are nonpositive, and since \(c>0\), it follows immediately that, \[\delta U = 0 \, . \label{eq:condition95of95delta95U}\tag{66}\] Substituting Eqs. 64 and 66 into Eqs. 10 and 62 , and after simplification, we finally obtain \[\delta^2 S = \frac{1}{4G} \delta^2 \int_{\mathcal{C}} dA_{\epsilon} \left(1 - v \theta_v\right) = \frac{1}{4G} \delta^2 A_\mathcal{T}(v) \, . \label{secondordervariationS}\tag{67}\] Therefore, we have shown that, under second-order perturbations around a spherically symmetric background, dynamical black hole entropy in Einstein’s gravity is proportional to the area of the apparent horizon. When \(\delta U=0\) and \(\delta^2 U > 0\) (so that \(U>0\)), the second relation in Eq. 6 implies \[\delta^2 \theta_{v}=(-D^2 + c) \delta^2 U \not\equiv0\, ,\] since integrating over the compact horizon cross section yields \[\int_{\mathcal{C}}\delta^2\theta_vdA =c\int_{\mathcal{C}}\delta^2UdA>0\,.\] So the second order variation in Eq.(67 ) is nontrivial.

In addition, from Eq. 65 , the difference of the modified canonical energy between \(\Sigma(t_2)\) and \(\Sigma(t_1)\) is negative; hence the dynamical black hole entropy is non-decreasing to second order in the perturbation. That is, \[\Delta\delta^2 S \geqslant 0 \, .\] Thus, under second-order perturbations, the dynamical black hole entropy satisfies the classical second law of black hole thermodynamics.

4.2.2 Case II↩︎

In this case, we no longer impose the null energy condition, thus the two conditions in Eqs. 63 and 64 may no longer hold. Comparing Eq. 62 with Eq. 9 , it appears that under second-order perturbations the area of the apparent horizon does not always proportional to the dynamical black hole entropy, since there remain extra terms that cannot be generally eliminated. According to [21], there is an ambiguity in \(\delta\xi^a\) or equivalently between \(\delta\kappa_2\) and \(\delta\kappa_3\). To ensure the area law (with respect to the apparent horizon) of the dynamical black hole entropy in the sense of second-order perturbation, \(\delta\kappa_2\), \(\delta\kappa_3\), and \(\delta\theta_v\) have to satisfy the constraint equation \[\left(v\delta\kappa_3 - \int_0^v \delta\kappa_2\,dv'\right)\delta\theta_v = \eta(v,x) \, , \label{eq:kappa95constraint95apparent95horizon95entropy}\tag{68}\] where \(\eta\) denotes \[\eta = \frac{\kappa}{2}\left\{\left(1 + \frac{\alpha v}{\theta_u}\right)(\theta_u\delta U)^2 - v\left[(D\delta U)^2\theta_u + (D^2\delta U)\gamma^{ab}\delta\gamma_{ab}\right]\right\} \, .\] According to Eq. 53 , we can express this equation in terms of \(\delta\xi^a\) as \[(vl_{[a}k_{b]}\nabla^a\delta\xi^b + \delta\xi\cdot l)\delta\theta_v = \eta \, . \label{eq:delta95xi95projection95constraint}\tag{69}\] After simplifying this relation in the chosen coordinate system, it takes the form \[\left[\partial_u(k\cdot\delta\xi) - v^2\partial_v\left(\frac{\delta\xi\cdot l}{v^2}\right)\right]\delta\theta_v = \eta \, . \label{eq:delta95xi95nonunique95constraint}\tag{70}\] By examining this equation more carefully, we find that there is no unique solution for \(\delta\xi^a\). Indeed, if a particular vector field \(\delta\xi^a\) solves the above equation, then it remains a solution at least under the transformation \[\begin{align} &\delta\xi^a \longrightarrow \delta\xi^a - v^2[\partial_u X(u,v,x)]k^a - v^2[\partial_v X(u,v,x)]l^a + Y^i(u,v,x) m_i^a \, ,\\ & Y^i(u,v,x) \overset{\mathcal{H}}{=}0 \, , \qquad \partial_v X(u,v,x) \overset{\mathcal{H}^+}{=}0 \, , \end{align}\] where \(X(u,v,x)\) and \(Y^i(u,v,x)\) are arbitrary smooth functions. The first two terms change the two projections \(k\cdot\delta\xi\) and \(l\cdot\delta\xi\) by a pure mixed-derivative contribution, which cancels in the combination appearing in the equation, while the transverse part \(Y^i(u,v,x) m_i^a\) does not contribute to these projections.

