Vesselin Petkov, 27.02.2016
Minkowski Institute, Montreal, Canada
This is another example of the kind of questions that are being analyzed at the Minkowski Institute – on the one hand, it is believed that black holes exist, especially after the recent detection of gravitational waves (initially almost unanimously regarded as a signature of black holes, but now it was shown that that might not be the case ); on the other hand, however, a careful (Minkowski-type) examination of the Schwarzschild solution of the Einstein equations in general relativity reveals that the two major problems with it cannot be easily brushed aside:
1. The Schwarzschild solution contains a singularity and perhaps most relativists feel very uneasy about this fact, because the existence of a singularity in a physical theory  is a clear sign that the theory breaks down in the circumstances where the singularity appears. The authors of a recent attempt  to free general relativity of singularities stated their motivation clearly: “We believe that no acceptable physical theory should have a singularity (!), not even a coordinate singularity of the type discussed above! The appearance of a singularity shows the limitations of the theory.”
2. The Schwarzschild solution implies that there exist two realities inside and outside the Schwarzschild sphere of radius r = 2m (where m is the gravitating mass)  – one for observers falling together with the constituents of the collapsing body (of mass m), and another for distant observers far away from the Schwarzschild sphere. The problem is most clearly formulated by Dirac :
“We see that the Schwarzschild solution for empty space can be extended to the region r < 2m. But this region cannot communicate with the space for which r > 2m. Any signal, even a light signal, would take an infinite time to cross the boundary r = 2m, as we can easily check. Thus we cannot have direct observational knowledge of the region r < 2m. Such a region is called a black hole, because things may fall into it (taking an infinite time, by our clocks, to do so) but nothing can come out.
The question arises whether such a region can actually exist. All we can say definitely is that the Einstein equations allow it. A massive stellar object may collapse to a very small radius and the gravitational forces then become so strong that no known physical forces can hold them in check and prevent further collapse. It would seem that it would have to collapse into a black hole. It would take an infinite time to do so by our clocks, but only a finite time relatively to the collapsing matter itself.”
So, here is the second major problem with black holes explicitly formulated – as we are (and always have been) distant observers it will take an infinite time for black holes to form, which, in ordinary language, means that black holes will never exist for us.
In the second reality of the falling observers, according to general relativity, black holes will form.
I wonder whether there will be physicists and (perhaps more probably) philosophers who would claim that reality is observer-dependent. But even if this were the case, black holes would never exist in our reality, because this follows from general relativity – the same theory that predicts the very existence of black holes. More rigorously stated, the Schwarzschild solution of the Einstein equations in general relativity predicts both (i) black holes, and (ii) that they will never form for distant observers.
What strikes me is that there are colleagues who freely talk about black holes perhaps without realizing that they are employing double standards with respect to the meaning of “take an infinite time”:
- From Dirac’s quote above: “Any signal, even a light signal, would take an infinite time to cross the boundary r = 2m” – in this case “take an infinite time” is interpreted to mean “never“; so only according to this interpretation even light will never escape from the Schwarzschild sphere, which allows one to talk about black holes.
- Again from Dirac’s quote above: “It would seem that it would have to collapse into a black hole. It would take an infinite time to do so by our clocks” – in this case “take an infinite time” inexplicably is not taken to mean “never” as above; instead, it is assumed without any justification that black holes exist.
For more on whether or not black holes exist see [7-9].
 Vitor Cardoso, Edgardo Franzin, and Paolo Pani, “Is the Gravitational-Wave Ringdown a Probe of the Event Horizon?” Phys. Rev. Lett. 116, 171101 (2016) and also Synopsis: Did Black Hole “Mimickers” Produce LIGO Signal?
 Robert Geroch, What is a Singularity in General Relativity? Annals of Physics 48 (1968) 526-540
 P.O. Hess, M. Schäfer, W. Greiner, Pseudo-Complex General Relativity (Springer, Heidelberg 2016). See the essence of the approach: W. Greiner, P.O. Hess, M. Schäfer,T. Schönenbach and G. Caspar, “There are No Black Holes – Pseudo-Complex General Relativity”, in: P. Nicolini, M. Kaminski, J. Mureika, M. Bleicher (eds), 1st Karl Schwarzschild Meeting on Gravitational Physics (Springer, Heidelberg 2016) pp. 33-42
 In addition to this problem, Papapetrou also and particularly emphasizes the serious anomaly on the Schwarzschild sphere, whose physical meaning, I think, has not been thoroughly examined – “But these geodesics are space-like for r > 2m and time-like for r < 2m. The tangent vector of a geodesic undergoes parallel transport along the geodesic and consequently it cannot change from a time-like to a space-like vector. It follows that the two regions r > 2m and r < 2m do not join smoothly on the surface r = 2m.” 
 A. Papapetrou, Lectures on General Relativity (Reidel, Dordrecht 1974) pp. 85-86
 P.A.M. Dirac, General theory of relativity (Princeton University Press, Princeton 1996) pp. 35-36
 Backreaction of Hawking Radiation on a Gravitationally Collapsing Star I: Black Holes? (Black holes cannot form)