Soft thermodynamics of gravitational shock wave

The gravitational shock waves have provided crucial insights into entanglement structures of black holes in the AdS/CFT correspondence. Recent progress on the soft hair physics suggests that these developments from holography may also be applicable to geometries beyond negatively curved spacetime. In this work, we derive a remarkably simple thermodynamic relation which relates the gravitational shock wave to a microscopic area deformation. Our treatment is based on the covariant phase space formalism and is applicable to any Killing horizon in generic static spacetime which is governed by arbitrary covariant theory of gravity. The central idea is to probe the gravitational shock wave, which shifts the horizon in the $u$ direction, by the Noether charge constructed from a vector field which shifts the horizon in the $v$ direction. As an application, we illustrate its use for the Gauss-Bonnet gravity. We also derive a simplified form of the gravitational scattering unitary matrix and show that its leading-order contribution is nothing but the exponential of the horizon area: $\mathcal{U}=\exp(i \text{Area})$.

1 Introduction naive application of holographic intuitions 1 2 . Hence, as a first step, it is desirable to develop a universal framework to characterize the gravitational shock waves through some microscopic lens in a way applicable to geometries beyond the AdS/CFT correspondence as well.
Recent rapid progress on the soft hair of black holes in an asymptotically flat space provide useful hints toward such a goal with successful derivations of the Bekenstein-Hawking entropy [39][40][41][42][43][44] 3 . It has been also pointed out that the linearized gravitational shockwaves can be realized as BMS supertranslations via certain choices of parameters [55][56][57]. Furthermore, some general expressions of thermodynamic relations for soft charges have been derived for the Schwarzschild black hole and several other geometries [58].
While these developments may provide a further insight into quantum aspects of the gravitational shock waves, it remains unclear how the soft charges may be measured in a concrete setting. In addition, the derived thermodynamic relations, associated with soft charges, are in rather abstract forms without concrete physical interpretations. As such, the implications of the soft hair physics on conceptual puzzles of quantum black holes still remain vague, in comparison with concrete developments within the framework of the AdS/CFT correspondence. What is currently missing is an effort to tie these two developments.
Hence, deriving a microscopic thermodynamic characterization of the gravitational shock waves on a unified footing may be the necessary first step.
In this paper, we derive the microscopic thermodynamic relations which are localized on the near horizon region and are sensitive to the gravitational shock wave in a concrete form. Our treatment is applicable to arbitrary gravitational shock waves at a bifurcating horizon in generic static spacetime (including AdS, dS, and asymptotically flat spaces), and is valid for arbitrary covariant theory of gravity which may be beyond general relativity (e.g. higher-derivative gravity theories). Furthermore, our thermodynamic relation provides a concrete physical interpretation of the gravitational shock waves, or soft charges. In particular, we derive a remarkably simple formula which relates the horizon area deformation induced by two intersecting gravitational shock waves to the incoming and outgoing energy sources. We will then find that the soft charge associated with the gravitational shock waves is nothing but the horizon area in the presence of two intersecting gravitational shock waves.
Our main technical machinery is the covariant phase space formalism, namely Wald's Noether charge method. It turns out that the original treatment by Wald, which focuses on contributions at the linear order of field variations, is not sufficient to probe the gravitational shock wave. For this reason, we will extend the Noether charge method to include variations of the matter fields at arbitrary order. This refinement enables us to obtain the microscopic thermodynamic relations (as in Eq. (7)) which directly associate the area deformation to the energy momentum tensor variation. One benefit of the covariant phase space formalism is that it is applicable to arbitrary covariant theory of gravity. As an application, we demonstrate that, in the Gauss-Bonnet gravity [59][60][61][62], the gravitational shock wave equations indeed follow from the microscopic thermodynamic relation with much simplified calculations. 1 For instance, the absence of the spatial infinity may suggest that the structure of quantum entanglement in the de Sitter space differs crucially from that of the AdS space or the asymptotically flat space.
Our result also suggests a concrete framework to physically characterize (or, even measure) soft charges via out-of-time order correlation (OTOC) functions. The leading-order (in 1/G N ) behaviours of OTOCs are dominated by the gravitational scattering unitary matrix which typically takes the following form; Here f (Ω 1 , Ω 2 ) is the green function for the shock wave equation, and P in and P out account for the incoming and outgoing energy fluxes. Our thermodynamic characterization enables us to rewrite the above scattering matrix into a remarkably simple form: up to some multiplicative factor in the phase. Here "Area" represents the horizon area in the presence of two intersecting gravitational shockwaves due to P in and P out . Thus, the phase factor of the scattering matrix is proportional to the horizon area (or equivalently, the soft charge of an infalling matter measured by an outgoing matter).