Even if one can choose certain vector fields \(\delta\xi^a\) satisfying Eq. 69 such that the dynamical black hole entropy remains proportional to the apparent-horizon area under second-order perturbations, this does not by itself guarantee that the entropy is nondecreasing at second order. Consequently, it also does not guarantee that the dynamical black hole entropy satisfies the classical second law of black hole thermodynamics.

5 Conclusion and discussion↩︎

In this work, we investigated the entropy of dynamical black holes arising from second-order perturbations of a stationary black hole with a spherically symmetric bifurcate Killing horizon. Working in Gaussian null coordinates, we constructed the apparent horizon perturbatively as a deformation of the event horizon and analyzed its geometric properties up to second order, including the associated null expansions and the area element. This framework allowed us to express the second-order variation of the apparent-horizon area entirely in terms of horizon quantities defined on the background event horizon.

To establish the relation between horizon dynamics and thermodynamic quantities, we employed the covariant phase space formalism and introduced a modified canonical energy that incorporates the contribution of external matter fields. This construction leads to a balance law relating the second-order variation of the dynamical black-hole entropy to the flux entering the horizon. By combining the geometric relations derived for the apparent horizon with the Raychaudhuri equation and the horizon boundary conditions, we obtained a significant simplification of the entropy formula. In particular, under the null energy condition, all additional contributions vanished that the entropy can be expressed solely in terms of the apparent-horizon area.

The main result of this work is that, when the null energy condition is satisfied, under second-order perturbations around a spherically symmetric background in Einstein gravity, the dynamical black hole entropy is given precisely by the area of the apparent horizon cross section. This provides a clear and explicit realization of the entropy proposal in a near-equilibrium setting, and provides further evidence for the idea that the apparent horizon, rather than the event horizon, captures the physically relevant notion of entropy in dynamical situations. In this sense, our result strengthens the connection between horizon geometry and thermodynamic behavior beyond strict stationary.

Several open questions remain. First, the derivation of the apparent-horizon area law relies on a consequence of the null energy condition. It would therefore be interesting to determine whether the area law survives, possibly in an altered form, when the null energy condition is violated. Clarifying these issues would be important for understanding the universality of dynamical black-hole entropy in the presence of quantum matter fields, where violations of the classical null energy condition can naturally arise and quantum corrections may break the relation between entropy and horizon area. Second, the uniqueness of the dynamical entropy beyond linear order remains to be clarified [12]. The entropy proposal is based on the existence of a local and covariant form \(\boldsymbol{B}_{\mathcal{H}^+}\) satisfying \(\underline{\boldsymbol{\Theta}}(g,\delta g)=\delta\boldsymbol{B}_{\mathcal{H}^+}(g)\) for first-order perturbations of a stationary black hole. However, this condition does not uniquely determine \(\boldsymbol{B}_{\mathcal{H}^+}\) beyond linear order when considering more general theories of gravity. Additional terms whose first variation vanishes on a stationary background can be added without affecting the entropy at first order, while potentially contributing to the entropy at second order. It would therefore be interesting to understand whether further physical requirements can uniquely determine the corresponding entropy functional. In particular, one may ask whether the validity of a second law beyond linear order can serve as a criterion for selecting a preferred choice of \(\boldsymbol{B}_{\mathcal{H}^+}\) and hence a preferred notion of dynamical black-hole entropy.

Beyond these issues, it is also important to understand to what extent the present result can be generalized. Our analysis was restricted to spherically symmetric, asymptotically flat black holes in Einstein gravity. it remains unclear whether the same apparent-horizon characterization of entropy remains valid in more general settings. For example, rotating backgrounds, asymptotically AdS spacetimes, and higher-curvature theories provide natural testing grounds. Determining whether the dynamical entropy can still be associated with the apparent horizon, and identifying the corresponding entropy functional when the Wald entropy differs from the area, these would offer valuable insight into the universality of dynamical black-hole thermodynamics. The extension to nonspherically symmetric configurations is currently under investigation.

Acknowledgement↩︎

This work is supported in part by the National Natural Science Foundation of China with grants No. 12475063, No. 12075232 and No. 12247103.

References↩︎

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1. Here we have omitted some technical details, such as the existence of null infinity [26].↩︎