Sharp Bounds on the Critical Stability Radius for Relativistic Charged Spheres

In a recent paper by Giuliani and Rothman [17], the problem of finding a lower bound on the radius R of a charged sphere with mass M and charge Q < M is addressed. Such a bound is referred to as the critical stability radius. Equivalently, it can be formulated as the problem of finding an upper bound on M for given radius and charge. This problem has resulted in a number of papers in recent years but neither a transparent nor a general inequality similar to the case without charge, i.e., M <= 4R/9, has been found. In this paper we derive the surprisingly transparent inequality root M <= root R/3 + root R/9 + Q(2)/3R. The inequality is shown to hold for any solution which satisfies p + 2pT <= rho, where p >= 0 and pT are the radial- and tangential pressures respectively and rho >= 0 is the energy density. In addition we show that the inequality is sharp, in particular we show that sharpness is attained by infinitely thin shell solutions.