The Mass of the Oppenheimer-Snyder Hole: Only Finite Mass Quasi-Black Holes

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 → r₀ = 2GM/c² in a proper time $τ ₀ ∝ r₀^-1/2 in the limit of large Schwarzschild time t → ∞. But Eq. (37) was approximated from Eq. (36) whose essential character is (t\sim r_0 \ln{\sqrt y+1\over \sqrt y-1}$) where, at the boundary of the star y = r_b/r₀ = r_bc²/2GM. And since the argument of a logarithmic function cannot be negative, one must have y ≥ 1 or 2GM/r_bc² ≤ 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].