Quasilocal Smarr relation for an asymptotically flat spacetime

A quasilocal Smarr relation is obtained from Euler's theorem for $n$-dimensional asymptotically flat spacetimes and it is checked through several examples by calculating quasilocal variables using the Brown-York quasilocal formalism, along with Mann-Marolf counterterms. To find the entropy in the quasilocal frame, we define a quasilocal free energy from the Euclidean gravity action with a Tolman temperature. We discovered that the quasilocal free energy definition is not altered by a surface pressure term, and the entropy obtained through this method agrees with the usual Bekenstein-Hawking entropy. We also check that the quasilocal free energy combined with the quasilocal Euclidean action value is consistent with our quasilocal Smarr relation.


Introduction
One of the aspects of General Relativity is that there is no local definition for the mass, angular momentum, charge, etc. of a gravitating object; these are inherently Newtonian concepts and do not transfer over to curved spacetime. Nevertheless, such quantities can be defined in a global sense as asymptotic charges of the spacetime.
In fact, we know how to calculate a conserved charge for matter fields in flat spacetime well. If the Lagrangian is invariant under spacetime transformation then a scalar field will change according to and the energy-momentum tensor T ab is calculated and satisfies Noether's theorem then ensures that there exist conserved quantities given by E = d 3 xT 00 , P i = d 3 xT 0i (1.4) where the scalar field is assumed to have compact support and the volume integrals are taken over that compact support. On the other hand, in curved spacetime the energy-momentum tensor is contained in Einstein's equation, where the left term contains matter fields only and also satisfies (1.3). Therefore a conserved charge for matter fields can be obtained by the energy momentum tensor from the left side of the Einstein equation. However, no information can be obtained for the gravitational field from the energymomentum tensor. Instead, information about the gravitational field is encoded in the Riemann tensor. Traditionally, the canonical formulation requires an arbitrary choice for the separation between space and time, while general relativity treats these as a unified spacetime. Through this formulation, constructing the concept of gravitational energy along with computing this energy was an arduous task. Another difficulty was that in curved spacetime physical quantities are computed locally, whereas there is no meaningful local notion of energy density of the gravitational field to construct a conserved charge in general relativity.
Even though these difficulties, the global charge for a gravitational field was first successfully obtained in 1959 by R. Arnowitt, S. Deser, and C. Misner, which is known as the ADM mass [1][2][3][4]. Through this approach, a spacetime is decomposed into spatial hypersurfaces foliated by time so that the traditional canonical method is applied for a gravitational field at spatial infinity. The ADM result implies that a global charge is independent of a choice of a coordinate. Based on this success Komar alternatively constructed a conserved charge formula by using a Killing vector field where c = −2 for a timelike Killing vector field and c = 1 for a rotational Killing vector field which give a mass and an angular momentum for a gravitational field respectively. Later, various methods such as AD(T) method [5,6], Brown-York's quasilocal method [7], covariant phase space method [8,9]  where T eff = κ 8π is an effective surface gravity, A is the black hole area, Ω is an angular velocity, L is an angular momentum, Φ is an electromagnetic potential, and Q is a charge of the electromagnetic field. It is important to note that the mass is expressed in a bilinear form of other physical variables. The