ar X iv : g r - qc / 9 90 30 11 v 2 2 9 D ec 1 99 9 Improved Upper Bound to the Entropy of a Charged System . II

Recently, we derived an improved universal upper bound to the entropy of a charged system S ≤ π(2Eb − q 2)/¯ h. There was, however, some uncertainty in the value of the numerical factor which multiplies the q 2 term. In this paper we remove this uncertainty; we rederive this upper bound from an application of the generalized second law of thermodynamics to a gedanken experiment in which an entropy-bearing charged system falls into a Schwarzschild black hole. A crucial step in the analysis is the inclusion of the effect of the spacetime curvature on the electrostatic self-interaction of the charged system. According to the thermodynamical analogy in black-hole physics, the entropy of a black hole [1–3] is given by S bh = A/4¯ h, where A is the black-hole surface area. (We use grav-itational units in which G = c = 1). Moreover, a system consisting of ordinary matter interacting with a black hole is widely believed to obey the generalized second law of ther-modynamics (GSL): " The sum of the black-hole entropy and the common (ordinary) entropy in the black-hole exterior never decreases ". This general conjecture is one of the corner stones of black-hole physics. It is well known, however, that the validity of the GSL depends on the (plausible) existence of a universal upper bound to the entropy of a bounded system [4]: Consider a box filled with matter of proper energy E and entropy S which is dropped into a black hole. The energy delivered to the black hole can be arbitrarily red-shifted by letting the assimilation 1 point approach the black-hole horizon. If the box is deposited with no radial momentum a proper distance R above the horizon, and then allowed to fall in such that R < ¯ hS/2πE , (1) then the black-hole area increase (or equivalently, the increase in black-hole entropy) is not large enough to compensate for the decrease of S in common (ordinary) entropy. Arguing from the GSL, Bekenstein [4] has proposed the existence of a universal upper bound to the entropy S of any system of total energy E and effective proper radius R: S ≤ 2πRE/¯ h , (2) where R is defined in terms of the area A of the spherical surface which circumscribe the system R = (A/4π) 1/2 [4]. This restriction is necessary for enforcement of the GSL; the box's entropy disappears but an …