Massive spheres in general relativity

Abstract

The exact relativistic form of the equation of hydrostatic support by an isotropic pressure is found in an especially convenient form. A quantity u , the natural generalization of Schwarzschild’s m/r ratio at the surface, is used, and it is proved that the critical value u = ½ cannot be attained except perhaps under conditions of severe tension. It is shown that if u is everywhere less than different physical restrictions are significant in well-defined inner and outer zones as far as the attainment of high u values is concerned. A number of limiting configurations are derived for physically significant restrictions. In particular it is shown that if the density is nowhere negative then u < 0.485 ..., while if in addition the density is nowhere less than three times the pressure and nowhere increasing outwards, then u < 0.319.