Main result
Here we present a summary of our main result. We study a bifurcating Killing horizon in generic static spacetime of the following form: where u = 0 and v = 0 corresponds to the bifurcating horizons. We assume that the timelike vector is a Killing vector of the spacetime. The gravitational shock wave can be generated by shifting this spacetime asṽ where θ(u) is a step function and x collectively denotes angular profiles. At the leading order, the metric variation can be computed by considering the horizon translation vectorα: where Lα is the Lie derivative with respect toα. To satisfy the Einstein equation, an additional energy momentum tensor needs to be introduced: For general relativity (i.e. L (G) = 1 16π (R − 2Λ)), Sfetsos derived the shock wave equation of motion which relates the shift profile α(x) and the energy momentum profile T (p) (x) via a certain differential equation [32]. In this paper, we will work on arbitrary covariant theory of gravity (i.e. arbitrary covariant Lagrangian L (G) ) which may be beyond general relativity.
We will investigate the shock wave geometry by using the covariant phase space formalism [63][64][65][66][67][68] 4 and derive a microscopic thermodynamic formula which captures low-energy excitations on the Killing horizon.
The standard method utilizes the timelike Killing vector ξ (t) = ∂ t to construct the Noether charge Q ξ and relate the horizon area (at u = v = 0) to the ADM thermodynamic parameters at asymptotic infinity. It turns out, however, that the area variation due to the gravitational shock wave vanishes at u → 0, and hence non-trivial thermodynamic relations cannot be derived.
In this paper, we study the gravitational shock wave by using the Noether charge Qβ constructed from another horizon translation vectorβ in the u direction: In particular, we probe the gravitational shock wave, which was generated by the shift v → v + α(x) in the v direction, by another shift u → u + β(x) in the u direction. For general relativity, this enables us to derive the following microscopic thermodynamic relation: up to a multiplicative factor. Here Area(β(x), α(x)) corresponds to the total area of the sphere at u = β(x) and v = α(x). For a generic covariant Lagrangian L (G) , the area term would correspond to the Wald's geometric entropy. This thermodynamic relation can be rewritten as such that E shock = 0 corresponds to the shock wave equation of motion for α(x) and T (p) (x).
We will also study the gravitational scattering unitary matrix: Let us assume that P out and P in induce the shifts α(x) and β(x). Then, for general relativity, the above thermodynamic characterization enables us to rewrite U as follows: up to a multiplicative factor for the area. Hence, the scattering matrix is the exponential of the horizon area (or equivalently, the soft charge).

Relation to previous works
The microscopic relation between the area deformation and the gravitational shock wave for the Schwarzschild black hole, for the case with T µν = 0 (i.e. a "sourceless" shock wave), was initially communicated to us from Yoni BenTov 5 . Our contribution is to include the energy momentum tensor and extend this observation to a bifurcate Killing horizon in generic static spacetime for arbitrary covariant theory of gravity. The covariant phase space formalism has been recently applied to the Jackiw-Teitelboim (JT) gravity which is a tractable model of the gravitational backreaction [70].
In an asymptotically flat space, Hawking, Perry and Strominger have identified a concrete way of generating the gravitational shock wave from BMS supertranslation and superrotation on the boundary of the asymptotic flat spacetime [57]. Namely they provided a generic formula to prove the shock wave can be created by a supertranslation, by using the Noether charge constructed from a supertoration. This result was further investigated and extended, see [58,[71][72][73] for instance. It is worth clarifying our contributions and novelties in comparison with these previous works. First, previous works study the linearized shock wave near the horizon whereas our treatment applies to the shock wave which runs exactly on the horizon as well. Second, our thermodynamic relation is cast in a concrete form, and hence its physical interpretation as a microscopic Hilbert space of a black hole becomes transparent. Third, as already noted above, our treatment applies to any bifurcate Killing horizon in generic static spacetime for arbitrary covariant theory of gravity.
It should be noted that, while our primary motivation stems from quantum aspects of black holes, the treatments in the present paper are entirely classical. Also, our treatment is localized near the horizon and does not consider asymptotic infinity or boundaries. As such, we do not discuss BMS transformations in a direct manner.

