Modification of a Lie-algebra-based approach and its application to asymptotic symmetries on a Killing horizon

We develop a new approach to find asymptotic symmetries in general relativity as a modification of the Lie-algebra-based approach proposed in T. Tomitsuka et al. [Classical Quantum Gravity 38, 225007 (2021)]. Those authors proposed an algorithmic protocol to investigate asymptotic symmetries. In particular, their guiding principle helps us to find a non-vanishing charge that generates an infinitesimal diffeomorphism. However, in order to check the integrability condition for the charges, it is necessary to solve differential equations to identify the integral curve of vector fields, which is usually quite hard. In this paper, we provide a sufficient condition of the integrability condition that can be checked without solving any differential equations, avoiding the difficulties in the approach in the above reference. As a demonstration, we investigate the asymptotic symmetries on a Killing horizon and find a new class of asymptotic symmetries. In 4D spacetimes with a spherical Killing horizon, we show that the algebra of the corresponding charges is a central extension of the algebra of vector fields.


Introduction
The uniqueness theorem [2][3][4] states that every 4-dimensional stationary black hole solution to the Einstein-Maxwell equations in general relativity is completely characterized by just three parameters, mass, angular momentum, and electric charge. On the other hand, Bekenstein [5] proposed that a black hole has entropy proportional to its horizon area A, and Hawking [6] showed that a black hole emits the thermal radiation and has what we call the Bekenstein-Hawking (BH) entropy A/4G, where G denotes the gravitational constant. It suggests that the black hole has a lot of microstates even though it can be characterized by the above three parameters. What is the origin of such microstates?
So far, a great deal of effort has been devoted to explaining the origin of the BH entropy. One possible origin is the so-called asymptotic symmetries on a horizon. General relativity is invariant under diffeomorphisms. Sometimes, it is argued that diffeomorphisms are gauge transformations in general relativity, which do not change the state of the system physically. If so, the metrics connected by diffeomorphisms cannot be distinguished from each other and hence diffeomorphisms may seem to have nothing to do with the origin of microstates.
However, in fact, not all diffeomorphisms generate gauge transformations. A way to judge whether a diffeomorphism is not a gauge transformation is to check the value of the charge generating the transformation. If the value of a charge is modified by a diffeomorphism, In particular, the algebra of the charges in 4-dimensional spacetimes with a spherical Killing horizon is calculated explicitly, which is shown to be a central extension of A.
This paper is organized as follows: In Sec. 2, we briefly review the covariant phase space method, which is adopted in this paper to construct the charges generating infinitesimal diffeomorphisms. In Sec. 3, we briefly review a Lie algebra based approach proposed in Ref. [1]. In Sec. 4, we provide a sufficient condition for the charges to be integrable. In Sec. 5, we find a new symmetry on the Killing horizon by using our approach and investigate the algebra of its charges. In Sec. 6, we present the summary of this paper. In this paper, we set the speed of light to unity: c = 1.

