The Statistical Mechanics of Black Hole Thermodynamics *

Although we have convincing evidence that a black hole bears an entropy proportional to its surface (horizon) area, the “statistical mechanical” explanation of this entropy remains unknown. Two basic questions in this connection are: what is the microscopic origin of the entropy, and why does the law of entropy increase continue to hold when the horizon entropy is included? After a review of some of the difficulties in answering these questions, I propose an explanation of the law of entropy increase which comes near to a proof in the context of the “semiclassical” approximation, and which also provides a proof in full quantum gravity under the assumption that the latter fulfills certain natural expectations, like the existence of a conserved energy definable at infinity. This explanation seems to require a fundamental spacetime discreteness in order for the entropy to be consistently finite, and I recall briefly some of the ideas for what the discreteness might be. If such ideas are right, then our knowledge of the horizon entropy will allow us to “count the atoms of spacetime”. When I first learned of the thermodynamics of black holes, and specifically of the fact that a black hole possesses an entropy proportional to its horizon area, my reaction (after thinking about it a while) was that this was just as if the horizon were divided into small “tiles” of a fixed size, with each tile carrying roughly one bit of information. To see that this would lead to the correct proportionality law, imagine a 0 or 1 engraved on each tile. If there were A tiles, then the number of possible configurations would be * To appear in the Proceedings of the Chandra Symposium: “Black Holes and Relativistic Stars”, ed. R. M. Wald, held December, 1996, Chicago, (U. of Chicago Press).