Horizon Constraints and Black Hole Entropy

A question about a black hole in quantum gravity is a conditional question: to obtain an answer, one must restrict initial or boundary data to ensure that a black hole is actually present. For two-dimensional dilaton gravity—and probably for a much wider class of theories—I show that the imposition of a spacelike “stretched horizon” constraint modifies the algebra of symmetries, inducing a central term. Standard conformal field theory techniques then fix the asymptotic density of states, successfully reproducing the BekensteinHawking entropy. The states responsible for black hole entropy can thus be viewed as “would-be gauge” states that become physical because the symmetries are altered. ∗email: [email protected] Suppose one wishes to ask a question about a quantum black hole. In a semiclassical theory, this is straightforward, at least in principle—one can look at quantum fields and gravitational perturbations around a black hole background. In a full quantum theory of gravity, though, such a procedure is no longer possible—there is no fixed background, and the theory contains both states with black holes and states with none. One must therefore make one’s question conditional: “If a black hole with property X is present. . . ” Equivalently, one must impose constraints, either on initial data or on boundaries, that restrict the theory to one containing an appropriate black hole [1]. Classical general relativity is characterized by a symmetry algebra, the algebra of diffeomorphisms. But it is well known that the introduction of new constraints can alter such an algebra [2–4]. For the simple model of two-dimensional dilaton gravity, I will show below that the imposition of suitable “stretched horizon” constraints has the effect of adding a central extension to the algebra of diffeomorphisms of the horizon. This is a strong result, because such a centrally extended algebra is powerful enough to almost completely fix the asymptotic behavior of the density of states, that is, the entropy [5,6]. Indeed, I will show that given a reasonable normalization of the “energy,” standard conformal field theoretical methods reproduce the correct Bekenstein-Hawking entropy. Moreover, while the restriction to two-dimensional dilaton gravity is a significant one, I will argue that the conclusions are likely to extend to much more general settings. These results suggest that the entropy of a black hole can be explained by two key features: the imposition of horizon boundary conditions, which can alter the physical content of the theory by promoting “pure gauge” fields to dynamical degrees of freedom, and the existence of a Virasoro algebra, which can control the asymptotic density of states. Both of these features are present for the (2+1)-dimensional black hole [7–9], and a number of authors—see, for example, [10–19]—have suggested that near-horizon symmetries may control generic black hole entropy. In particular, we shall see that the near-horizon conformal symmetry of [20] is closely related to the horizon constraint introduced here. 1. Dilaton Gravity in a Null Frame We start with canonical two-dimensional dilaton gravity in a null frame, that is, expressed in terms of a null dyad {la, na} with l · n = −1. The metric is then gab = −lanb − lbna, (1.1) and “surface gravities” κ and κ̄ may be defined by ∇alb = −κnalb − κ̄lalb ∇anb = κnanb + κ̄lanb, (1.2)