Covariant phase space method
Let us briefly review the covariant phase space method developed in [13][14][15][16][17][18][19], which will be adopted in the rest of this paper. This method enables us to investigate and construct the algebra of the charges in an independent way of the choice of a local coordinate system.
We here focus on the gravitational system without any matter field. The Einstein-Hilbert action with the cosmological constant Λ is given by where M d D x denotes the integral over a D-dimensional spacetime M, g and R are the determinant of the metric g µν and the Ricci scalar, respectively. The variation of the integrand is decomposed into two parts as where G µν is the Einstein tensor defined by while Θ is called the pre-symplectic potential, which is defined by For an infinitesimal transformation of metric δ ξ g µν := £ ξ g µν , where ξ is a vector field and £ ξ denotes the Lie derivative along it, we have since L EH is a scalar density. Defining the Noether current Eqs. (2) and (5) imply that holds. From this equation, one can see that if g µν satisfies of the equation of motion, i.e., the Einstein equations G µν + Λg µν = 0, the current is conserved: where ≈ denotes the equality which holds for any solution of the equation of motion. In fact, we can decompose the current into two parts [18]: Here, the bracket [ , ] for indices denotes an anti-symmetric symbol which is defined by where S d is the permutation group and sgn(σ) denotes the signature of σ ∈ S d . The Noether charge is defined as the integral of the current over a (D − 1)-dinensional submanifold Σ in M and given by where ∂Σ is the boundary of Σ. Note that the integral measure is given by where ǫ µ1...µD is the D-dimensional Levi-Civita symbol.
For any linear perturbations of metric δ 1 g µν and δ 2 g µν , the pre-symplectic current is defined as The integral of the pre-symplectic current over Σ is called the pre-symplectic form, denoted by Let H[ξ] denote the charge generating an infinitesimal transformation such that g µν → g µν + £ ξ g µν for a vector field ξ. For a linear perturbation δg µν , it is known [15][16][17][18][19] that the variation of the charge is given by From Eqs. (2), (6) and (9), we get where S µν (g, δg, £ ξ g) is an anti-symmetric tensor defined by Therefore, if the generator H[ξ] exists, its variation satisfies Assuming δg µν satisfies the linearized Einstein equations, the first term vanishes and we get This equation implies that the values of charges are characterized by the asymptotic behaviors of the metric g µν , its perturbation δg µν and the vector field ξ.

4/21
Let us now investigate under what condition the charge exists. Given the formula for the variation in Eq. (19), it must hold (20) for any variation δ 1 and δ 2 as the partial derivatives of a multivariable function commute.
Since it holds equation (20) is equivalent to Although this is a necessary condition for the charge to exist, it is also a sufficient condition as long as the space of g µν has no topological obstruction [19]. Therefore, we call Eq. (22) the integrability condition.
If the integrability condition is satisfied, the charge at a metric g µν can be evaluated by the integral along a smooth path from a reference metric g (0) µν to g µν in the space of metrics. More precisely, by using an arbitrary one-parameter set of metrics g µν (λ) such that g µν (λ = 0) = g (0) µν and g µν (λ = 1) = g µν , the charge at g µν is evaluated as where we have set the reference of the charge H[ξ] so that it vanishes at g (0) µν . Since Eq (20) is satisfied, the charge in Eq.(23) is independent of the choice of path g µν (λ).

Review on a Lie algebra based approach
In this section, we review the our approach developed in [1], where we proposed a guiding principle which helps us to find a non-trivial algebra of the charges. This principle ensures the existence of two elements in the algebra such that their Poisson bracket does not vanish. Therefore, as long as the integrability condition of the charges is satisfied, the transformation generated by the algebra cannot be gauged away.
In order to investigate the asymptotic symmetries of a background metricḡ µν of interest with the covariant phase space method, we have to specify (i) the set of metrics which includesḡ µν and (ii) the set of vector fields which forms a closed algebra. In the following, they are denoted by S and A, respectively. These sets must be chosen such that an element of S is mapped into itself under any infinitesimal diffeomorphism generated by A. Note that only the asymptotic behaviors of the metrics and the vector fields are relevant for the charges. In prior studies, such as [20], it is common to fix the algebra A as the set of vectors satisfying the asymptotic Killing equation for a given S. In this approach, lots of trials and errors are required to find S such that the integrability condition is satisfied and that the charges form a non-trivial algebra.

5/21
In the Lie algebra based approach proposed in [1], an alternative way is adopted to fix S and A; given an algebra A, we define S by where φ * denotes the pullback. In this case, we need to choose A carefully so that the resulting charges are integrable and form a non-trivial algebra. In the rest of this paper, the set S is always defined by Eq.(24).
There are advantages to adopt the set S defined in Eq.(24). First, ifḡ µν is a solution of the Einstein equations, then any element of S automatically satisfies the Einstein equations. In addition, a linearized perturbation δg µν is generated by an infinitesimal diffeomorphism and can be written as with a vector field χ ∈ A. In the following, the variation corresponding to such a perturbation is denoted by δ χ . This property is particularly important to find a candidate of A with the Lie algebra based method as we will see soon. A schematic picture of the set of metrics S is shown in FIG. 1.