Organization
The paper is organized as follows. In section 2, we present a review of Wald's Noether charge method.
In section 3, we will make certain refinements to the Wald's entropy formula so that it is valid up to higher order of matter field variations. In section 4, we present a review of gravitational shock wave on a bifurcate Killing horizon in generic static spacetime. In section 5, we investigate the variations of the metric due to horizon translations. In section 6, we derive a microscopic thermodynamic relation on the horizon. In section 7, we demonstrate that the shockwave equations of motion indeed follows from the area deformation. In section 8, we apply our formalism to the Gauss-Bonnet gravity. In section 9, we briefly discuss the gravitational scattering matrix. In section 10, we conclude with discussions.
In this section, we present a self-contained review of the Noether charge method in the presence of matter fields. Readers who are familiar with the method may skip to the next section.

Variation of Lagrangian
Bold letters are used to represent differential forms. Consider the following diffeomorphism covariant Lagrangian d-form on a d-dimensional manifold 6 : where L (G) is the gravitational Lagrangian d-form and L (M ) is the matter Lagrangian d-form. Here we assumed that the Lagrangian can be separated into two parts L (G) and L (M ) 7 . At the linear order in δg µν , δφ, the variations can be expressed as where φ collectively denotes all the matter fields. Here denotes the volume form = d d x √ −g and Θ (G) , Θ (M ) are called symplectic potentials 8 . These variations lead to equations of motion: where the energy momentum tensor is defined by 9 Field configurations {g µν , φ} which satisfy equations of motion Eq. (13) are called on shell.
Now consider variations under an infinitesimal diffeomorphism by an arbitrary vector field ξ. According 6 The diffeomorphism covariance condition is L ξ L(ψ) = ∂L ∂ψ L ξ ψ. 7 More precisely, we assume that L (G) consists of terms which do not involve matter fields φ and their covariant derivatives ∇µ 1 . . . ∇µ m φ whereas L (M ) consists of terms which do not involve Riemann tensors R abcd and their covariant derivatives ∇µ 1 . . . ∇µ m R abcd . As such, Riemann tensors R abcd do not directly couple to matter fields (e.g. terms like Rφ 2 do not appear) in the Lagrangian. Such matter fields are often said to be minimally coupled to gravity. Gravitational shock waves with non-minimal couplings have been studied in [74]. 8 Since the matter Lagrangian does not involve R abcd and its covariant derivatives, the matter symplectic potential Θ (M ) can be constructed so that it does not depend on δgµν . See Section 3 of Iyer and Wald for details [67]. 9 The Euler-Lagrange derivative is defined by to Eq. (12), the total on shell variation can be expressed bŷ Here L ξ denotes the Lie derivative with respect to ξ. It is worth emphasizing that the variationδ ξ L is induced by a diffeomorphismδ ξ acting on each field g µν , φ.
The variationδ ξ L can be expressed in another form due to the diffeomorphism covariance. The covariance of the gravity and matter Lagrangians implieŝ where where we made use of the Cartan's magic formula 10 Note thatδ ξ L (G) ,δ ξ L (M ) must be total derivatives according to Eq. (17).

Wald's entropy formula
Wald introduced the following Noether current (d − 1)-form One can verify Here we used g to denote the metric tensor g µν for brevity of notation. One can actually show that J ξ,Wald is exact due to the fact that J ξ,Wald is closed for all ξ [65]. As such, the Noether charge (d − 2)-form Q ξ,Wald can be constructed 11 Let us now fix ξ and consider the variation of the Noether current Eq. (19) at the linear orders of 10 Here ξ · Λ indicates insertion of ξ into the first argument of Λ. It is also called interior product and denoted as ι ξ Λ. 11 A systematic algorithm to construct the Noether charge Q ξ,Wald can be found in Iyer and Wald [67].
δg µν , δφ: where we used the on shell conditions Eq. (13) in the second line. Let us define the symplectic currents by ] By using the Cartan's formula Eq. (18), we obtain Finally we shall focus on variations δg, δφ which satisfy the linearized equations of motion. This allows us to replace J ξ,Wald and its variation δJ ξ,Wald with the Noether charge dQ ξ,Wald and its variation δdQ ξ,Wald .
Hence, by focusing on on shell variations, we obtain The above expression can be further simplified when ξ is a Killing vector: Let us further assume a similar relation for the matter field: Then at the linear order in δg and δφ, we have Hence we arrive at the well-celebrated result of Wald:

