Symmetries play fundamental role in the development of the fundamental laws of nature. Thanks to the celebrated Noether's theorem, which says that for every continuous global symmetry of a physical system, there exits associated conserved charges. These global symmetries are typically represented by the finite-dimensional group of transformation and associated Lie algebra. In the context of gravitational theory, it is the diffeomorphism symmetry which plays fundamental role. The associated Noether charges [1, 2] are believed to be encoded into the thermodynamics properties of gravity [3] [4] [5] . One of the most striking object of gravity is black hole which is manifestly endowed with thermodynamics properties specially attributed to the horizon [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] . Horizon is a null hyper-surface located at a special point in radial coordinate of the black hole spacetime. Out of infinite dimensional diffeomorphism symmetry, it has been shown to exist a special class called asymptotic symmetry which forms a closed Virasoro algebra [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] (an extensive list of works in this direction can be obtained from reference list of [14] ). An important thermodynamic quantity namely entropy of the horizon can be expressed in terms of the central charge of this algebra by using well known Cardy formula [22] . This approach was first formulated in the seminal work by Brown and Henneaux for the (1+2) dimensional gravity theory [23] . Later it was proved that the entropy and various other thermodynamic quantities can be attributed to any generic null surface [21, [24] [25] [26] .
In recent years the understanding of asymptotic symmetries has received widespread interests in the realm of infrared properties of gravity, particularly focusing on * maitra.91@iitg.ac.in † debu@iitg.ac.in ‡ bibhas.majhi@iitg.ac.in scattering phenomena near the null boundaries of spacetime [27] - [32] (for a review, see [30] ). It has been shown that these low energy scattering phenomena can be beautifully described by aforementioned non-trivial group of infinite-dimensional symmetries first appeared in the pioneering works by Bondi, van der Berg, Metzner and Sachs (BMS) [33] - [38] . In these works, the attempt was to figure out the isometries of asymptotically flat spacetime at null infinity. To their surprise the symmetry group turned out to be an infinite dimensional one with Poincaré as sub group. Apart from usual Poincaré transformation, the additional infinite number of generators of symmetry transformation are coined as "supertranslations". The crucial point in all these discussion was related to the boundary conditions. Significant efforts have been devoted in specifying the appropriate set of boundary conditioBibhas Ranjan Majhins on the gravitational field configuration. One of the popular procedure is to identify the fall off behavior of the background metric components expressed in Bondi-Sachs coordinates at null infinity. The basic criterion of identifying those boundary conditions was that under arbitrary diffeomorphism the form of the background metric must be preserved at its asymptotic null boundaries. Subsequently analogous asymptotic symmetry analysis has been explored near another null surface namely, horizon of black hole [39] - [42] . Further analysis have been done near both extremal and non-extremal null surfaces in a more general setting which include Lanczos-Lovelock (LL) gravity, non-gravitational fields such as U(1) electromagnetism [43] . Also, the symmetry parameters are shown to behave as Goldstone modes which leads to thermal nature of the horizon at the semi-classical level [44] .
As has been pointed out already most of the studies on asymptotic symmetry has been performed either at asymptotic null infinity or near the black hole horizon. A natural question arises, what will be the physical implication of the diffeomorphism summetries on a physical boundary located in the bulk spacetime? To explore the full power of this asymptotic symmetries in understand-ing the nature of gravity, we believe it is timely to explore the asymptotic symmetries near any arbitrary non-null hypersurface embedded in the bulk spacetime manifold. The hypersurface can be thought of as domain wall dividing the spacetime into two regions. In our present discussion we will consider a timelike hypersurface situated at a finite radial distance outside the black hole horizon.
To motivate further with regard to our choice of the timelike boundary in the bulk spacetime, it is important to mention the recent developments on the correspondence between gravity and fluid dynamics, where timelike surface away from the horizon play important part. The interesting connection has been established between Einstein field equations of gravity and Navier-Stokes (NS) equation of any fluid on a timelike surface [45] - [50] . Wherein the (p + 2)-dimensional metric solution of Einstein's equations of motion describes a (p + 1)dimensional non-relativistic incompressible fluid on the timelike surface. This is obtained from the conservation Brown-York surface stress energy tensor on the timelike hypersurface. Interesting connection has been uncovered between the fluid dynamics on the timelike membrane and infinite set of symmetries and associated conserved charges which are similar to the BMS symmetries and charges [51] . All these important results motivate us to perform the detail investigation on the diffeomorphism symmetries of the timelike surface under consideration.
