Semiclassical Black Hole States and Entropy Typeset Using Revt E X

We discuss semiclassical states in quantum gravity corresponding to Schwarzschild as well as Reissner-Nordström black holes. We show that reduced quantisation of these models is equivalent to Wheeler-DeWitt quantisation with a particular factor ordering. We then demonstrate how the entropy of black holes can be consistently calculated from these states. While this leads to the Bekenstein-Hawking entropy in the Schwarzschild and nonextreme Reissner-Nordström cases, the entropy for the extreme ReissnerNordström case turns out to be zero. Typeset using REVTEX 1 The issues of black hole entropy and Hawking radiation play a key role in any attempt to quantise the gravitational field. For a deeper understanding it is of central importance to provide a satisfactory interpretation of black hole entropy from statistical mechanics. Recently, progress on this question has been achieved in the context of string theory [1], but the more conservative framework of quantum general relativity provides interesting insight into this question, too. In 2 + 1 dimensions a statistical interpretation has been suggested using the Chern-Simons form [2], but in 3 + 1 dimensions this remains still elusive. On the level of the semiclassical approximation, black hole entropy has been discussed in the framework of path integrals [3,4]. Such a treatment exploits the formal analogy of euclidean path integrals in standard quantum field theory to partition sums in statistical mechanics. The entropy is then calculated as the logarithm of the density of states in the partition function which is found from an appropriate saddle point approximation to the path integral. To ensure thermodynamical stability, the black hole has to be enclosed in a spatially finite box (or, alternatively, has to be embedded in an anti-de Sitter spacetime [5]). If appropriate boundary conditions are imposed at the wall of the box and at the black hole horizon, the partition function can be evaluated, and the black hole entropy is found from the boundary term at the horizon to take the Bekenstein-Hawking value A/4h̄G, where A is the area of the horizon [4]. The purpose of the present paper is to investigate how the black hole entropy can be consistently found from semiclassical solutions to the Wheeler-DeWitt equation in quantum gravity. Since from a physical point of view there is a close connection between the WKB approximation for wave functionals and the saddle point approximation for path integrals, this should be possible to achieve. Some interesting new aspects will turn out in this discussion which thus complements the standard treatment in the path integral context. In the following we shall consider spherically symmetric gravitational systems which include the important cases of the Schwarzschild and the Reissner-Nordström black holes. A WKB solution of the Wheeler-DeWitt equation for the former case was given in [6] (see also [7]). On the other hand, a reduced quantisation was performed in [8] with the mass 2 of the black hole as the only remaining configuration variable (see [9] for an analogous discussion in the framework of connection dynamics). We shall show the equivalence of reduced quantisation to Wheeler-DeWitt quantisation in a particular factor ordering. We extend the discussion to include the Reissner-Nordström case, where we present a careful investigation into the notions of ‘classically allowed’ and ‘classically forbidden’ regions. Our main point then will be the recovery of the black hole entropy from the Hamilton-Jacobi functional in a consistent way. In the course of this discussion it will turn out in a natural way that the entropy of the extreme Reissner-Nordström black hole vanishes. We start from the ADM form of the general spherically symmetric spacetime metric on the manifold IR× IR× S: ds = −Ndt + Λ(r, t)(dr +N dt) +R(r, t)dΩ, (1) where dΩ denotes the standard metric on S, and N (N ) is the lapse function (shift function). Inserting the ansatz (1) into the Einstein-Hilbert action and varying with respect to N and N r leads to the Hamiltonian constraint and the radial momentum constraint [6–8], HG ≡ G 2 ΛP 2 Λ R −G PΛPR R + VG G ≈ 0, (2) Hr ≡ PRR ′ − ΛP ′ Λ ≈ 0, (3) where the gravitational potential term VG reads explicitly VG ≡ RR Λ − RRΛ Λ + R 2Λ − Λ 2 . (4) The inclusion of the cosmological constant λ is straightforward and would lead to an additional term λΛR/2 in (4). In the following we shall include in addition a spherically symmetric electromagnetic field [5]. The corresponding vector potential is written in the form A = φ(r, t)dt+ Γ(r, t)dr. (5) This, then, leads to the addition of the kinetic term 3