Example: general relativity
We illustrate the covariant phase space formalism and the Noether charge by looking at general relativity with scalar fields. The Lagrangian is given by The equations of motion are where G µν ≡ R µν − 1 2 Rg µν + Λg µν . The symplectic potentials are given by and the energy momentum tensor is For variations by diffeomorphism, we have To compute the Noether current, it is useful to make use of the following relation: We can then rewrite Θ and obtain By using the Einstein equation: we arrive at Hence the expression of J ξ,Wald is written entirely with the metric and does not contain Λ, φ or T µν explicitly. Finally we obtain the Noether charge 3 Two refinements to Wald's entropy formula The original Noether charge method is not particularly suitable for studying the gravitational shock wave geometries. In this section, we will make two refinements to the Wald's entropy formula.
First, we will derive an alternative expression of the Noether current J ξ,field , which differs from the Wald's construction J ξ,Wald , by evaluating the Lie derivative of the matter Lagrangian L ξ L (M ) explicitly and relating it to the energy momentum tensor T µν . Our construction of the Noether charge Q ξ,field is particularly useful for studying the gravitational backreaction as it relates the energy momentum tensor variation directly to the metric variation.
Second, we will extend the linear order analysis by Wald to the higher order matter field variations δφ while keeping the metric variation δg at the linear order. This is due to the difficulty that contributions from the linear order variation δφ vanish for the gravitational shock wave geometries. Our construction of the Noether current Q ξ,field enables us to evaluate its variation δQ ξ,field under the matter field variation δφ at any order in a systematic manner.

Matter current
Our derivation deviates from Wald's where we evaluate the Lie derivative L ξ L (M ) of the matter Lagrangian in Eq. (17). Diffeomorphism variation of the matter Lagrangian can be explicitly evaluated as follows: where we made use of the metric compatibility the evaluation of the integral by parts, and on shell condition of matter fields. Let us define the matter Here We then obtain Recall that L ξ L (M ) can be expressed as a total derivative, according to Eq. (17). Thus, from Eq. (45), we can deduce the conservation of the energy momentum tensor: Then, the Lie derivative of the matter Lagrangian L ξ L (M ) can be expressed as Also, the Lie derivative of the gravity Lagrangian is given by Here it is worth emphasizing that L ξ L (G) = dΘ due to the matter current.
We define the Noether current (d − 1)-form by One can verify that J ξ,field is closed when {g, φ} are on shell: 12 Explicitly, we have where is the Levi-Civita symbol. Here J (M ) ξ should be understood as the Hodge dual of the 1-form current.
Then one can construct the Noether charge Q ξ,field : ∃Q ξ,field such that J ξ,field = dQ ξ,field {g, φ} on shell. (51) Note that our definition J ξ,field differs from Wald's J ξ,Wald since, instead of Θ ξ,field . We will compare two constructions in section 3.3

Higher-order matter variation
Until this point, our derivation of the Noether current Eq. (49) is an exact calculation which does not rely on perturbative analysis 13 . The remaining task is to evaluate the variation of the Noether current Eq. (49) under δg µν , δφ.
We will focus on the linear order metric variation δg µν while we do not impose such restriction on the matter field variation δφ. We then obtain where we used E (G)µν = − 1 2 T µν . By using the gravity part of the symplectic current It is worth emphasizing again that Eq. (53) is valid up to the linear order in δg µν and up to any order in δφ. Note that the matter field variation δφ enters through the matter current variation δJ Finally, we arrive at We will use this formula to derive the microscopic thermodynamic relation. 13 Recall that it suffices to use the linearized variation formula Eq. (12) to evaluate the Lie derivative. 14 In fact, we do not need the on shell condition for the matter fields φ for this equation to hold. We would need it to make sure dJ Then we have Ω (G) (δg, L ξ g) = 0 in the linear order in δg.
When ξ is a Killing vector, the matter current is conserved: or equivalently Thus, we can construct the matter charge (d − 2)-form Q

Comparison
Let us explicitly compare two possible constructions of the matter current To begin, we compare their total derivatives: and Their difference indeed vanishes for matter fields on shell: Hence both J ξ,field lead to valid constructions of the Noether charge Q ξ,Wald and Q ξ,field . Next, let us look at the scalar field Lagrangian We have Hence, we have For the electromagnetic Lagrangian we have and where we used the Cartan's magic formula for the Lie derivative. Hence we obtain which, by imposing on shell conditions ∇ α F µα = 0, becomes Note that two definitions of the matter current for the electromagnetic field differ by a total derivative.
Thus we can write which leads to the following difference of two constructions: Here it is worth emphasizing that the charge difference Q For our application to the gravitational shock wave geometries, terms which vanish as ξ → 0 do not contribute to the surface integral of the Noether charge. As such, our construction of the Noether charge Q ξ,field bears the same geometrical meaning as the Wald's construction Q ξ,Wald .