A timelike hypersurface defined by the radial coordinate r = r c > r H is assumed to be situated outside the black hole horizon. In the present paper we consider both Schwarzschild and Kerr black hole spacetimes. First step would be to express the metric near the surface in Gaussian normal coordinate. In order to do that we have followed the methodology outlined in [52] . Once the metric is derived we will follow the usual procedure for the asymptotic symmetry analysis. By imposing appropriate boundary conditions near the surface we determine the diffeomorphism symmetry parameters. The boundary conditions are such that the form of the metric near the timelike surface must remain invariant under arbitrary diffeomorphism. The conserved charges due to the diffeomorphism symmetries are calculated and the associated symmetry algebra are constructed thereafter.
In this section we give a quick overview on the construction of the Gaussian normal coordinate (GNC) system which will be used in the later part of our analysis. These coordinate system is suitable to explain physical quantities on an (p + 1)-dimensional hypersuface S, embedded in the (p + 2)-dimensional manifold N . For the present discussion we consider sub-manifold S as a timelike hypersurface. One can choose a unique vector N a in the tangent space of the manifold N in such a way that the vector will be orthogonal to all the vectors in the tangent space of the submanifold S. So for S, we can normalize N a such that N a N a = 1; i.e. N a is spacelike corresponding to signature (−, +, +, +) of the metric. Thus the vector N a is unit normal to S. Now we can construct unique geodesic with tangent N a emanating from an arbitrary point P on the aforesaid hypersurface. So the geodesic must pass through another arbitrary point Q situated in the small neighbourhood of the hypersurface S. In this setup one defines a coordinate system in the form (x 1 , x 2 , x 3 , ρ), among them (x 1 , x 2 , x 3 ) denote the coordinates of the point P on the surface. On the other hand fourth coordinate ρ labels the point Q which lies on the geodesic, very near to S. Here ρ is a spacelike coordinate while any one of (x 1 , x 2 , x 3 ) is timelike and rest are spacelike coordinates. So the GNC system (x 1 , x 2 , x 3 , ρ) will be well defined in the small neighbourhood of the point P in such a way that the congruence of spacelike geodesics will be orthogonal to the hypersurface. The spacetime metric in the neighbourhood of the surface can be expressed in the following form (more details in this regard can be found in [53] ):
Here µ, ν indices are defined on the hypersurface S whose induced metric is given by h µν . Whereas, a, b correspond to spacetime indices. Here we have metric coefficient g ρρ = 1 due to the normalization of the vector field N a . Furthermore g ρµ = 0 since vector N a is orthogonal to the surface. Hence in this newly formed coordinate system, ρ = constant hypersurface describes timelike hypersurface S in GNC. Next in the upcoming sections we will express Schwarzschild and Kerr geometry in this coordinate system in order to study the symmetry properties of an arbitrary timelike surface under consideration.
The Schwarzschild metric in original Schwarzschild coordinates is given by,
where, dΩ 2 = (dθ 2 + sin 2 θdφ 2 ). Since our main objective is to study the symmetry of this spacetime near a timelike surface, we shall express the above metric in GNC near this hypersurface. We choose S as r = r c (constant) hypersurface lies outside the horizon r = 2M . Below we shall follow the procedure, described in the last section, to express metric (2) in GNC (more detail in this regard can be found in [52] ). The unit spacelike normal vector to S for Eq. (2) is given by, N a = (0, 1 − 2M r , 0, 0). The spacelike geodesic congruence that cross the surface orthogonally will posses N a as a tangential vector near the surface. Near the surface we can parametrize the geodesics with affine parameter ρ = r−r c . The timelike surface is identified as ρ = 0. Therefore, any geodesics curve X a (ρ) with X a = (t, r, θ, φ), crossing the surface can be expressed through the following Taylor series expansion,
The first term in the right hand side is identified as X a ρ=0 = (t, r c , θ, φ). The second one is given by the tangent vector to the curve, dX a dρ ρ=0 = N a ρ=0 . The third term can be evaluated by considering the geodesic equation N a ∇ a N b = 0 at ρ = 0. This yields
Where all the other component equations identically vanish. Using these in Eq.