differential form of mass, dM , is also shown in the paper as dM = T eff dA + ΩdL + ΦdQ. (1.8) Soon after J. Bardeen, B. Carter, and S. Hawking published the same results of (1.7) and they interpreted that (1.8) is analogous with a first law of thermodynamics by corresponding 4T eff = T = κ 2π to a real temperature and S = A 4 to entropy [11]. At that time, however, those authors pointed out that T and S should be distinct from real temperature and entropy. Later Jacob Beskenstein boldly insisted that S is the real entropy of black holes [12][13][14], and Steven Hawking further corroborated this by finding that black holes radiate as thermal objects at a temperature of precisely κ 2π [15,16]. All of this work was developed using asymptotic charges, but it is important to describe the physical variables of a gravitational system in a quasilocal frame for several reasons. First, the quasilocal quantities could describe more realistic and detailed physical situations, such as binary stars or black hole mergers. Second, spatial infinity cannot by realized in numerical works, but rather finite domains are always required. Numerical studies of collapse also often track apparent horizons which are quasilocal in nature compared to event horizons [17,18]. Several quasilocal formalisms have been suggested [19], and the first law of black hole thermodynamics in a quasilocal frame is studied in Brown-York's work [7]. In such work, the subtraction background method is employed to render the gravity action finite, and the boundary energy-momentum stress tensor is constructed so as to define quasilocal quantities. They showed that the first law of thermodynamics in a finite domain for the four-dimensional Schwarzchild black hole is yielded as where the Bekenstein-Hawking entropy S is used, satisfying the area law. Here new definitions are introduced different from (1.8). P is surface pressure and A is an area of a quasilocal surface with a certain radius, for example r = R. Their definitions will be explained in the next section. E is a quasilocal energy and T Tolman is the Tolman temperature defined by where κ is the event horizon surface gravity and 1 N (R) is the lapse function evaluated on the quasilocal boundary. Hereafter we denote the Tolman temperature T Tolman as T R which means the temperature measured at r = R.
Historically the first law of the black hole thermodynamics was introduced by varying the Smarr relation which is the bilinear form of the thermodynamic variables, but the Smarr relation in a quasilocal frame has not been explored yet. To complete thermodynamics of the black hole system it is important to obtain the quasilocal Smarr relation. In this paper, we confine ourselves in an asymptoically flat spacetime for a general dimension of n. We find a quasilocal Smarr relation from Eulers' theorem and check it with several examples. To do so, we calculate the quasilocal variables by employing Brown-York's quasilocal formalism with a Mann-Marolf counterterm. Entropy is found by defining a quasilocal free energy through the Euclidean gravity action with the Tolman temperature. We found that the quasilocal free energy expression is not altered by the surface pressure term and entropy is the same as Bekenstein-Hawking entropy in our examples. This paper is organized as follows. In section 2, the Mann-Marolf counterterm and Brown-York's formalism are briefly explained along with the construction of the renormalised gravity action. In section 3, the free energy in a quasilocal frame is defined by Euclidean formulation and an entropy expression is found in a quasilocal frame. In section 4, a quasilocal Smarr relation is derived from the Eulerian theorem for a charged black holes in n dimensional spacetime. In section 5, we check this quasilocal Smarr relation with several black hole spacetimes. Lastly, we summarize our results and discuss the future works.

