WHY THERE CANNOT BE ANY FINITE MASS ``Black Hole''
First recall, the gravitational mass appearing in the vacuum Schwarzschild solution appears as an `Integration Constant'' (IC) $$\alpha = 2 M$$ (G=c=1).
For an object with finite radius (Sun, Earth, Galaxy), this IC ($\alpha$) is obviously finite (and positive):
\begin{equation}
M = \int_0^{R_0} 4 \pi \rho R^2 dR
\end{equation}
But is there a guarantee that $\alpha$ would remain finite in case the body would contract to a ``Point Mass'', i.e., $R_0 \to 0$. BH paradigm is based on the HYPOTHESIS that this IC must be finite even when the upper limit of the integration would merge with the lower one ($R_0=0$). However, it was shown by me that this hypothesis was incorrect, i.e., the gravitational mass of a neutral ``Point Mass'', the source of BH solution is zero:
Ref. 1. A. Mitra, Journal ofhttp: Mathematical Physics, Volume 50, Issue 4, pp. 042502-042502-3 (2009): //arxiv.org/abs/0904.4754
Such a result was earlier inferred (not proved) by French mathematical relativist Luis Bel
Ref. 2. L. Bel, "Schwarzschild Singularity," Journal of Mathematical Physics, Vol. 10, No. 8, 1969 pp. 1501-1503. doi:10.1063/1.1664997
The abstract of the above paper is a one liner ``A new point of view is presented for which the Schwarzschild singularity becomes a real point singularity on which the sources of Schwarzschild's exterior solution are localized.''
The Sch singularity becomes a ``point singularity'' only when M=0 for a ``point particle'' and for the Schwarzschild BH.
WHY THIS RESULT IS INEVITABLE?
Suppose the point particle has a mass $M_0$. Then one would expect a Ricci Scalar ($\cal R}$ at $R=0$
\begin{equation}
{\cal R} (R=0) = - 4 \pi \delta{R=0} M_0/R^2
\end{equation}
where $\delta{R}$ is Dirac delta function. But the BH solution yields
$\cal R} =0$. These two results can be reconciled iff $M_0=0$.
There for massive ``Black Hole Candidates'' (or any thing else with finite gravitational mass) CANNOT be true BHs.
Therefore we could have bade farewell to BH paradigm in 2009. In another post I will analyze this result from the view point of GR collapse, no numerical hanky panky, but by generic or exact means.