Fig. 1
A schematic picture of the set of metrics S defined in Eq.(24). Vector fields ξ and η are elements of a Lie algebra A. All metrics in S are connected to the background metric g µν by diffeomorphisms generated by A. For any metric g µν ∈ S, there exists a smooth path g µν (λ) fromḡ µν to g µν . For any tangent δg µν (λ) at a point g µν (λ) in S, there is a vector field χ ∈ A such that δg µν (λ) = £ χ g µν (λ).
Now, let us review the key idea in [1], which is helpful to find A yielding a non-trivial algebra of charges. The algebra is non-trivial if The diffeomorphism associated with the algebra cannot be gauged away if Eq. (27) is satisfied. Otherwise, all the charges vanish for any metric, implying that the metrics in S cannot 6/21 be discriminated by the value of charges and that the diffeomorphisms generated by A may be gauged away. Note that it may be hard to check the condition in Eq. (27) directly since the set of metrics S depends on A. Instead, we adopt a sufficient condition as a guiding principle to fix A. More precisely, we first derive a formula for for arbitrary vector fields ξ and η. Since Eq. (28) can be calculated atḡ µν , we do not need to specify S nor A at this point. By using it, we then fix two vector fields ξ and η so that Eq.(29) does not vanish. We define A as a closed algebra containing η and ξ, which can be obtained by calculating the commutators of ξ and η. The algebra A defined in this way trivially satisfies Eq.(27) and hence the diffeomorphisms generated by A cannot be gauged away by construction.
Of course, we also need to impose Eq. (22) to get integrable charges. This condition can be recast into where we have used Eq. (25). For a given background metricḡ µν , Eq. (28) works as a guiding principle to find non-trivial charges. However, there still remains a difficulty in finding integrable charges since we have to choose ξ and η so that Eq. (30) is also satisfied, which requires trials and errors. It often takes an effort to check Eq. (30) for an arbitrary g µν ∈ S since we have to calculate the asymptotic behaviors of g µν near the boundary. As a necessary condition, in Ref. [1], we adopted Eq. (22) at the background metric, i.e., before checking Eq. (30) directly. This condition can be checked relatively easily since we only need the background metricḡ µν and the algebra A. The approach proposed in Ref. [1] can be summarized as the following six steps: Step 1 Fix a background metricḡ µν of interest.
Step 2 For the background metric, find two vector fields ξ and η satisfying Eq. (28). These are the candidates generating non-trivial diffeomorphisms whose charges are integrable.
Step 3 Introduce the minimal Lie algebra A including ξ and η by calculating their commutators. Check whether the integrability condition at the background metric, i.e., Eq. (31), is satisfied for the algebra A as a necessary condition for Eq. (30). If it holds, go to the next step. Otherwise, go back to Step 2.
Step 4 Construct the set S of metrics g µν which are connected to the background metricḡ µν via diffeomorphisms generated by A.
Step 5 Check the integrability condition in Eq. (22). If it is satisfied, then go to the following step. If not, go back to Step 2.
Step 6 Calculate the charges by using Eq. (23). Here, we fix the reference metric as the background metric: g µν =ḡ µν .