(3) and keeping upto second order in ρ, we find the transformation of coordinates from (t, r, θ, φ) to (t ′ , ρ, θ ′ , φ ′ ) as,
Application of these transformations into Eq. (2) leads to the following form of the metric:
where the argument x corresponds to the angular coordinates (θ, φ). Now the metric coefficients can be obtained by simply using tensor transformation and keeping up to O(ρ) as follows g ρρ (ρ) = g rr ∂r ∂ρ ∂r ∂ρ ,
Note that in the sufficiently small neighbourhood of the surface the metric Eq. (6) assumes the form Eq. (1) to the leading order in ρ. In our following analysis of asymptotic symmetries we consider this metric near the timelike hypersurface r = r c in this background.
The key ideas in identifying the asymptotic symmetries are intertwined with the appropriate set of boundary conditions on the variation of the metric coefficients. Due to diffeomorphism the structure of the metric under consideration is assumed to remain same in our region of interest. Generically the conditions on the metric coefficients are of two different categories. In one category, the gauge fixed conditions are strictly assumed to be invariant under diffeomporphism transformation x a → x a + ζ a and those are
In the above expression £ ζ implies the Lie derivative along the vector field ζ a . We call the above set as "strong" conditions. These four conditions Eq. (9) essentially provide residual diffeomorhism transformation generators. The functional form of those generators can possibly be restricted by the other category of transformations under which the metric coefficients are assumed to change with the appropriate fall-off conditions as
These weak fall off conditions are postulated considering the leading order behaviour of the metric coefficients g tt and g AB in terms of ρ. Note here that the Lie variation of the component g tA is considered as O(ρ), although this component is zero in the original metric Eq. (6) in GNC. This is allowed as near S the metric remains same and so consistent with our imposed symmetry. We now first solve the gauge fixed condition Eq. (9) to identify the appropriate symmetry parameter, and the form of the solutions are as follows,
where T, F and R A are the integration constants. These are the residual diffeomorphism symmetry parameters under which the gauge choice remains the same. To proceed further, we chose R A = 0 as Schwarzschild spacetime does not posses any angular momentum to begin with. Further, we also set T (t, x) = 0 which keeps the position of the hyperfurface intact at ρ = 0. Therefore, we are left with one diffeomorpism parameter F which is similar to the super-translation discussed in the literature for the asymptotic null boundary and null horizon. The functional form of F will be further constrained by the fall off condition Eq. (10) particularly for g tA component which enforces the following constraints,
This feature is completely different from what was obtained for the case of a general null-surface (see e.g. [43] ). The other two fall off conditions on metric components g tt and g AB do not give any further constraint relations. Finally we can write symmetry parameters very near to the surface as;
Analogous to the near horizon symmetry, we call F (t) as supertranslation parameter. Unlike near horizon symmetries [39] and asymptotic null infinity BMS symmetries [34] , symmetry generator near the timelike hypersurface embedded in the Schwarzschild spacetime, the supertranslation parameter turned out to be dependent only on time coordinate.
Now we want to construct symmetry algebra followed by the super-translation generator constructed in the last section. We have only one non-zero component of ζ a = (F (t), 0, 0) which will generate the symmetry algebra of the residual symmetry transformation. Since the g tt component of the metric does not diverge near ρ = 0, the periodicity of Euclidean time near the surface in principle is infinite. Therefore, associated Fourier mode of F will be continuous in frequency space ,
Here α(ω) represents individual Fourier mode of F (t).
