Time-Dependent Internal Solutions for Spherically Symmetrical Bodies in General Relativity: II. Adiabatic Radial Motions of Uniformly Dense Spheres

A differential equation is derived for the radius of a spherically symmetrical body of uniform density as a function of time t , for any arbitrary equation of state of the material at the centre, the equation of state elsewhere being determined by that at the centre through the condition $$\rho = \rho(t)$$ . This differential equation can be applied to cases where the body may oscillate or continually contract or expand. In Paper I of this series we overlooked the possibility of asymptotic contraction to a radius $$R\gt 9GM/4c^2$$ , where M is the mass. If, however, a stage is reached for which $$R\lt 9GM/4c^2$$ collapse to zero volume must occur. We have applied our analysis to bodies in which the material at the centre obeys a polytropic equation of state and obtained general formulae for determining the radial motion uniquely for any polytropic index and initial conditions.