Gravitational shock wave
Having reviewed and refined the Noether charge formula, let us shift the gear a bit. In this section, we present a brief review of gravitational shock wave geometries by following the works by Dray, 't Hooft and Sfetsos [29,32] who derived the gravitational shock wave equations of motion from the Einstein equation.
Consider the following family of d-dimensional static spacetime, expressed in the Kruskal-type coordinate and its corresponding energy momentum tensor: Following Sfetsos [32], we assume that the metric and matter fields satisfy the following conditions: The above conditions follow from the assumption that the timelike vector ∂ t is a Killing vector of the spacetime as we will see in the next subsection. Non-vanishing Christoffel symbols are Let us shift this spacetime by where x represents x j collectively and θ(u) is a step function. The resulting metric and energy momentum tensor are Note that the shifted metric and the shifted energy momentum tensor still satisfy the Einstein equation except at u = 0 where an additional source of the energy momentum tensor is needed. Consider the extra contribution to the energy momentum tensor from a massless particle, located at u = 0 and moving with the speed of light in the v-direction: where p (v) is the momentum of the particle; T vv = 4p (v) δ(u)δ(x) 15 .

The Einstein equation is satisfied if
Eq. (77) holds and the following equation holds: Here the Laplacian is defined as We will derive this equation by using the Noether charge method later 16 .

Horizon translation vector
In this section, we investigate the symmetry properties of the spacetime. The gravitational shock wave solutions require certain consistency conditions on the fields g µν , T µν as we reviewed in the previous section.
These conditions actually follow from the existence of the timelike Killing vector ξ (t) = ∂ t as we shall show in this section. We will also study the effect of a vector fieldβ which introduce the horizon translation.
In later sections, we will use the horizon translation vectorβ to construct the Noether charge which is sensitive to the gravitational shock wave.

Timelike vector
In this subsection, we verify that Eq. (77) and Eq. (78) follow from the fact that the timelike ξ (t) = ∂ t vector is a Killing vector.
A vector field ξ is a Killing vector if L ξ g µν = 0. Let us begin by computing L ξ g µν for the metric given in Eq. (76). Looking at (µ, ν) = (u, v), we have Looking at (µ, ν) = (i, j), we have Looking at (µ, ν) = (u, i), we have Looking at (µ, ν) = (v, i), we have Consider the following timelike vector field: where κ represents the surface gravity. One can verify and L ξ (t) g ui , L ξ (t) g vi = 0, suggesting that ξ (t) asymptotically becomes a Killing vector on the horizon at

Horizon translation
Next, consider the following horizon translation vector field: where β(x) is an arbitrary function of x i . With direct calculations, one finds (92) Namely, at the limit of v → 0, we have In the next section, we will use this horizon shift vectorβ to construct the Noether charge and derive the microscopic thermodynamic relations as in Eq. (7). It is worth recalling that, in the original treatment by Wald, the vector field ξ in the Noether charge Q ξ was chosen to be a Killing vector. This was crucial in the derivation of Wald's Noether charge relation since the pre-symplectic form Ω (G) (g, δg, L ξ g) vanishes when ξ is a Killing vector. In the above calculation, we found that the horizon shift vectorβ is not a Killing vector at v = 0 (unless β(x) does not depend on x). Nevertheless, the pre-symplectic form Ω (G) (g, δg, Lβg) vanishes at v = 0 as we shall see in section 6.3.

Microscopic thermodynamics on horizon
In this section, we will derive microscopic thermodynamic relations which are localized on the horizon.

Noether charge from shifted vector
Let us briefly recall the setup. The original static spacetime was given by Eq. (76) which is reprinted below The shifted metric and the energy momentum tensor are given by Eq. (81) and Eq. (82) which are obtained by the substitution v →ṽ = v + θ(u)α(x). It is worth noting that, at the leading order in α(x), the metric variation is given by Here 0 − means that u approaches zero from below. For our purpose of deriving the shock wave equations of motion, it actually suffices to choose Σ A as any boundary with u < 0.
Finally, let us construct the Noether charge. As a vector field, we shall take the shifted timelike vector: in order to construct the Noether charge. It is essential to observe which will be useful in simplifying the thermodynamic relation.