The kernel functionF (ω, t) = exp (−iωt) will form the complete orthogonal basis with continuous frequency ω. Now Let us consider two parameters ζ a 1 = (F 1 , 0, 0) and ζ a 2 = (F 2 , 0, 0) associated with the super-translation generators. The standard Lie bracket algebra between these two diffeomorphism vectors is given by [54] ,
where,
Here δ g ζ1 ζ 2 denotes the variation in ζ 2 under the variation of the metric induced by ζ 1 . Second and third terms in the definition of Lie bracket (15) are considered in order to take into account the variation of the vector fields due to the higher order variation of the metric. But we will ignore higher order effect on the variation of diffeomorphism vectors. Therefore, Eq. (16) will be our main equation to construct the symmetry algebra. In terms of F , the Lie bracket takes the following form,
In terms of Fourier modes, the left hand side becomes,
with α(ω 1 , ω 2 ) = α(ω 1 )α(ω 2 ), and the right hand side of the equation Eq. (17) becomes,
Comparing both the equations Eq. (18) and Eq. (19), the algebra among the basis functions becomes,
Here the important point to mention is that although the supertranslation parameter defined on the timelike surface in Schwarzschild spacetime came out to be dependent only on time, but the aforementioned bracket algebra (20) is similar to the that obtained for a generic null surface ( [43] ).
For any diffeomorphsim invariant gravity theory conserved charges corresponding to symmetries play crucial role in understanding the thermodynamics of the black hole. Innumerable studies have been done where some specific set of diffeomorphism symmetric charges have been shown to be related to the black hole entropy [1]- [19] . In all these works, the charges have been calculated on the horizon two dimensional null surface. Here we are dealing with timelike surface which is three dimensional. In the present context we shall calculate the charge at a particular instant of coordinate time.
In Einstein gravity conserved charges due to diffeomorphism symmetry on a two-surface is expressed in the following way [55] ,
where J ab is the Noether potential, given by
The integration is being done on the two-surface. In the present situation we shall perform this on the ρ = 0 surface at a particular time slice. Therefore this subspace is two dimensional and can be indicated by two unit normals: one is timelike and another is spacelike. Hence this particular subspace is null, but not the horizon as r c = 2M . So the charge (21) corresponds to any null surface other than horizon. The surface element on the 2-surface is
where σ is the determinant of the induced metric on the hypersurface under consideration. M a is the unit spacelike normal vector defined on the timelike surface (ρ = constant), and is given by, M a = (0, 1, 0, 0). Similarly n a is the unit timelike normal on the spacelike hypersurface (t = constant) taking the following form,
As the timelike surface is situated at ρ = 0 the only non-vanishing component of the surface element in Eq. (23) is,
Using Eq. (13) and Eq. (25) one obtains the diffeomorphisms charge to be,
As expected form the spherically symmetric background, the charge comes out to be independent of the position of the timelike surface under study. For Kerr black hole background, we will see this does not hold true. Next in order to compute bracket algebra between charges we express the Noether charge Eq. (26) in terms of the Fourier transform of the supertranslation parameter given in (14):
The Fourier mode of the charge is defined as
Therefore, the algebra among the charges directly follows from the that of the supertranslation F as follows,
which leads to,
Using Eq. (20), the final expression becomes,
Before we end the discussion, let us try to explore the thermodynamic interpretation of the super-translation charge Eq. (28) if any in a more general setting. Let us start with the general static, spherically symmetric metric of the form ds 2 = −f 1 (r)dt 2 + f 2 (r)dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ) . (32) In this case the metric near timelike hypersurface r c in GNC is determined to be as
The above can be obtained by using the following transformations,
Now like earlier the charge for the symmetries near r c , given by (13), for F (t) = 1 at a particular time turns out to be
where A rc = 4πr 2 c is the area of our subspace of timelike hypersurface.
