International Journal of Modern Physics D Vol. 22, No. 9 (2013) 1350054 (18 pages) c  World Scientific Publishing Company DOI: 10.1142/S0218271813500545 THE MASS OF THE OPPENHEIMER–SNYDER-BLACK HOLE: ONLY FINITE MASS QUASI-BLACK HOLES ABHAS MITRA ∗ and K. K. SINGH Astrophysical Sciences Division, Bhabha Atomic Research Centre, Mumbai, India ∗ amitra@barc.gov.in Received 2 April 2013 Revised 3 May 2013 Accepted 3 May 2013 Published 11 June 2013 Oppenheimer and Snyder (OS) in their paper apparently showed the formation of an event horizon [see Eq. (37) in Phys. Rev. 56 (1939) 455] for a collapsing homogeneous dust ball of mass M as the circumference radius of the outermost surface, r b → r 0 = 2GM/c 2 in a proper time τ 0 ∝ r -1/2 0 in the limit of large Schwarzschild time t →∞. But Eq. (37) was approximated from Eq. (36) whose essential character is (t ∼ r 0 ln √ y+1 √ y-1 ) where, at the boundary of the star y = r b /r 0 = r b c 2 /2GM. And since the argument of a logarithmic function cannot be negative, one must have y ≥ 1 or 2GM/r b c 2 ≤ 1. This shows that, at least, 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 τ →∞. Thus, the gravitational mass of OS-black holes (OS-BHs), is unique and equal to zero. In fact, by invoking Birkhoff’s theorem, it has been found that the OS collapse is only a fictitious mathematical artifact because it corresponds to a matter density ρ = 0 [Mitra, Astrophys. Space Sci. 332 (2011) 43, arXiv:1101.0601]. Further, this is also in agreement with the proof that Schwarzschild BHs have the unique gravitational mass M = 0 [Mitra, J. Math. Phys. 50 (2009), arXiv:0904.4754], and they represent asymptotic final state of physical collapse for which entire mass-energy is radiated out [Mitra and Glendenning, Mon. Not. R. Astron. Soc. Lett. 404 (2010) L50, arXiv:1003.3518]. Finally this is in agreement with the conclusion that “the discussion of physical behavior of black holes, classical or quantum, is only of academic interest — we wonder whether nature allows gravitational collapse to continue inside the EH at all” [Narlikar and Padmanabhan, Found. Phys. 18 (1989) 659, doi:10.1007/BF00734568]. Keywords : General relativity gravitational collapse; relativistic astrophysics; black hole; black hole alternatives; eternally collapsing objects. PACS Number: 04.20.Cv 1. Introduction One of the oldest and most fundamental problems in physics and astrophysics is that of gravitational collapse and, specifically, that of the ultimate fate of a sufficiently 1350054-1 Int. J. Mod. Phys. D Downloaded from www.worldscientific.com by TATA INSTITUTE OF FUNDAMENTAL RESEARCH on 06/18/13. For personal use only.