Thermodynamic relation
Our central formula is Eq. (54) which is reprinted below: Integrating it on the v = 0 surface generates Two comments follow. First, the last term 1 2 (χ· )T µν δg µν vanishes since the integral is along the vector field χ and thus Λ (χ · ) = 0. Second, we have introduced the matter charge Q (M ) χ,field . Here it is useful to note that, whileβ is not a Killing vector of the spacetime, one can still construct the matter charge on the v = 0 surface since Then, by using the Stokes' theorem, we obtain Below, we will further simplify this expression by using the properties of the shifted vector field χ.
Let us evaluate each term in Eq. (102) at Σ A and Σ B . First, we observe by noting that χ = 0 at Σ B and δg = 0 at Σ A . Second, we observe since δg = 0 at Σ A . Third, we can construct the matter charge Q (M ) χ,field as an integral over v = 0 surface in the u direction : Due to the constraint T vv = 0 at v = 0 in Eq. (78), we have There are two potential contributions to the variation δT vv . The first contribution is due to the metric shift which is explicitly given by It turns out that this contribution is subleading in α. Recall that the timelike vector ξ (t) is the symmetry of the spacetime. We then have Let us Taylor expand T vv for small u and v. Then we find that only the terms of the form u a v a+2 are allowed. This suggests that T vv is ∼ v 2 at most, and hence δT vv is ∼ vα + α 2 . Integrating it along v = 0 only generates a contribution of O(α 2 ). The second contribution comes from the shock wave. Considering and δQ (M ) Note that √ G is the volume element induced on the (d − 2)-dimensional surface (u, v) = (0, 0). Hence, we obtain

Pre-symplectic form
The remaining task is to evaluate the pre-symplectic form Ω (G) (δg, L χ g). Here we will show that Since the timelike vector ξ (t) is the symmetry of the original spacetime, we have and where denotes that we evaluate the differential form on the v = 0 surface. Recalling δg = θ(u)Lαg at the linear order in α(x), we arrive at Let us recall that the variation of the gravity Lagrangian is given by δL (G) = E (G) µν δg µν + dΘ(δg).
Using the equations of motion and taking δ = Lβ, we obtain Taking another variation Lα, integrating it and using the Cartan's formula, we have Similarly, we have Since Lα and Lβ commute, we have LβLαL (G) = LαLβL (G) . Hence we obtain 19 By integrating Ω (G) (g, Lαg, Lβg) on the v = 0 surface, we obtain u=0 Ω (G) (Lαg, Lβg) = 0 (120) since the term with (dx) u does not contribute to the integral.
This enables us to obtain the following simple constraint that governs microscopic thermodynamics of 19 For general relativity, we have checked that LαΘβ LβΘα 0 via brute force calculation.
gravitational shock waves on the horizon: This is the central result of this paper.
As we will see in the next section, the first term corresponds to the area deformation under two shifts u → u + β(x) and v → v + α(x). The second term corresponds to the matter charge variation due to the gravitational shock wave, probed by the horizon translation vector fieldβ. It is worth emphasizing that, if the original unshifted timelike vector ξ (t) were used, we would not have any interesting thermodynamic relation since δQ ξ,field = δQ ξ,field = 0. This is related to the fact that the area variation by v → v + α(x) vanishes at the bifurcate surface as we shall explicitly see in the next section.

Shock wave from area minimization
In this section, we derive the gravitational shock wave equations of motion by using the microscopic thermodynamic relation on the horizon.

Area variation
We have evaluated the matter charge δQ (M ) χ,field in the previous section. Here we evaluate δQ χ,field at Σ B . For general relativity with scalar fields, the Noether charge is given by 20 where a = a 1 · · · a d−2 collectively denotes (d − 2) indices. Recall that χ is held fixed in variations. Also Non-trivial contributions come from b, c = u, v. We thus have It is worth relating the variation of the Noether charge explicitly to the area deformation (Fig. 4). Let Area(β, α) denote the total area of the hypersphere at (u, v) = (β, α) in the original unperturbed metric Figure 4: The area deformation in the original unperturbed geometry.
where we have used Area(β, 0) = Area(0, 0), since (u, v) = (β, 0) is still on the horizon. Hence the Noether charge variation is proportional to the increase of the area of the bifurcating horizon by two null-shifts u → u + β and v → v + α. Note that α comes from the shift by the gravitational shock wave while β is set by the shift in the vector field χ.
One comment follows. If we used the timelike Killing vector ξ instead of the shifted one χ, we would have the contribution proportional to Area(0, α) − Area(0, 0) which vanishes since (u, v) = (0, α) is still on the horizon. Hence the Noether charge relation would be trivial.

