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 − q2)/h̄. There was, however, some uncertainty in the value of the numerical factor which multiply the q2 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 Sbh = A/4h̄, where A is the black-hole surface area. (We use gravitational 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 thermodynamics (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