Abstracts of talks

A variety of examples support the conjecture that the horizon area of a near-equilibrium black hole is an adiabatic invariant in the sense that slow perturbations of the hole leave the area invariant. In physics the quantum operators corresponding to adiabatic invariants often have a discrete spectrum. By analogy the Kerr black hole mass M might have a discrete spectrum provided the classical relation between horizon area, mass and angular momentum goes over into the quantum theory. Semiclassical evidence is presented that there exists a quantum of horizon area independent of black hole scale so that the black hole horizon area eigenvalues are uniformly spaced, or equivalently, the spacings between black hole mass levels go roughly like 1/M. Black hole entropy can then be interpreted, as first done by Mukhanov, as quantifying the degeneracy of these leveles. Quantization of horizon area suggests that the area operator is part of an algebra of black hole observables and other operators. I describe such an algebra, delineated by Mukhanov and myself, which, based on a few assumptions, leads to the uniformly spaced area eigenvalues. It is crudely reminiscent of Pauli's algebraic quantization of the H atom, so that the black hole seems to play the same role in gravitation that the atom played in the nascent quantum mechanics.

It is becoming increasingly plausible that black hole thermodynamics can be understood as the statistical mechanics of an induced quantum field theory on the horizon. For the (2+1)-dimensional black hole, this theory is a conformal field theory, and powerful tools are available. A number of "state-counting" methods have been proposed over the past few years, but none is entirely satisfactory. In particular, Strominger has recently suggested a simple approach to obtaining the entropy of the (2+1)-dimensional black hole. While his argument is unfortunately not quite right, it points us toward properties of a conformal field theory needed to understand black hole entropy, and suggests strong ties between the (2+1)-dimensional model and higher dimensional black holes. I will review the current status of (2+1)-dimensional black hole thermodynamics and suggest directions for future research.

The entropy of the Schwarzschild black hole in four dimensional space-time is computed by first mapping it onto a near extremal charged black hole with four charges having the same thermodynamic entropy. This step is classical but can be interpreted in string theory language. Performing a well defined limiting process, the charges appear then as carried by a finite number of extremal BPS branes and all non BPS excitations vanish. The number of extremal branes is expressed in terms of the gravitationnal radius of the Schwarzschild black hole and the degeneracy of the brane configuration is computable from string theory. The counting reproduces exactly the Hawking-Beckenstein entropy of the four dimensional Schwarzschild black hole. The role of string theory in the derivation is discussed.

In this talk I shall summarize attempts to explain black hole entropy by counting the states of black-hole quantum excitations. I shall demonstrate that this idea can be succesfully developed for special types of theories, so called models of induced gravity. The approach based on the induced gravity gives a natural explanation of the universality of the black hole entropy, that is its independence of details of a background microscopic theory.

The idea of induced gravity was proposed by Sakharov in 1968. According to this idea the Einstein gravity is a low-energy effective theory induced by quantum effects of heavy constituents. Jacobson in 1994 proposed that in the induced gravity the Bekenstein-Hawking entropy might be related to counting the states of constituents. This proposal was recently verified and proved for a class of the induced gravity models. Moreover, it was shown that the statistical-mechanical entropy obtained by counting the available states of constituents always exactly coincides with the Bekenstein-Hawking expression. The obtained entropy is universal, that is it does not depend on the concrete choice of the model. Possible relations of results for black hole entropy in the induced gravity to calculations of black hole entropy in fundamental theory of quantum gravity (such as the string theory) is discussed.

An infinite density of states just outside a black hole is required in ordinary field theory to account for the outgoing modes that carry the Hawking radiation. If there is a physical cutoff these modes must come from somewhere else, either from ingoing modes that are turned back at the horizon or from superluminal modes originating inside the black hole. The first of these possibilities occurs in field theory on a lattice falling freely into a black hole, and the second occurs in superfluid helium-3 in a moving texture simulating a black hole. Both mechanisms involve a transmutation of short wavelength into long wavelength degrees of freedom.

Generalizing previous canonical quantum gravity results for Schwarzschild black holes from 4 to D › 3 spacetime dimensions yields an energy spectrum

Assuming the degeneracies d_n of these levels to be given by d_n = gⁿ, g › 1, leads to a partition function which is mathematically the same as that of the primitive droplet nucleation model for 1st-order phase transitions in D-2 spatial dimensions.
Exploiting the well-known properties of the so-called critical droplets of this model immediately leads to the Hawking temperature and the Bekenstein-Hawking entropy of Schwarzschild black holes. Thus, the "holographic principle" of 't Hooft and Susskind is naturally realised. The values of temperature and entropy appear closely related to the imaginary part of the partition function which describes metastable states.
Finally some striking conceptual similarities ("correspondence point" etc.) between the droplet nucleation picture and the very recent approach to the quantum statistics of Schwarzschild black holes in the framework of the DLCQ Matrix theory are pointed out. (hep-th/9803180)

Trans-Planckian particles are elementary particles accelerated such that their energies surpass the Planck value. There are several reasons to believe that trans-Planckian particles do not represent independent degrees of freedom in Hilbert space, but they are controlled by the cis-Planckian particles. A way to learn more about the mechanisms at work here, is to study black hole horizons, starting from the scattering matrix Ansatz.

By compactifying one of the three physical spacial dimensions, the scattering matrix Ansatz can be exploited more efficiently than before. The algebra of operators on a black hole horizon allows for a few distinct representations. It is found that this horizon can be seen as being built up from string bits with unit lengths, each of which being described by a representation of the SO(2,1) Lorentz group. We then demonstrate how the holographic principle works for this case, by constructing the operators corresponding to a field F(x,t). The parameter t turns out to be quantized in units t_Planck/R , where R is the period of the compactified dimension.

I will describe a Gedankenexperiment (developed with Daniel Sudarsky) in which the area of a black hole is induced to fluctuate by about 50% of its magnitude. Two conclusions will be drawn: that the - p log p form of the entropy is the only one compatible with the second law of thermodynamics, and that only a spacetime ("history") approach to quantum gravity can do justice to black hole entropy. In a more general vein, I will also offer comments to the effect that it is not enough to get the numerical value of the entropy, one must also explain why the second law continues to work when horizons are present. In case any time remains (and I have the requisite expression by then!), I will write down an integral for the horizon area in causal set theory.

Date last revised: June 19, 1998.