Classification theorem and properties of singular solutions to the Tolman-Oppenheimer-Volkoff equation

TL;DR

This paper proves a classification theorem for singular solutions of the TOV equation, showing their properties, causal structure, and how to modify equations of state to avoid pathologies, with implications for stellar models.

Contribution

It establishes a comprehensive classification of singular solutions to the TOV equation under thermodynamic constraints and analyzes their physical and causal properties.

Findings

Singular solutions have negative-mass Schwarzschild metrics at the center.

Thermodynamic consistency prevents finite-radius pathological solutions.

Singular solutions are causally well behaved and conformally globally hyperbolic.

Abstract

The Tolman-Oppenheimer-Volkoff (TOV) equation admits singular solutions in addition to regular ones. Here, we prove the following theorem. For any equation of state that (i) is obtained from an entropy function, (ii) has positive pressure and (iii) satisfies the dominant energy condition, the TOV equation can be integrated from a boundary inwards to the center. Hence, thermodynamic consistency of the EoS precludes pathological solutions, in which the integration terminates at finite radius (because of horizons, or divergences / zeroes of energy density). At the center, the mass function either vanishes (regular solutions) or it is negative (singular solutions). For singular solutions, the metric at the center is locally isomorphic to negative-mass Schwarzschild spacetime. This means that matter is stabilized because the singularity is strongly repulsive. We show that singular solutions are causally well behaved: they are bounded-acceleration complete, and they are conformal to a globally hyperbolic spacetime with boundary. Finally, we show how to modify unphysical equations of state in order to obtain non-pathological solutions, and we undertake a preliminary investigation of dynamical stability for singular solutions.