On the maximization of the central gravitational redshift of a schwarzschild sphere with a given surface redshift

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-shift z_s≤4.77 arbitrarily large central red-shifts z_c 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-shift z_s(λ)≤1.71. (ii) A configuration with z_s>z_s(λ) is not possible, (iii) A configuration with z_s=z_s(λ) has a unique and finite z_c=z_c(λ). (iv) For z_s<z_s(λ) arbitrarily large central red-shifts can be obtained for configurations with arbitrarily large radii.