✒ ✑
In our previous paper [1], we only considered the vacuum solutions to the Einstein equation without the cosmological constant. We can easily extend the analysis to the solutions with the cosmological constant. In this case, Eq.(27) is recast into where C µν αβ := g µγ g νδ C αβγδ is the Weyl tensor and ǫ µν := √ −g(d D−2 x) µν . Note that it can be checked that Eq. (32) is equivalent to Eq. (39) in Ref. [1] if the cosmological constant Λ vanishes. In Step 2 of the above algorithmic protocol, Eq. (28) plays a role of a guiding principle to find non-trivial charges. In addition, Eq. (31) in Step 3 helps to reduce useless calculations on the charges which turn out not to be integrable. An advantage of the above algorithmic protocol is the fact that calculations in Steps 2 and 3 can be done by using only the background metricḡ µν . By using this protocol, we have found a new class of symmetries on a Rindler horizon in Ref. [1], which generates position dependent dilatations in time and in the direction perpendicular to the horizon. We have termed such a transformation superdilatation. However, there still remain the following hard tasks: In Step 4, it is required to identify all diffeomorphisms generated by vector fields in A to obtain S, which is usually difficult. Only after this step is completed, the integrability condition can be checked for all metrics in S in Step 5.
To overcome this issue, in the next section, we propose a sufficient condition for the charges to be integrable, which can be checked at the background metricḡ µν . It enables us to find an algebra A yielding non-trivial and integrable charges without explicitly calculating diffeomorphisms generated by A or the metrics in S. This is a key advantage of the new approach in this article. To calculate the charges explicitly, we still need to identify A and S. However, since the sufficient condition ensures that the charges are integrable, there is no possibility that the efforts in calculating A and S are wasted. 8/21 It should be noted that the algebra of charges can be identified without calculating the values of the charges explicitly. In fact, the Poisson bracket of the charges satisfies where [ξ, η] is a commutator of ξ, η and K(ξ, η) is a constant dependent not on g µν but on g µν (see e.g., Ref. [21]). Evaluating the left hand side of Eq. (33) at the background metric g µν , we get K(ξ, η) since it is always possible to make the values of charges H[χ] ḡµν at the background metricḡ µν vanish for all χ ∈ A. If K(ξ, η) can be absorbed into charges by shifting them by constants, then the algebra of the charges is isomorphic to A. If not, the algebra of the charges is a central extension of A. Therefore, we can fully characterize the algebra of charges itself without calculating the diffeomorphisms generated by A explicitly, overcoming the difficulties in the approach in Ref. [1].

Integrability condition
In this section, we provide a sufficient condition for the charges to be integrable. This condition can be checked at the background metric, implying that we can obtain integrable charges without calculating the family of metrics S directly.
Given an algebra A, the integrability condition that the second line in Eq. (21) equals to zero is recast to where we have used Eq. (25), S is the set of metrics defined in Eq. (24) and ω ν (g, δ 1 g, δ 2 g; x) is given by for a solution g µν of the Einstein equations and linearized perturbations δ 1 g µν and δ 2 g µν satisfying the linearized Einstein equations. To check whether Eq.(34) is satisfied directly, we need the asymptotic behavior of the integrand near the boundary ∂Σ. By using the wellknown duality between a diffeomorphism and a coordinate transformation of tensor fields (see Appendix A for the details), we derive a formula to calculate the asymptotic behaviors under certain assumptions which will be made below. First we introduce our set-up and several assumptions to derive the sufficient condition for the charges to be integrable. We fix a D-dimensional background spacetime (M,ḡ) and a Cauchy surface Σ. For notational simplicity, we fix a specific coordinate system ψ : M → R D in such a way that the Cauchy surface is characterized by t = const. and that its boundary is specified by ρ = 0, where we have defined ψ(p) = (y 0 (p), y 1 (p), y M (p)) = (t, ρ, σ M ) (M = 2, · · · , D − 1).
Let H denote the union of the boundary for all t: 9/21 or equivalently, In this set-up, the integrability condition evaluated at the background metric is given by We assume that any diffeomorphism generated by A does not map a point in the outside (resp. inside) of {Σ t } t to a point in the inside (resp. outside) of {Σ t } t . Then, the ρ-component of the vector fields generating the diffeomorphisms must vanish on the boundary. Thus, we impose the following condition on the asymptotic behaviors of the vector fields: Let us assume that hold. Under these assumptions, we get ∀ξ,η,χ ∈ A,ξ(y) [µ ω ν] (ḡ, £ηḡ, £χḡ; y) =   (41) are a sufficient condition for the charges to be integrable at an arbitrary metric, i.e., Eq. (34). Fix a diffeomorphism φ : M → M generated by A. The integrability condition (34) at g = φ * ḡ is written as where we have adopted another coordinate system ϕ, which is related with ψ by By using Eqs. (A6), (A8), (A13) and (A14), we have where the vector fieldξ is defined byξ := (φ * ) −1 ξ. On the other hand, for the algebra A whose elements satisfy the asymptotic condition in Eq. (40), we have where e α M := ∂x ′α ∂σ M . By using Eq. (46), the asymptotic behavior of e α M is given by for any M = 2, 3, · · · D − 1. By using Eqs. (45) and (47), the left hand side of Eq. (43) is proportional to From the asymptotic behaviors of the coordinates in Eq. (46), any points in H is mapped into itself by a diffeomorphism φ generated by A. Therefore, the integral region φ(∂Σ) corresponds to the limit of ρ → 0. Note that, since ǫ µνα2···αD−1 is anti-symmetric under the change in its indices, the integrand in Eq. (49) vanishes except for the contributions coming from the contractions of indices where one of (µ, ν, α M2 , · · · , α MD−1 ) is ρ. Such a contribution is always O(ρ) since Eqs. (42) and (48) hold. Thus, we finally get and conclude that Eqs. (40) is also a sufficient condition for the integrability condition to be satisfied at any metric g µν in S. Our approach adopted in this paper is summarized in the following 4 steps: Step 1 Fix a background metricḡ µν of interest.
Step 2 For the background metricḡ µν , find two vector fields ξ and η with the asymptotic form in Eq. (40) satisfying Eq. (28). These are the candidates of the vector fields which generate non-trivial diffeomorphisms whose charges are integrable.
Step 3 Introduce the minimal Lie algebra A including ξ and η by calculating their commutators. Check whether Eq. (41) holds. If it does, go to the next step since the charges are integrable. Otherwise, go back to Step 2.
Step 4 Investigate the algebra of the charges for A via (28).

