Kruskal Coordinates and Mass of Schwarzschild Black Holes

Abstract

Schwarzschild coordinates (r,t) fail to describe the region within the event horizon (EH), (r <= 2 M), of a Black Hole (BH) because the metric coefficients exhibit singularity at r=2 M, and the radial geodesic of a particle appears to be null (ds^2 =0) when actually it must be timelike (ds^2 >0), if, indeed, r_g =2 M >0. Thus, both the exterior and the interior regions of BHs are described by singularity free Kruskal-Szekeres coordinates. However, we show that, in this case too, ds^2= 0 for r=2M. And this result can be physically reconciled only if the EH coincides with the central singularity or if the mass of Schwarzschild black holes M= 0. This conclusion agrees with the result of our following paper (astro-ph/9904163) which points out that the Oppenheimer-Snyder paper showing formation of a BH, too, corresponds to a BH of mass M=0. It means that Schwarzschild singularity and the central singularity are synonymous, and there is no spacetime beneath the EH. Hence the concept of ``White Holes'' and ``Other Universes'' suggested by the Kruskal diagram naturally ceases to exist. We insist that our derivation is straightforward and there is no scope for any ambiguity, any reader may be able to verify it. The proper time for formation of a zero mass BH may be seen to be infinite. Therefore at any finite epoch there would not be any BH. On the other hand, there would be such eternally collapsing configurations with arbitrary high but finite surface red-shifts. If accretion onto such collapsing configurations is advection dominated, there would be little radiation output. Observationally, such collapsing configurations may be identified as ``Black Holes''.