Eternal Black Holes and Quasilocal Energy *

We present the gravitational action and Hamiltonian for a spatially bounded region of an eternal black hole. The Hamiltonian is of the general form H = H+ − H−, where H+ and H− are respectively the Hamiltonian for the regions M+ and M− located in the left and right wedges of the spacetime. We construct explicitly the quasilocal energy for the system and discuss its dependence on the time direction induced at the boundaries of the manifold. This paper extends the analysis of Ref. [1] to spacetimes possesing a bifurcation surface and two timelike boundaries. The construction suggests that an interpretation of black hole thermodynamics based on thermofield dynamics ideas can be generalized beyond perturbations to the gravitational field itself of a bounded spacetime region. Based on the talk presented by E.A. Martinez at the Lake Louise Winter School on Particle Physics and Cosmology, February 20-26, 1994. electronic addresses: [email protected], [email protected] 1 Recent work on black hole thermodynamics has suggested an definition of quasilocal energy for spatially bounded systems in general relativity [1]. In thermodynamic applications this energy plays the role of the thermodynamical internal energy conjugate to inverse temperature [2]. This quasilocal energy is obtained directly from the gravitational action and Hamiltonian and includes negative contributions due to gravitational binding. In the present paper we will present the action and Hamiltonian for an eternal black hole and discuss in detail the quasilocal energy for the system. The motivation for studying the action of a fully extended black hole is at least twofold: in the first place, it is important to understand the general properties of the boundary Hamiltonians when the spacetime is bounded not by one but by two timelike boundaries separated by a bifurcation surface. In the second place, the action for eternal black holes will play a fundamental part in the study and generalization to the full gravitational field of interpretations of black hole thermodynamics based on the existence of spacetime regions M+ and M−. The density matrix describing internal states of a black hole can be obtained by tracing over degrees of freedom in M+. The density matrix can be used to calculate the entropy connected with the region M− (black hole entropy) [3]. In this way pure states for an eternal black hole would result in mixed thermal states for observers located on either disconnected region. One can relate the density of states with the action for an eternal black hole and consider its thermodynamical implications. In this approach an interesting relation with thermofield dynamics arises (for more details see Ref. [4]). We start by discussing the kinematics of a completely extended stationary spacetime (t denoting its Killing vector) with the topology of an eternal black hole. The corresponding Kruskal spacetime is the union of four sectors R+, R−, T+, and T− [5]. The regions R+ and R− are asymptotically flat and t μ is timelike at their spatial infinities. We concentrate attention in two wedges M+ and M− located in the right (R+) and left (R−) sectors of the Kruskal diagram. Denote by Σ± the spacelike boundaries of M± and by B± their timelike boundaries. The regions M+ and M− intersect at a two-dimensional spacelike surface (bifurcation surface). 2 The spacetime line element can be written in the general form ds = −Ndt + hij(dx + V dt)(dx + V dt) , (1) where N is the lapse function. The four-velocity is defined by uμ = −N ∂μt, where the lapse function N is defined so that u · u = −1. The four-velocity u is the timelike unit vector that is normal to the spacelike hypersurfaces (t = constant) denoted by Σt. The symbol Σt (±) indicates the part of Σt located in M±. The Killing vector t μ and the four-velocity u are related by