Induced metric
The remaining task is to evaluate the induced metric on Σ B . A spacetime point on Σ B can be expressed as X µ ( y) ≡ (β( y), α( y), y) where α( y) results from the metric variation. The induced metric on Σ B is given by where G ij ( y) = g µν (X( y)) ∂X µ ∂y i ∂X ν ∂y j .
We shall Taylor expand G ij ( y) for small α, β. Looking at (µ, ν) = (u, v), we have the following contribution to G ij ( y): g uv (X( y)) ∂β ∂y i ∂α ∂y j = F (0, 0) + Looking at (µ, ν) = (i, j), we have the following contribution to G ij ( y): We observe that the leading correction to G ij ( y) is at the order of αβ. In the expansion of Eq. (129), the term ∂ 2 F (u,v) ∂u∂v αβ ∂β ∂y i ∂α ∂y j is at the order of α 2 β 2 , and hence is negligible. Then we find where dσ 2 * represents the unperturbed induced metric when (α, β) = (0, 0): To summarize, we have obtained the variation of the induced metric: where F and G ,uv are evaluated at u = v = 0.
Recall that the first order variation of the determinant is Hence the variation of the determinant at Σ B is given by integrating by parts, we obtain Note that total derivatives do not contribute to the surface integral of the Noether charge.
Finally, we derive the shock wave equations of motion. Using the expression of deformed area density Eq. (136), we obtain then the constraint Eq. (121) leads to We can arrange it into the following form Hence we recover the shock wave equation of motion 8 Gauss-Bonnet gravity The microscopic thermodynamic relation can be generalized to arbitrary covariant theory of gravity. The gravitational shock wave equations in the Gauss-Bonnet gravity have been previously studied for a certain black hole solution in [20] while exhaustive studies have been presented in [75,76] with rather involved calculations. In this section, we present a simpler derivation of the gravitational shock wave equations of motion in the Gauss-Bonnet gravity based on the Noether charge method.

Noether charge in Gauss-Bonnet gravity
The Lagrangian d-form of the Gauss-Bonnet gravity is This yields the Noether charge (d − 2)-form [67] Q χ a = − ade 1 16π We shall focus on a family of the metric of Eq. (76). The main task is to evaluate the variation δQ χ under (u, v) = (β, 0) → (β, α) at Σ B . It suffices to compute the contribution to Q χ a from (d, e) = (u, v) in Eq. (142) 21 : and its variation δQ χ a : To evaluate H, we need the following non-vanishing Ricci tensors: and the following Riemann tensor: Putting all these together, we obtain the following simple result where R (d−2) is the scalar curvature computed from h ij .
As for δH, the following observation simplifies the calculation. Let us focus on the variation of the first term R µν g µν in H: δ(R µν g µν ) = δ(R µν )g µν + R µν δ(g µν ).
By utilizing the following relation δ(R µν )g µν can be expressed as a total derivative. In a similar manner, we can show the following Note that total derivatives do not contribute to the surface integral of the Noether charge. By dropping these terms, we obtain 21 Note there is a factor of 2 which comes from the contribution of (d, e) = (v, u) √ GG ij δG ij , the total variation Eq. (144) can be expressed as Finally, by making use of Eq. (133) and integrating by parts, one can obtain the gravitational shock wave equations of motion in the Gauss-Bonnet gravity.

Isotropic space
So far, our discussions are applicable to generic static spacetime in the Gauss-Bonnet gravity. By specializing in isotropic spaces, one obtains further simplified expression of δQ χ . Let us consider the following metric where the metric of h ij depends on k = 1, 0, −1 and looks like We then have Plugging these into Eq. (152), we obtain where r = r H is the horizon radius. By setting , we see that the Noether charge variation δQ χ,field is rescaled by a factor of (1 + 2λ GB ). This recovers the result from [20] (