We shall now show that the above result may be interpreted as the amount of heat content in the hypersurface under study at a particular instant of time. This can be straight forwardly seen for the simple Schwarzschild case, where the metric coefficient is f 1 = 1/f 2 = 1 − (2M/r). The local gravitational acceleration at the radial coordinate r c is given by g(r c ) = M/r 2 c = f ′ (r c )/2. Hence, the choice of global super-translation generator, F (t) = 1, yields
For general spherically symmetric black hole Eq. (32), it can be proved that local acceleration takes the from g(r c ) = f ′ (r c )/ 4f 1 (r c )f 2 (r c ). A discussion regarding this has been presented in Appendix A. Hence, the quantity Q may be interpreted as the local heat content on the hypersurface at a fixed time if one identifies the surface entropy S c ∝ A c and local temperature T c ∝ g(r c ). The temperature is identified as the Unruh temperature which is directly connected with the local gravitational acceleration g(r c ). Hence, with respect to the local observer hovering near the r = r c hypersurface will see the surface as thermal object with the aforementioned Unruh temperature T c . The same conclusion can also be obtained though an interesting perspective considering the Tolman relation [56] - [58] , which gives rise to temperature gradient of a thermodynamic equilibrium system in the gravitational field. The relation entails T (r) g ab k a k b = T (r) −f 1 (r) = T 0 , where k a = (1, 0, 0, 0) is the timelike Killing vector of the background under consideration and T (r) is called Tolman temperature. T 0 is arbitrary constant equilibrium temperature. Using the relation f ′ 1 = −2f 1 (T ′ /T ) and F (t) = 1 which precisely corresponds to the time-like Killing vector k a , we obtain the following interesting relation,
where we have chosen the normal to r = constant surface in such a way thatn a = (0, √ f 2 , 0, 0) and consequentlŷ n a = (0, 1/ √ f 2 , 0, 0). Now it has been shown in [59] that a classical gas of radiation under Newtonian acceleration satisfies ∇ a T /T = −g a . Therefore the temperature gradient part of (37) leads to gravitational acceleration along the normal to hypersurface and hence we obtain (36) . It would be interesting to explore this relation Eq. (37) further in the context thermodynamic origin of gravitational theory.
So far we have tried to explore the symmetry of a generic non-null hypersurface situated at r c > r H in the Schwarzschild black hole background. In this section we consider its flat space limit M → 0. One expected outcome naturally follows from the Eq. (27) is that the symmetry charge vanishes on the timelike hypersurface. This can also be straightforwardly calculated considering a generic timelike surface in the Minkowski background. In spherical polar coordinates the metric is ds 2 = −dt 2 + dr 2 + r 2 (dθ 2 + sin θ 2 dφ 2 ) . (38) Following the same procedure as discussed in detail we can obtain the induced metric on a generic time like surface defined by r = r c in Gaussian normal coordinates as,
The Lie variation of the gauge conditions and other metric components as fall-off conditions will remain the same as given in Eq. (9) and Eq. (10) . Therefore, the solution of the gauge choices will be same as found in Eq. (11) with the limit M → 0. The constraints coming from the fall-off condition will be same as before Eq. (12) .
Hence symmetry algebra will remain same Eq. (20) as for the Schwarzschild black hole background. However, as mentioned in the beginning, our explicit computation and also obvious from Eq. (27) the associated conserved charge becomes zero. As we have already pointed out in the previous section, the super-translation charge is directly related to an important thermodynamics quantity called heat content associated with the global parameter F = 1. Hence, our explicit computation for the flat spacetime gives consistent null results for the same.
We will now extend earlier discussion for a timelike surface located outside the horizon of Kerr black hole. The Kerr metric in Boyer-Lindquist coordinate is expressed as
where, ∆ = r 2 + a 2 − 2M r and Σ = r 2 + a 2 cos 2 θ . We will follow the same methodology as elaborately discussed for the Schwarzchild black hole. Our task is to express the metric in the neighbourhood of the time like surface r = r c > r H in Gaussian normal coordinate in the form of Eq. (1). The detail procedure in this regard has been discussed in the last section in case of Schwarzschild spacetime.
In the Kerr background spacetime, the non-vanishing component of the unit space like normal vector N a to the hypersurface under consideration is given by,
Now following the same procedure discussed in the previous section, and Taylor expanding the geodesics as given in Eq. (3), we can define the transformation between spacetime coordinates (t, r, θ, φ) and GN coordi-
where we define s 1 (θ) = N r r=rc , and
The above transformations upto O(ρ 2 ) will be sufficient for our subsequent discussion. Using the above transformation of coordinates, the induced metric near the spacelike hypersurface under consideration will assume the GN form as Eq. (1) where metric components upto O(ρ) are given by,
g θθ = Σ rc + 2ρ r c s 1 (θ) ;
Here Σ rc and ∆ rc are the corresponding quantities defined at r = r c . These are given by,
Therefore, the above form (44) in GN coordinate properly describes the spacetime geometry very near to the timelike surface situated at ρ = 0 (where r − r c ≈ ρ).
