Symmetries, Horizons, and Black Hole Entropy
Author: Steve Carlip
Affiliation: Department of Physics, University of California at Davis
[This is an invited article from Prof. Steve Carlip who received this year's Gravity Research Foundation award for his essay on this topic. The award-winning essay will be published in future issue of General Relativity and Gravitation and International Journal of Modern Physics D.
-- 2Physics.com Team]
Drop a box of hot gas into a black hole. The initial state is gas plus a black hole; the final state is a slightly larger black hole, and nothing else. If the second law of thermodynamics -- which requires that entropy never decrease -- is to hold, the final black hole had better have enough entropy to account for the entropy of the gas it swallowed up.
Thirty-five years ago, Bekenstein used such thought experiments to show that a black hole should have an entropy proportional to the area of its event horizon in Planck units . Soon afterwards,Hawking demonstrated that black holes are, indeed, thermodynamic objects, radiating as black bodies with characteristic temperatures and entropies that match Bekenstein's estimates . In every other thermodynamic system we know, thermal properties reflect the statistical mechanics of underlying microscopic states. Entropy,for example, counts the microstates, while temperature measures their energy. Since the Bekenstein-Hawking entropy involves both Planck's constant and Newton's constant, a statistical mechanical description would have to involve quantum gravity, and might teach us something about the unsolved problem of how to quantize general relativity.
Until fairly recently, no one had a clear idea of the microscopic states responsible for black hole entropy. Today, we suffer the opposite problem: we have many explanations, each describing a different set of states but all agreeing on the final numbers. String theory, for instance, gives us three ways to count black hole states (as excitations of weakly bound branes, as horizonless "fuzzball" geometries, and as states in a dual field theory "at infinity"); loop quantum gravity provides two more; others come from induced gravity, causal set theory, holographic entanglement entropy, and global geometry . None of these approaches is complete, but within its realm of applicability, each seems to work. The new puzzle -- the "problem of universality" -- is to understand why everyone gets the same answer.
One attractive possibility is that a hidden symmetry of classical general relativity controls the thermodynamic properties of black holes. Near the horizon, a black hole looks nearly scale-invariant (technically, conformally invariant) and nearly two-dimensional; quantities such as masses get red-shifted away, as do excitations transverse to the r-t plane. Cardy showed twenty years ago that the thermodynamic properties of a two-dimensional conformal field theory are completely determined by a few parameters that describe its symmetries . Two-dimensional conformal descriptions of matter in a black hole background can be used to derive the spectrum of Hawking radiation ; perhaps similar reasoning can be applied to the degrees of freedom of the black hole itself.
To see whether such an explanation makes sense, we must first figure out what it means to ask a question about a black hole in quantum gravity. The uncertainty principle prevents us from simply saying,"A black hole is present." Instead, we must find a way to impose constraints strong enough to ensure the presence of a black hole,but weak enough to be allowed by quantum mechanics. My most recent work has focused on the possibility of introducing such "horizon constraints" as ordinary constraints in the Hamiltonian formulation of general relativity . The results so far are promising: one can obtain the correct Bekenstein-Hawking entropy for a wide class of black holes from the constraints and the symmetry alone. Moreover, there is some evidence of "universality": at least one string theory approach can be understood as a special case of the horizon constraint method, and there are tantalizing hints of a connection with loop quantum gravity. If this explanation is really universal, the horizon constraints should be hidden in other derivations of blackhole entropy as well. We're looking...
A universal explanation of black hole thermodynamics should not, of course, give a complete description of the underlying microstates -- that would ruin its universal character. Still, the horizon constraint method suggests a new way of looking at the degrees of freedom of a black hole. The key point is that the horizon constraints break the fundamental symmetry of general relativity, general covariance (technically, diffeomorphism invariance). As a result, states that would normally be considered equivalent, differing only by a "gauge" transformation, are now physically distinct. This is roughly analogous to the Goldstone mechanism in condensed matter and particle physics, in which a broken symmetry gives rise to new degrees of freedom. In a few cases [7,8], this can be made explicit; it is an open question whether the description works more generally.
The other crucial open question is whether the horizon constraint method can also describe Hawking radiation and other thermodynamic properties of the black hole. To answer, we need to understand how the horizon constraints affect matter near the horizon. This is a hard question, but shouldn't be an impossible one.
 J.D. Bekenstein, Phys. Rev. D7 (1973) 2333.
 S.W. Hawking, Nature 248 (1974) 30.
 For some references, see S. Carlip, J. Phys. Conference Series 67 (2007) 012022, arXiv:gr-qc/0702094.
 J.A. Cardy, Nucl. Phys. B270 (1986) 186.
 S. Iso, T. Morita, and H. Umetsu, arXiv:hep-th/0701272.
 S. Carlip, arXiv:gr-qc/0702107, to appear in Phys. Rev. Lett.
 S. Carlip, Class. Quant. Grav. 22 (2005) 3055, arXiv:gr-qc/0501033.
 R. Aros, M. Romo, and N. Zamorano, Phys. Rev. D75 (2007) 067501, arXiv:hep-th/0612028.