Revisiting the old problem of general-relativistic adiabatic collapse of a uniform-density self-gravitating sphere

Abhas Mitra1

Abstract

The problem of general-relativistic adiabatic collapse of a uniform-density perfect sphere has been studied since Wyman (Phys. Rev. 70, 396, (1946)) [1]. Apparently, there could be bouncing and oscillating solutions in such a case, as claimed by numerous authors since then. Consequently, various authors invoked such models for explaining pulsations of compact objects. However, here, for this age-old problem, we prove that for an assumed nonstatic adiabatically evolving sphere, density homogeneity implies (isotropic) pressure homogeneity too. This proof is based on the simple fact that in general relativity (GR), given one time coordinate t, one can employ another time coordinate t t* = f(t) without any loss of generality. Since this proof does not use any exterior boundary condition, it is valid in a cosmological scenario too. However, here we focus on the evolution of an isolated sphere having a boundary. And the proof obtained here shows that a uniform-density perfect fluid collapse can occur only if the (isotropic) pressure is p = 0, i.e., only when the problem is reduced to the one treated by Oppenheimer and Snyder. For such an isolated sphere, we offer a supporting proof. This result is important and non-trivial because in the past 65 years innumerable authors working on this problem failed to see that the collapse of a supposed homogeneous sphere is (actually) synonymous to the old Oppenheimer-Snyder problem.

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