Statistical Mechanics and Black Hole Entropy

I review a new (and still tentative) approach to black hole thermodynamics that seeks to explain black hole entropy in terms of microscopic quantum gravitational boundary states induced on the black hole horizon. (Talk given at CAM ’95, joint meeting of the Canadian Association of Physicists, the American Physical Society, and the Mexican Physical Society, Quebec City, Canada.) ∗email: [email protected] It has been over twenty years since we first learned from Bekenstein [1] and Hawking [2] that black holes are thermodynamic systems, characterized by temperatures and entropies. But despite considerable progress in the field, we still lack a convincing “statistical mechanical” picture of black hole thermodynamics. Indeed, black hole entropy remains rather fundamentally paradoxical. On the one hand, given the macroscopic parameters of mass, charge, and angular momentum, a black hole configuration has the highest obtainable entropy, implying at least naively that the black hole has a large number of macroscopically indistinguishable microscopic states. On the other hand, a black hole has no hair: given the same macroscopic parameters, there is, in fact, only one classical black hole state. A number of attempts have been made to resolve this paradox (see [3] for a review), but none is yet generally accepted. In this paper, I would like to advocate a new, still rather tentative approach, which seeks to explain black hole entropy in terms of microscopic quantum gravitational states on the horizon. I do not yet know how to apply this picture to realistic (3+1)-dimensional black holes, but at least in the simpler case of 2+1 dimensions, it has been shown (modulo some reasonable assumptions about quantization) to lead to the correct entropy [4]. My starting point is the observation that any quantum mechanical statement about black holes is necessarily a statement about conditional probabilities: for instance, “If spacetime contains an event horizon of a certain size, then we should see Hawking radiation with a certain spectrum.” So the first question we must ask is how to impose such a condition—a restriction on the geometry of a spacetime hypersurface—in a quantum mechanical computation. One obvious answer is to start with a path integral formalism, split spacetime M along a hypersurface Σ into two pieces, say M1 and M2, and perform separate path integrals over M1 and M2 with suitable boundary conditions on Σ (figure 1). We are therefore naturally led to consider the problem of “sewing” path integrals. M1 M2 M Figure 1: The manifold M is formed by “sewing” M1 and M2 along Σ.