The quasilocal method by Brown-York with Mann-Marolf counterterm
We briefly introduce the Mann-Marolf counterterm [20][21][22] that will be used for obtaining the renormalized gravity action, and then review the quasilocal method developed by BY in [7]. We further discuss an electrically charged case and its quasilocal description.
Through this paper we consider n-dimensional spacetime (M, g) and denote its index µ = 0, · · · , n − 1. The timelike boundary (∂M, h) is a timelike hypersurface defined by a spacelike normal vector n µ and we denote its index a = 0, 2, · · · , n − 1. Finally, the spatial boundary of a constant time slice (B, σ) is a spacelike hypersurface defined by n µ and a timelike normal vector u a and we denote its index A = 2, · · · , n − 1. Physically, B is the geometry of the quasilocal boundary and ∂M is the evolution of B through time.

Mann-Marolf counterterm
It is known that a gravity action diverges for a non-compact spacetime as r goes to infinity, while a compact spacetime does not have such a divergence. For an asymptotically flat spacetime, this divergence is mainly due to considering a surface term. This surface term is called the Gibbons-Hawking (GH) term and when it is introduced into the gravity action, it grows with r in four-dimensions. As a remedy, a non-dynamical term is introduced in the action as to remove the divergence. Here the non-dynamical term should not alter the equation of motion but make the total gravity action finite To generate this non-dynamical term, the reference background approach was suggested in [23]. However, if the dimension of a spacetime is higher than three, the existence or uniqueness of such embeddings of a hypersurface (∂M, h) into a proper reference frame On the other hand, as increasing the interest of AdS/CFT, the algorithm to generate counterterms as a non-dynamical term have been well constructed for AdS spacetime [24][25][26][27]. The same algorithm for generating counterterms, unfortunately, is not applicable to an asymptotically flat spacetime. However, the Mann-Marolf(MM) counterterm method provides one way to generate the counterterms for an asymptotically flat spacetime. The procedure for generating MM-counterterms is described as follows.
The gravity action is written as whereK is a solution of where R ab is the Ricci tensor on the boundary ∂M. This counterterm is known to be local and covariant. The relation (2.3) is motivated from the Gauss-Codazzi equation for a timelike hypersurface which is described as follows where Considering a general form of the metric in n dimensional spacetime Choosing ∂M to be cylindrical Ω cyl such as the solution of (2.3) is calculated each for four dimensions and higher than four dimensions in which higher sub-leading terms of order of r are omitted here, but can be calculated in principle. Up to the sub-sub-leading term of order of r is computed in [22]. In the asymptotic flat case, the leading term of the counterterm removes divergence of the gravity action and sub-leading terms give corrections to finite parts of the action. In this paper we consider up to the first sub-leading term of the MM-counterterm solutions, as displayed in (2.8) and (2.9). Indeed, our metric ansatz (2.5) for MM-counterterm is not general to cover a charged black hole solution since the sub-sub leading order of (2.5) is more constrained. However, as the electric field in a n-dimensional asymptotically flat spacetime does not show the divergent behaviour, we can safely use the MM-counterterm solution (2.9) to obtain the renormalised action for charged black hole cases as well.

Brown-York's Quasilocal variables
The Hawking temperature is defined relative to an observer located where the timelike Killing vector field has unit norm −ξ µ ξ µ = 1. In an asymptotically flat spacetimes, this means the observer is located at infinity. As a consequence, any stationary observer at finite radius where |ξ| < 1 will measure a redshifted temperature known as Tolman temperature, as mentioned in the introduction (1.10). Associated with this local temperature, quasilocal thermodynamic quantities should be defined accordingly.
One of the quasilocal formalism is constructed by Brown and York in [7] which will be used in this paper. Here the MM-counterterm is taken to be the non-dynamical term (2.10) and this renormalised gravity action constructs the Brown-York's boundary stress energymomentum tensor. This is calculated as where I cl is the on-shell action of I remormailzed and π ab = K ab − Kh ab andπ ab =K ab −Kh ab . This tensor yields a quasilocal energy density , proper momentum surface density j A and boundary stress s AB as follows where u a is a timelike normal vector field, σ AB is an induced metric on the hypersurface B. In n-dimensional spacetime, the surface pressure is defined by where σ ab is the pull-back of σ AB . The quasilocal energy contained within B is obtained by the surface integration 14) and the conserved quantity along the Killing vector field, ξ, is gained by In addition, we consider a charged case with the electric potential to be A = (Φ(r), 0, 0, · · · , 0), (2.16) and the total electric charge Q which is confined inside of an event horizon is defined as where F = dA. The total charge is independent of r at the end. This indicates that different from the quasilocal energy (2.14) the total electric charge is independent of an observer. The electric potential measured by an observer with a four velocity u µ obs is given by [28] A obs = A µ u µ obs (2.18) so the electric potential measured on the quasilocal boundary becomes where again, N (R) is the lapse function.

