A new proof for non-occurrence of trapped surfaces in spherical collapse

We present here a very simple, short and new proof which shows that no trapped surface is ever formed in spherical gravitational collapse of isolated bodies. Although this derivation is of purely mathematical nature and without any assumption, it is shown, in the Appendix, that, physically, trapped surfaces do not form in order that the 3 speed of the fluid as measured by an observer at a fixed circumference coordinate R (a scalar), is less than the speed of light c. The consequence of this result is that, mathematically, even if there would be Schwarzschild Black Holes, they would have unique gravitational mass M = 0. Recall that Schwazschild BHs may be considered as a special case of rotating Kerr BHs with rotation parameter a = 0. If one would derive the Boyer-Lindquist metric [1] in a straight forward manner by using the Backlund transformation[2], one would obtain a = M sinφ where φ is the azimuth angle. This relation demands that BHs have unique massM = 0 (along with a = 0) which in turn confirms that there cannot be any trapped surface in realistic gravitational collapse where the fluid has real pressure and density. Since there is no trapped surface and horizon, there is no Information Paradox in the first place. When a self-gravitating fluid undergoes gravitational contraction, by virtue of Virial Theorem, part of the gravitational energy must be radiated out. Thus the total mass energy, M , (c = 1) of a body decreases as its radius R decreases. But in Newtonian regime (2M/R ≪ 1, G = 1), M is almost fixed and the evolution of the ratio, 2M/R, is practically dictated entirely by R. If it is assumed that even in the extreme general relativistic case 2M/R would behave in the same Newtonian manner, then for sufficiently small R, it would be possible to have 2M/R > 1, i.e, trapped surfaces would form[3,4].