On the Oppenheimer‐Volkoff Equations in General Relativity

Abstract

We introduce a new formulation of the Oppenheimer‐Volkoff (O‐V) equations, a system of ordinary differential equations that models the interior of a star in general relativity, and we use this to give a completely rigorous mathematical analysis of solutions. In particular, we prove that, under mild assumptions on the equation of state, black holes never form in solutions of the O‐V equations. As a corollary, this implies that the portion of the empty‐space Schwarzschild solution inside the Schwarzschild radius cannot be obtained as a limit of O‐V solutions having non‐zero density. We also prove that if the density ρ at radius r is ever larger than where M ( r ) is the total mass inside radius r , then M must become negative for some positive radius. We interpret M <0 as a condition for instability because we show that if the pressureis a decreasing function of r , then M ( r )<0 at some r >0 implies that the pressure tends to infinity before r =0.