Euclidean Formulation in a quasilocal frame
Here we argue about black hole entropy in a quasilocal frame from the Euclidean formulation. Black hole thermodynamics can be understood by a path integral method in quantum field theory at finite temperature where Euclidean action is used by doing a Wick rotation τ → it from the Lorentzian one. Here we take a semiclassical context by considering the saddle point approximation, in which the classical Euclidean solutions of the Einstein equations dominate the path integral. The partition function is written as where g is a fluctuation of the metric and Ψ are matter fields. Taking a saddle point of the bulk action the partition function is approximated to Since the free energy is defined by we can find entropy from the free energy expression. Here in I E the Euclidean time integration gives the period β and it yields an inverse of Hawking temperature, dτ = β = 1 T , and the coordinate of r is integrated from the black hole horizon to infinity.
For the n-dimensional charged black hole case, the Euclidean gravity action is written as where in the second line, the Einstein equation (1.5) is used with and in the third line, Maxwell equation and Gauss theorem are applied. The Euclidean gravity action (3.6) is associated with the Euclidean metric and Euclidean electric potential. The Euclidean gravity action times temperature gives free energy which takes a form of and then entropy is computed as which is known to agree with black hole entropy from the area law. Now we consider a quasilocal frame with a finite domain of r = R. Then we could define a quasilocal free energy to be (3.10) Since the Euclidean action I E is integrated from the black hole horizon to a finite distance of R, the Euclidean time integration, which has the period β and dτ = β = 1 T , should be adjusted by −ξ µ (t) ξ µ(t) = −g tt (R). Thus the Tolman temperature is used in F R . From the definition (3.10) we discovered that the free energy for a charged black hole case is yielded as (3.11) Seeing (1.9), a new term P dA was introduced in the first law of thermodynamics in a quasilocal frame, but P A does not participate in the free energy expression. Then the entropy can be obtained by We found that (3.12) agrees with Bekenstein-Hawking entropy in our examples.

Quasilocal Smarr relation by Euler's Theorem
In [10], Eulers theorem on homogeneous functions is used to obtain the Smarr relation. Eulers theorem states that if a function f (x, y, z) obeys the scaling relation Let us extend this argument into a quasilocal frame. In this case a family of timelike hypersurfaces exists for r = constant and one specific hypersurface can be chosen, for example r = R. Thus the quasilocal energy E is related to not only an entropy S and a charge Q, but also a quasilocal area A whose radius is R. They have scaling dimensions such as So the Eulers theorem states The difference between (4.4) and (4.6) is the presence of the PA term, but a surface pressure P vanishes when r goes to infinity. The electric potential (2.19) should be measured by an observer placed at r = R, where Tolman temperature (1.10) is used. We expect this quasilocal Smarr relation (4.6) to hold for observers placed at r = R in an asymptotically flat spacetime. If we expand this argument into n-dimensional spacetime, each variable has a length dimension as and the Eulerian theorem yields the relation as follows which is the n-dimensional quasilocal Smarr relation. Next section we examine this quasilocal Smarr relation with the Schwarzschild black holes, Reissner-Nordström black holes and n-dimensional charged black hole solutions.

Thermodynamics in a Quasilocal Frame
In this section, we consider various black hole solutions in an asymptotically flat spacetime and set up the ansatz for the quasilocal Smarr relation with coefficients to be determined. For each case, we calculate the quasilocal quantities obtained from the boundary stress energy-momentum tensor (2.12) and then determine the coefficients of the ansatz. We find that the quasilocal Smarr relation ansatz with the determined coefficient agrees with (4.9). Firstly we start with a four dimensional Schwarzschild black hole solution, and find their a quasilocal first law along with the quasilocal Smarr relation. We also explore the Reissner-Nordström black hole and generalize it to a higher dimensional case.

Schwarzschild Black Holes
Let us write the Schwarzschild black holes spacetime (M, g) in the form of Assuming an observer on a hypersurface (∂M, h) at r = R, the temperature measured by this observer is given by (1.10) The quasilocal energy density and surface pressure are obtained as follows and M = r h 2 is replaced from N (r h ) = 0. The total quasilocal energy is obtained by integrating the quasilocal energy density (5.4) over the hypersurface at a constant time (B, σ) and is yielded as The total quasilocal energy is given by a function of r h and R. This agrees with a conserved charge M when taking R → ∞.
To obtain the entropy in a quasilocal frame, we calculate the quasilocal free energy from the Euclidean action by (3.10) and then the entropy is yielded as Now let us make a variation of E with respect to r h and R. Formulating the expression to the variation of entropy dS and to the variation of the quasilocal area dA it simply takes a form of where S = πr 2 h and A = 4πR 2 , and T Tolman is replaced by (5.3) and P is replaced by (5.5). To find out an algebraic relation between these thermodynamic variables (E, T Tolman , S, P, A), let us take an ansatz E = aT R S + bP A (5.12) where a and b can be determined by plugging each value as follows Then we obtain a = 2 and b = −2 which yields This relation agrees with the quasilocal Smarr relation in (4.6) with Q = 0.