Unlike Schwarzschild black holes, Kerr spacetime is endowed with angular momentum which play characteristically different role in defining the properties of the associated symmetry algebra. Main objective of the whole symmetry analysis program is to keep the structure of the metric under investigation invariant under diffeomorphism transformation. Here also the following constraints on the behaviour of some of the metric components which are associated with the gauge condition, need to be maintained as before, and those are,
where A stands for the transverse coordinates θ and φ.
Because of the inherent rotational angular momentum of the spacetime, the weak fall off conditions of the remaining metric component in terms of ρ coordinate are assumed to follow,
The fall-off conditions directly follow from the leading order behaviour of the metric coefficients g tt , g tA and g AB . Solving the three strong form of gauge conditions expressed in Eq. (46) we determine the general solutions of the symmetry vectors as,
where x corresponds to the angular coordinates θ, φ.
Here T F , and R A are the unknown integration constants. These are the diffeomorphism vectors under which the form of the metric near the time like surface under consideration will remain invariant. Now further restrict on the above vectors under which the position of the surface does not change will lead to the following additional constraint T (t, x) = 0. The fall-off conditions expressed in Eq. (47) have been observed to be automatically satisfied by the solution space Eq. (48) . Finally the non-vanishing components of the symmetry vectors ζ = ζ a ∂ a are given by,
In the above expression R A denotes the rotation parameter. For generality we assume all the differmorphism parameters to be function of all the coordinates namely ζ θ = R θ (t, θ, φ) and ζ φ = R φ (t, θ, φ). Following the conventional definition we describe the parameters F as the generator of the supertranslation and R A as the superrotation. Important to note that for Kerr black hole background, the supertranslation generator turned out to be both time and angular coordinate dependent as opposed to that of the Schwarzschild black hole background. Following the same procedure discussed for the Schwarzchild black hole, we now explore the bracket algebra among the three symmetry generators χ a = (F, 0, 0) and ξ a = (0, 0, R A ). The generators are expressed in terms of following Fourier expansion,
In the above expression α(ω) andᾱ A (ω) represent the Fourier modes in frequency domain of the functions F (t, x) and R A (t, x) respectively. The argument x corresponds to angular coordinates (θ, φ).F k (ω, t, x) and R l A (ω, t, x) are identified as basis generators among which the commutator algebra will be as follows, By using Eq. (16) , the required bracket for the supertranslation vector with itself can be calculated as,
After expanding the bracket we obtain,
Comparing expression (52) with (53) we get,
By using the similar manner we can calculate the commutator algebra between supertranslation and superrotation parameters as
The Lie bracket for the superrotation diffeomorphism vector with itself will take the following form,
Let us point out important difference between the results obtained for the time like surface embedded in Kerr spacetime from that of the null surface [43] . For the null surface case, the notable difference is that the general super-rotation generators is time independent as opposed to the present case. Therefore, this fact leads to difference result for the [R(t, x), F (t, x)] commutator. On the null surface the aforementioned commutator generates super-translation only where as on the time like surface it generates both super-rotation and super-translation depending upon the parameter values of the transformation.