✒ ✑
A crucial difference between the approach in Ref. [1] and the one proposed in this paper is the step where we check the integrability condition. In Ref. [1], we checked whether Eq. (22) holds for candidates of vector fields satisfying Eq.(28). It takes efforts in this step since we need to calculate all the diffeomorphisms generated by the algebra of the vector fields. Furthermore, these efforts may be wasted since the charges sometimes turn out not to be integrable. In contrast, in our new approach, we adopted Eq.(41) as a sufficient condition for the charges to be integrable, which can be checked at the bachground metric. It is much easier to check Eq.(41) than Eq.(22) since we do not need to identify the diffeomorphisms generated by the algebra of the vector fields.
As a demonstration, we investigate asymptotic symmetries on Killing horizons in the following section. Adopting our approach, we find that a class of supertranslation, superrotation and superdilatation yields a non-trivial and integrable algebra of charges with a central extension.

Asymptotic symmetries on Killing horizon
Let us investigate the asymptotic symmetries at a Killing horizon of a spacetime with our new approach developed in the last section. We will find a new class of asymptotic symmetries and show that the algebra of the corresponding charges is a central extension of the algebra of vector fields generating the transformations of the symmetries.
Step 1 Here, we adopt the following D-dimensional metric as the background metric: in the coordinate (t, ρ, ψ, θ A ) for A = 3, · · · , D − 1, where all coefficient functions f tψ , f tA , f ψψ and Ω AB depend on θ A while κ is a constant. We assume that the coefficient functions and κ are fixed so that the metric satisfies the Einstein equations. This class of metrics contains important spacetimes, for example, de-Sitter spacetime and the Kerr spacetime. It is known that the asymptotic behavior of the metric near the Killing horizon located at ρ = 0 is given by Eq.(51) and that the Cauchy surface is characterized by t = const. [12].
Step 2 Next we consider two vector fields ξ and η which have the asymptotic forms given by Eq. (40): as ρ → 0, where all coefficients are arbitrary functions of t, ψ and θ A . For the metric (51), vector fields (52) and (53), our guiding principle in Eq. (32) can be calculated as follows: where M, N = 2, · · · , D − 1 and D M denotes the covariant derivative on the (D − 2)dimensional hypersurface characterized by t = const. and ρ = const.. In Ref. [1], we investigated the following set of vector fields also satisfy Eq.(54). In fact, this set of vector fields generates a well-known class of transformations called supertranslations and superrotations. See Appendix C for a comment on the integrability of the charges for this algebra. As a first trial, let us analyze a simple algebra containing the above two known cases, which is given by in the rest of this section, where Step 3 For an arbitrary set of vector fields with asymptotic behavior in Eq. (53), the presymplectic current at the background metric given in Eq. (16) can be calculated as for ρ → 0. The components of the commutators of the vector fields in Eqs. (59) and (60) are calculated as 13/21 for ρ → 0. Thus, let us define the closed algebra A ′ including ξ, η In this case, since we have then we get ω ρ (ḡ, £ ηḡ , £ ξḡ ) = O(ρ) and hence Eq. (41) is satisfied. This condition in Eq. (65) means that we pick up only a divergenceless part in superrotation. Since holds, the algebra is closed. Therefore, instead of A ′ , we hereafter adopt A. Since Eqs. (28) and (41) are satisfied for A, the charges are integrable and form a non-trivial algebra.
Step 4 Let us investigate the algebra of charges for A. For simplicity, in the following, we will analyze as ρ → 0 in the coordinate system (t, ρ, θ, φ) (0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π) for D = 4. In this case, the induced metric on the horizon is given by ds 2 | ∂Σ = A(dθ 2 + sin 2 θdφ 2 ), where A > 0 is a parameter describing the area of the horizon.