Reissner-Nordström Black Holes
Solving Einsteins equation with a Maxwell field for the metric ansatz (5.1) and the electric potential ansatz (2.16), the metric function for a charged black hole which is written as the electric potential becomes where the regularity condition is imposed at black hole event horizon. Then from (2.17) the total electric charge is computed as Considering a timelike hypersurface (∂M, h) at r = R, the Tolman temperature turns to To compute the boundary stress energy-momentum tensor (2.11), we employ the MMcounterterm. The metric ansatz for the MM-counterterm has the same fall-off behaviour of r with this case (5.15), but is more constrained in the coefficient at the sub-sub-leading order of r. However, as we mentioned, we here only consider the effect of the MM-counter term up to the sub-leading order and the MM-counterterm (2.8) is yielded aŝ where the sub-leading term vanishes. Then like the previous subsection, we compute the boundary stress momentum tensor from the renormalized action and calculate the quasilocal energy density (2.12), surface pressure (2.12), and electric potential (2.19) which are computed as , (5.20) and M = q 2 2r h + r h 2 is replaced from N (r h ) = 0. The quasilocal energy is written as The quasilocal free energy is obtained from (3.10) and the entropy is computed as Taking an ansatz for an algebraic relation between these quasilocal thermodynamic variables (E, T Tolman , S, P, A) E = aT R S + bP A + cΦ R Q (5.29) and plugging each variable, the coefficients turn to This result agrees with (4.6).

Higher dimensional Charged Black Holes
Let us consider a higher dimensional charged black hole with the metric ansatz ds 2 = −N (r) 2 dt 2 + f (r) 2 dr 2 + r 2 dΩ n−2 , (5.31) solving the Einstein equation, we obtain metric functions and the electric potential where the regularity condition is also imposed at the black hole horizon for the electric potential, and the total electric charge is computed as From the metric, Tolman temperature at r = R can be read as To construct the renormalized gravity action, the metric ansatz (2.5) for the counterterm and the MM-counterterm solutions are taken up to the first sub-leading order of r. Then the MM-counterterm (2.9) with the metric (5.31) is yielded aŝ where the first sub-leading term vanishes since α = −γ (1) in this case. The boundary stress energy-momentum tensor becomes Then the energy density and surface pressure are computed along with the quasilocal electric potential at r = R. This is written as The total energy is and when taking R to infinity we obtain E ≈ (n−2)ω n−2 16π µ which agrees with one in [29]. The quasilocal free energy from (3.10) is (5.43) and the entropy (3.12) is computed as  Indeed, the quasilocal Smarr relation also can be reproduced from the quasilocal free energy value (5.43). Reformulating (5.43), it is written as and this should be equivalent to the free energy definition (3.11) Equating two expressions (3.11) and (5.51) agrees with the n-dimensional quasilocal Smarr relation (4.9).

Summary and Future work
We found a quasilocal Smarr relation (4.9) in n-dimensional asymptotically flat spacetime by using Eulers theorem. To check, we employed Brown-York's quasilocal formalism with the Mann-Marolf counterterm which renders the gravity action finite and calculated quasilocal quantities from the boundary stress energy-momentum tensor which are associated with the Tolman temperature. We defined a quasilocal free energy by the Euclidean gravity action with the Tolman temperature in order to find entropy in a quasilocal frame. We discovered that the quasilocal free energy expression is not altered by the surface pressure term and the entropy is the same as the Bekenstein-Hawking entropy in our cases. For several black hole solutions, we observed that quasilocal thermodynamic values satisfy the quasilocal Smarr relation (4.9). Furthermore, the Euclidean action value with the Tolman temperature and the quasilocal free energy definition are also consistent with the quasilocal Smarr relation.