Gravitation, the Quantum, and Bohr’s Correspondence Principle

The black hole combines in some sense both the “hydrogen atom” and the “black-body radiation” problems of quantum gravity. This analogy suggests that black-hole quantization may be the key to a quantum theory of gravity. During the last twenty-five years evidence has been mounting that black-hole surface area is indeed quantized, with uniformally spaced area eigenvalues. There is, however, no general agreement on the spacing of the levels. In this essay we use Bohr’s correspondence principle to provide this missing link. We conclude that the fundamental area unit is 4h̄ ln 3. This is the unique spacing consistent both with the area-entropy thermodynamic relation for black holes, with Boltzmann-Einstein formula in statistical physics and with Bohr’s correspondence principle. Everything in our past experience in physics tells us that general relativity and quantum theory must be approximations, special limits of a single, universal theory. However, despite the flurry of research, which dates back to the 1930s, we still lack a complete theory of quantum gravity. It is believed that black holes may play a major role in our attempts to shed some light on the nature of a quantum theory of gravity (such as the role played by atoms in the early development of quantum mechanics). The quantization of black holes was proposed long ago in the pioneering work of Bekenstein [1]. The idea was based on the remarkable observation that the horizon area of nonextremal black holes behaves as a classical adiabatic invariant. In the spirit of Ehrenfest 1 principle, any classical adiabatic invariant corresponds to a quantum entity with a discrete spectrum, Bekenstein conjectured that the horizon area of a quantum black hole should have a discrete eigenvalue spectrum. To elucidate the spacing of the area levels it is instructive to use a semiclassical version of Christodoulou’s reversible processes. Christodoulou [2] showed that the assimilation of a neutral (point) particle by a (nonextremal) black hole is reversible if it is injected at the horizon from a radial turning point of its motion. In this case the black-hole surface area is left unchanged and the changes in the other black-hole parameters (mass, charge, and angular momentum) can be undone by another suitable (reversible) process. (This result was later generalized by Christodoulou and Ruffini for charged point particles [3]). However, in a quantum theory the particle cannot be both at the horizon and at a turning point of its motion; this contradicts the Heisenberg quantum uncertainty principle. As a concession to a quantum theory Bekenstein [4] ascribes to the particle a finite effective proper radius b. This implies that the capture process (of a neutral particle) involves an unavoidable increase (∆A)min in the horizon area [4]: (∆A)min = 8π(μ 2 + P )b , (1) where μ and P are the rest mass and physical radial momentum (in an orthonormal tetrad) of the particle, respectively. In the classical case the limit b → 0 recovers Christodoulou’s result (∆A)min = 0 for a reversible process. However, a quantum particle is subjected to a quantum uncertainty – the particle’s center of mass cannot be placed at the horizon with accuracy better than the radial position uncertainty h̄/(2δP ). This yields a lower bound on the increase in the black-hole surface area due to the assimilation of a (neutral) test particle (∆A)min = 4πlp 2 , (2) where lp = ( G c )1/2 h̄ is the Planck length (we use gravitational units in which G = c = 1). Thus, for nonextremal black holes there is a universal (i.e., independent of the black-hole parameters) minimum area increase as soon as one introduces quantum nuances to the problem. 2 The universal lower bound Eq. (2) derived by Bekenstein is valid only for neutral particles [4]. Expression (1) can be generalized for a charged particle of rest mass μ and charge e. Here we obtain