The Mass of the Oppenheimer-Snyder Black Hole

TL;DR

This paper revisits the classical Oppenheimer-Snyder model of gravitational collapse, arguing that the mass of the resulting black hole must be zero, challenging the conventional understanding of black hole mass.

Contribution

The paper provides a detailed analysis showing that the gravitational mass of the Oppenheimer-Snyder black hole is necessarily zero, offering a new perspective on black hole formation.

Findings

Black hole mass in the Oppenheimer-Snyder model is zero.

Trapped surfaces are not formed during collapse.

Proper collapse time tends to infinity.

Abstract

The only instance when the General Relativistic (GTR) collapse equations have been solved (almost) exactly to explicitly find the metric coefficients is the case of a homogeneous spherical dust (Oppenheimer and Snyder in 1939 in Phy. Rev. 56, 455). The equation (37) of their paper showed the formation of an event horizon for a collapsing homogeneous dust ball of mass M in that the circumference radius the outermost surface, r_b = r_0= 2M in a proper time proportional to r_0^{-1/2} in the limit of large Schrarzschild time t= infinity. But Eq.(37) was approximated from the Eq. (36) whose essential character is t = log (y+1/y-1), where, at the boundary of the star y=r_b/ r_0 = r_b /2 M. And since the argument of a logarithmic function can not be negative, one must have y >= 1 or 2M/r_b <= 1. This shows that, atleast, in this case (i) trapped surfaces are not formed, (ii) if the collapse indeed proceeds upto r=0, we must have M=0, and (iii) proper time taken for collapse =infinity. Thus, the gravitational mass of OS black holes are unique and equal to zero. In the preceding paper (astro-ph/9904162), we assumed the existence of a finite mass BH and studied its properties in terms of Kruskal-Szekeres coordinates. And we showed that the radial geodesic of a material particle which must be timelike at r=2M, if M >0, actually becomes null. And this independently showed that BHs have unique mass, M=0.