On the maximization of the central gravitational red-shift of a schwarzschild sphere with a given surface gravitational red-shift

Abstract

Following Bondi static, spherically symmetric equilibrium configurations with a core and an envelope have been considered. It has been shown that for any configurations with nonnegative pressure and density and with a surface red-shiftzs ⩽ 4.77 arbitrarily large central red-shiftszc are possible in the limiting case of arbitrarily large radius. The effects of imposition of further constraints in the form of a real speed of sound not exceeding the speed of light are also examined. It is seen that for a given limiting sound-to-light-speed ratio √λ. (i) There exists a limiting surface red-shiftzs(λ) ⩽ 1.71. (ii) A configuration withzs >zs(λ) is not possible, (iii) A configuration withzs=zs(λ) has a unique and finitezc=zc(λ). (iv) Forzs<zs(λ) arbitrarily large central red-shifts can be obtained for configurations with arbitrarily large radii.