Particle absorption by black holes and the generalized second law of thermodynamics
BY SCOTT FUNKHOUSER
*
National Oceanic
Atmospheric Administration
South Hobson Avenue
Charleston
SC
USA
The change in entropy, S , associated with the quasi-static absorption of a particle of energy by a Schwarzschild black hole (ScBH) is approximately (/T ) s, where T is the Hawking temperature of the black hole and s is the entropy of the particle. Motivated by the statistical interpretation of entropy, it is proposed here that the absorption should be suppressed, but not forbidden, when S < 0, which requires the absorption cross section to be sensitive to S . A purely thermodynamic formulation of the probability for the absorption is obtained from the standard relationship between microstates and entropy. If S 1 and s /T , then the probability for the particle not to be absorbed is approximately exp[/T ], which is identical to the probability for quantum mechanical reflection by the horizon of an ScBH. The manifestation of quantum behaviours in the new probability function may intimate a fundamental physical unity between thermodynamics and quantum mechanics. Consider an isolated system X , subject only to internal forces, that consists of a Schwarzschild black hole (ScBH) and a stable test particle that is initially approaching the black hole. Let the energy of the particle be very small in comparison to the total energy of the black hole. Note that is defined when the particle is, effectively, infinitely far from the black hole, and in a frame that is stationary relative to the black hole. According to the classical description, the Schwarzschild metric sufficiently characterizes the relevant properties of the black hole, and the incident particle is essentially point-like. The classical probability for the black hole to absorb the particle may differ substantially, therefore, from the probability determined according to any model that provides a field-theoretic description of the particle. The purpose of this present work is to demonstrate that the thermodynamic properties of the black hole also generate a departure from the classical description of absorption, independent of the quantum behaviours of the particle. A surprising consistency emerges between the quantum theory of particles and the thermodynamics of black holes.
1. Introduction
It is instructive to review the basic components of the classical and quantum
mechanical models for the evolution of a system like X . Let the black hole have
a mass M , and the magnitude v of the velocity of the incident particle be defined
as (1 m2/2)1/2, where m is the mass of the particle. The units here are natural
units in which the Newtonian gravitational coupling G, the Planck constant h ,
the vacuum speed of light c and the Boltzmann constant kB are equal to 1. The
probability for the black hole to absorb the particle is, classically, characterized
by the cross section (Bogorodski 1962; Unruh 1976)
512(1 v2)3
4(1 4v2 + [1 + 8v2]1/2)(3 [1 + 8v2]1/2)2 .
The slowly varying quantity in brackets is equal to 16 when v = 0, and
approaches 27 in the limit as v approaches 1. The cross section (1.1) is therefore
approximately proportional to 1/v2 for all v < 1, and diverges as v vanishes
(Unruh 1976).
The first major revision of the classical model was, historically, the
incorporation of a field-theoretic treatment of the particle, which was most notably
presented by Unruh (1976). By solving the KleinGordon and Dirac equations in
the Schwarzschild metric, it is possible to determine the absorption cross sections
S and D for scalar and Dirac particles, respectively. Although S and D are
both similar to C in the high-energy regime 1, significant deviations from
the classical description occur at low energies, where (Unruh 1976)
In all of the cases discussed here, S = 8D (Unruh 1976). If the wavelength
= 1/ of the particle is larger than the horizon R = 2M of the black hole, then
equation (1.2) is approximately
v
for v > 2 Mm (Unruh 1976). The most important features distinguishing the
lowenergy quantum mechanical cross sections (1.3) and (1.4) from C are the factor
2Mm in equation (1.3) and the 1/v dependence in equation (1.4). The term
2Mm = Rm in equation (1.3) is approximately equal to R/ at low velocities
where m. As increases with respect to R, the factor 2Mm decreases
and absorption is thus attenuated, which represents one of the basic wave-like
behaviours of quantum theory. Note that equation (1.4), which characterizes
the regime of high velocity and low energy, does not exhibit the characteristic
quantum mechanical dependence on R/ (Unruh 1976).
The seminal field-theoretic descriptions of particle absorption in the
Schwarzschild metric have been expanded to model scenarios where the charge
and angular momentum of the incident particle may be important, and where the
Kerr, ReissnerNordstrom (RN) or KerrNewman (KN) metric is necessary
to describe the black hole (Frolov & Novikov 1998). There exists, however,
a latent singularity in the origi (...truncated)