Measuring shock waves
In this section, we present a brief discussion on the connection between OTOCs and the thermodynamic relation. Here we shall focus on the cases where the infalling and outgoing matters consist of scalar fields for simplicity of discussion.
So far, we have studied geometries with single gravitational shock waves running in the v direction and derived the thermodynamic relations by using shifted vector fields in the u direction. A naturally arising question concerns geometries with two intersecting gravitational shock waves running in the u and v directions. Such geometries can be generated by using shifts α(x) and β(x) in the v and u coordinates respectively, and have been explicitly considered by Kiem, Verlinde and Verlinde [77]. Readers might wonder if similar analyses based on the covariant phase space formalism would reveal thermodynamic relations for two gravitational shock waves. Unfortunately, the horizon area deformation induced by two gravitational shock waves are the second-order contributions, and as such, naive application of the covariant phase space formalism does not appear to give rise to non-trivial thermodynamic relations 22 .
Nevertheless, the microscopic thermodynamic relation can be generalized to geometries with two shock waves in a straightforward manner. This is due to the fact that, in a geometry with two shock waves, each shock wave needs to satisfy the corresponding gravitational equation of motion, which is identical to the one with a single shock wave [77]. Then the effective Lagrangian for two shock waves can be constructed as follows: up to a multiplicative factor for the area term 23 . Here T (u) (x) and T (v) (x) are energy-momentum tensor profiles running in the u and v directions respectively. By taking the variation of the Lagrangian, we indeed recover equations of motion for α(x) and β(x): Here one may interpret α(x) and β(x) as scalar fields which live on the bifurcating sphere, and the Area(α(x), β(x)) as interacting massive scalar fields. Then, T (u) (x) and T (v) (x) can be interpreted as source terms coupled with fields α(x) and β(x).
Let us now turn our attention to the problem of measuring gravitational shock waves. Recall that the gravitational shock waves can be probed by out-of-time order correlation (OTOC) functions of the form V (0)W (t)V (0)W (t) within the AdS/CFT correspondence. When the time separation between two perturbations V (0) and W (t) is larger than the scrambling time, two gravitational shock waves will intersect with each other near the black hole horizon. It can then be approximated that two shock waves are running on the black hole horizon and intersect with each other. Such an approximation would be valid if the values of u, v at the intersecting location are smaller than the effective shifts α, β.
At the heart of the calculation of OTOCs is the gravitational scattering unitary matrix. In his pioneering work, 't Hooft [78] derived the following scattering unitary matrix for the Schwarzschild black hole (often U = exp i dΩ 1 dΩ 2 P out (Ω 1 )f (Ω 1 , Ω 2 )P in (Ω 2 ) (160) where f (Ω 1 , Ω 2 ) is the green function for the shock wave equation 25 . Here P in and P out account for the incoming and outgoing energy flux. One can cast the above scattering matrix in an illuminating form which makes the relation to the soft thermodynamics more explicit. Here we consider the limit where two shock waves collide near the horizon. Note that one can identify P out (Ω 1 ) and P in (Ω 2 ) as T (u) (x) and T (v) (x) respectively. Then, one obtains α(x) = dΩ 2 f (Ω 1 , Ω 2 )P in (Ω 2 ), β(x) = dΩ 1 f (Ω 1 , Ω 2 )P out (Ω 1 ).
This expression suggests that, at the leading order, the decorrelation of OTOCs is given by the area deformation 26 It is worth emphasizing that this expression of the scattering unitary matrix can be generalized to any bifurcating horizon in generic static spacetime in arbitrary covariant theory of gravity 27 . The above expression of the scattering matrix can be viewed as a concrete method of measuring the soft charge, which is realized as the gravitational shock wave. We hope to further expand this observation in the future.
Also note that matter contents render rich additional structures in the scattering matrix, see the table 24 see [79] also. 25 Quantum mechanical aspects of the scattering matrix have been further studied in recent years, see [80][81][82] for instance. 26 Direct measurement of OTOCs in many-body quantum systems is challenging with inverse time-evolution, but an alternative formulation of OTOCs using entanglement does not require inverse time-evolution. Various experimental protocols for measuring OTOCs in controlled quantum simulators have been proposed [83][84][85][86][87], with concrete experimental demonstrations [88][89][90][91]. It is still premature to envision measurement of OTOCs in gravitational systems, but at the conceptual level, the smoothness of the black hole horizon for an infalling observer is an indirect evidence of decorrelation of OTOCs, see [26,72] for details. 27 While OTOCs are concepts primarily studied within the framework of the AdS/CFT correspondence, they can be generically defined as correlation functions of matter fields which were initially located away from the horizon. Hence, we speculate that our observation may be generalized to asymptotically flat spaces as well as the de-Sitter horizon.
in [78] for instance. It will be interesting to extend our analysis to the cases where the infalling and outgoing matters are not scalar fields.

Outlook
In this work, we have derived a simple microscopic thermodynamic relation for the gravitational shock waves by providing a generic framework to construct the Noether charge for them.
We anticipate that our formalism can be applied to a wide variety of geometries in order to study the effect of the gravitational backreaction. One interesting future problem is to apply a similar method to stationary black holes where thermodynamic treatment involves angular momentum [36].
We did not follow the standard treatment of the soft hair physics in characterizing the gravitational shock waves. Instead, we opted to work on the gravitational shock wave solutions on a more generic setting. It would be interesting to translate our results to the language of the soft hair physics and study the consequence of the finiteness of the black hole entropy. Also, it is an interesting future problem to generalize the expression of the gravitational scattering matrix so that it would be applicable to generic soft charges which may be beyond the gravitational shock waves. Developing a generic framework for measurement of soft charges is also an interesting problem.