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

Abstract

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+1where, 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]. 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 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 interestwe wonder whether nature allows gravitational collapse to continue inside the EH at all"

Mass of Schwarzschild Black Holes Is Indeed Zero And Black Hole Candidates Are Quasi Black Holes

“…But in the limit ǫ → 0, one has M → 0, so that, the gravitational mass of a neutral point particle M = 0. If so, the density of a dust, a collection of point particles, is zero and which explains why we obtained M = 0 for the Oppenheimer -Snyder black hole [10][11][12].…”

“…It was indeed shown that Oppenheimer & Snyder black hole tacitly corresponds to M = 0 and which never forms as the comoving proper time τ = ∞ in such a case [10,11]. Even when one would use Kruskal coordinates, it has been shown that, Schwarzschild BHs corresponds to M = 0 [12].…”

Properties of white dwarf in the binary system AR Scorpii and its observed features

“…The core of such exhausted star is no longer able to produce sufficient radiation pressure to prevent its outer layer from gravitational collapse. In Newtonian gravitation, the mass of collapsing star ( ) is constant even if it radiates (Mitra & Singh, 2013). Therefore, as radius of star ( ) decreases during the collapse, the ratio of mass to radius increases and leads to the extreme density in the final stable stage of the stellar evolution.…”