By Markus Heusler
Publisher: Cambridge University Press
Print Publication Year: 1996
Online Publication Date:March 2010
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511661396.008
The area theorem is probably one of the most important results in classical black hole physics. It asserts that (under certain conditions which we specify below) the area of the event horizon of a predictable black hole spacetime cannot decrease. This result bears a resemblance to the second law of thermodynamics. The analogy is reinforced by the similarity of the mass variation formula to the first law of ordinary thermodynamics. Within the classical framework the analogy is basically of a formal, mathematical nature. There exists, for instance, no physical relationship between the surface gravity, κ, and the classical temperature of a black hole, which must be assigned the value of absolute zero. Nevertheless, on account of the Hawking effect, the relationship between the laws of black hole physics and thermodynamics gains a deep physical significance: The temperature of the black–body spectrum of particles created by a black hole is κ/2π. This also sheds light on the analogy between the entropy and the area of a black hole.
The Killing property of a stationary event horizon implies that its surface gravity is constant. If the Killing fields are integrable (that is, in static or circular spacetimes), the zeroth law of black hole physics is a purely geometrical property of Killing horizons. Otherwise, it is a consequence of Einstein's equations and the dominant energy condition.
The Komar expression for the mass of a stationary spacetime provides a formula giving the mass in terms of the total angular momentum, the angular velocity, the surface gravity and the area of the horizon.