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

*Abhas Mitra*^{1}

(1) Theoretical Astrophysics Section, Bhabha Atomic Research Centre, Mumbai, India

### 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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