1999/04/13 by Abhas Mitra, Mitra, Abhas
Physics and Astronomy · #Astrophysics (astro-ph) #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #High Energy Physics - Theory (hep-th) #astro-ph #gr-qc #hep-th
paper · pdf · doi:10.48550/arxiv.astro-ph/9904163
6 pages using Revtex; graphically analyzes the Oppenheimer-Snyder paper to show that the resulting BH has zero mass
arxiv created 1999/04/15 · arxiv updated 2009/11/30
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, rb = r0= 2M in a proper time proportional to r0-1/2 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=rb/ r0 = rb /2 M. And since the argument of a logarithmic function can not be negative, one must have y >= 1 or 2M/rb <= 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.