14/21
Functions characterizing an element in A in Eq. (66) can be expanded as follows: where is the spherical harmonics, P m l (cos θ) is the associated Legendre polynomials and holds. Thus, these constants cannot be absorbed to the generators by redefinition. They are calculated as 16/21 Summarizing the above arguments, we finally get the following charge algebra: others are isomorphic to A in Eqs. (75a)-(75k) except for (75e).
Since A = 0, the algebra of the charges is a central extension of A. Equations (86) and (87) are the main results in this section.

Summary
In this paper, we developed a new approach to investigate asymptotic symmetries by modifying the protocol proposed in Ref. [1] by the authors of this paper and a collaborator. The key ingredient of our approach is making use of Eqs. (28) and (41) to find the algebra A of vector fields that generates transformations of asymptotic symmetries with non-trivial and integrable charges. As we have seen in Sec. 4, Eq. (41) provides a sufficient condition for the charges to be integrable, which can be checked at the background metric. This is a significant difference between the modified approach and the original one in Ref. [1], which saves the efforts of calculating all the diffeomorphisms generated by A required in the latter approach. As is mentioned in Sec. 3, the Poisson brackets of the charges can be calculated at the background metric and hence the algebra of the charges can be fully identified without calculating the diffeomorphisms generated by A explicitly. In Sec. 5, as a demonstration of our approach, we have investigated asymptotic symmetries of spacetimes with the Killing horizon with metrics in Eq. (51). We found that a new algebra of supertranslations, superrotations and superdilatations in Eq. (66) yields a nontrivial algebra of integrable charges. It is proven that for the algebra in Eq. (66), we have to eliminate rotationless part of superrotations to obtain integrable charges. As a particular example, for (1 + 3)-dimensional spacetime with metrics in Eq.(67), we explicitly calculated the algebra of charges, which is shown to be a central extension of the algebra of the vector fields.
It should be emphasized that our approach can be applied to any spacetime as long as we consider the diffeomorphisms which do not shift the boundary on which charges are defined. In particular, as we have mentioned in Introduction, microstates classified by asymptotic symmetries on a horizon are a possible origin of the Bekenstein-Hawking entropy. Our algorithmic approach is powerful to list such asymptotic symmetries. A discovery of new asymptotic symmetries will lead a better understanding of the nature of gravity and the spacetime structures, as the asymptotic symmetries in anti-de Sitter spacetime found in Ref. [20] led to the development of the AdS/CFT correspondence [22].
Of course, it should be noted that there may be asymptotic symmetries which cannot be found in our approach since Eqs. (28) and (41) are sufficient conditions for the charges to be integrable and form a non-trivial algebra. Nevertheless, we expect that our approach proposed in this paper is helpful to find new asymptotic symmetries as we have demonstrated the example in Sec. 5.