Associated with the symmetries discussed in the previous section, we now explicitly compute the conserved charges defined on the two dimensional r = constant, t = constant surface which is essentially a two sphere. The sphere at a particular instant of time is characterized by a spacelike and a timelike normal vectors defined as M a and n a respectively. For the metric Eq. (44), the unit spacelike normal vector M a assumes M a = (0, 1, 0, 0) and the non-zero components of the unit timelike normal vector n a are:
where we have defined the following quantities in GN coordinate, Σ(ρ, θ) = (r c + ρ s 1 − ρ 2 s 2 ) 2 + a 2 cos 2 θ, ∆(ρ, θ) = (r c + ρ s 1 − ρ 2 s 2 ) 2 + a 2 − 2M (r c + ρ s 1 − ρ 2 s 2 ), α(ρ, θ) = ((r c + ρ s 1 − ρ 2 s 2 ) 2 + a 2 ) 2 − a 2 sin 2 θ∆(ρ, θ),
Following the same procedure as discussed for the Schwarzschild case, the area element dΣ ab survives with two non vanishing components: dΣ tρ and dΣ ρφ . With all these ingredients the diffeomorphism charges can be expressed near the surface ρ = 0, as follows,
The first term is super-translation charge with the following explicit form,
the second term is attributed to the charge associated with the superrotation,
Now we are in a position to inspect the physical interpretation of the conserved supertranslation and superrotation charges. For simplest choice, F (t, x) = 1 and R A (t, x) = 1 the results (60) and (61) yield the followings form of the charges,
which can be identified as the Komar conserved quantities [60] [61] [62] defined on the timelike surface at a particular instant of time in the Kerr black hole spactime. However, with the given diffeomorphism vectors, we have not found any obvious thermodynamic interpretation as has been found for the Schwarzchild case. However, if we consider a slightly more general diffeomorphism vectors
where
is the the angular velocity on our timelike surface. In that case we have
To find similar relation obtained in Eq. (37), following the reference [63] , we generalize the Tolman relation for stationary spacetime as T (r) −g ab ζ a ζ b = T 0 where ζ a is the timelike vector, constructed out from the linear combination of two existing Killing vectors (∂ t ) a and (∂ φ ) a of the spacetime. Now, in this case the form of ζ a is taken to be ζ a = (1, 0, 0, Ω c ) and we define |K| = −(g tt + 2Ω c g tφ + Ω 2 c g φφ ) .
Then the Tolman temperature gradient leads to the gravitational acceleration along the normal to the surface as follows,
wheren a is the normal to the r =constant surface given by, (0, 1/ √ g rr , 0, 0). Associated expression for the conserved charge (64), in differential form, will then take the identical form as,
Area element δA c is given by √ α c sin θ dθdφ. It is now easy to check that after integration over transverse coordinates the expression (67) reduces to (64).
The charges can now be expressed in terms of the mode function of the symmetry generators as,
where each mode can be written as follows,
d 2 x 2M (sin θ)F k (ω, t, x) (a 2 + r 2 c )(r 4 c − a 4 cos 4 θ) (r 2 c + a 2 cos 2 θ) 3 ,
and Q A [R l (ω, t, x)] = 1 16πG
d 2 x 2aM (sin θ)R l A (ω, t, x) (3r 6 c + a 2 r 4 c + 4a 2 r 4 c cos 2 θ − a 4 (a 2 − r 2 c ) cos 4 θ) (r 2 c + a 2 cos 2 θ) 3
.
Then using Eq. (29) the algebra between modes of charges are computed as follows:
It is clear from the above Eq. (71) that the symmetry bracket among the charges are isomorphic to that among diffeomorphism vectors. Hence we can conclude that the components of the diffeomorphism symmetry generators together form a closed algebra which is quite different from the standard near horizon BMS algera.
In this paper we have studied in detail the symmetries of a generic timelike hypersurface embedded in black hole spacetime. We have considered two different black hole spacetime, where symmetries near any arbitrary rconstant hyper-surface have been observed to follow similar algebra as that of the asymptotic infinity or near black hole horizon. Symmetries have been identified by considering the class of diffeomorphisms which keeps the form of the metric near the time like surface invariant. Symmetry algebra turned out to be similar to that of the surfaces located at the horizon and at the asymptotic infinity of the black hole. The associated symmetry charges have also been computed and shown to follow the same algebra as differomorphism transformation. We have also established an interesting connection between the symmetry charges with the local heat content on the surface under the study.
The timelike surface divides the space into two complementary regions which are causally disconnected at any instant of time. Therefore, the symmetries near the surface could play interesting role in understanding the entanglement phenomena in quantum theory. Usually all the symmetry analysis have been studied so far in the asymptotic infinity or near the horizon keeping in mind the causal physical phenomena such as scatting of massive or massless particles. Role of symmetries in acausal phenomena such as entanglement has not been discussed in the literature. Our present analysis of symmetries near timelike surfaces could be useful to understand this acausal phenomena which will be taken up in future pub-lication.
the magnitude of the proper acceleration, defined as a prop = a i prop a iprop with a i prop = dX a /dτ , from (A3) is obtained as,
Therefore the local acceleration is given by a(x) = a prop / √ x. At r = r c , from (A4), this is given by
. (A6)