Interior of a Schwarzschild Black Hole Revisited

Abstract

The Schwarzschild solution has played a fundamental conceptual role in general relativity, and beyond, for instance, regarding event horizons, spacetime singularities and aspects of quantum field theory in curved spacetimes. However, one still encounters the existence of misconceptions and a certain ambiguity inherent in the Schwarzschild solution in the literature. By taking into account the point of view of an observer in the interior of the event horizon, one verifies that new conceptual difficulties arise. In this work, besides providing a very brief pedagogical review, we further analyze the interior Schwarzschild black hole solution. Firstly, by deducing the interior metric by considering time-dependent metric coefficients, the interior region is analyzed without the prejudices inherited from the exterior geometry. We also pay close attention to several respective cosmological interpretations, and briefly address some of the difficulties associated to spacetime singularities. Secondly, we deduce the conserved quantities of null and timelike geodesics, and discuss several particular cases in some detail. Thirdly, we examine the Eddington-Finkelstein and Kruskal coordinates directly from the interior solution. In concluding, it is important to emphasize that the interior structure of realistic black holes has not been satisfactorily determined, and is still open to considerable debate.

On Time-Dependent Black Holes and Cosmological Models From a Kaluza–klein Mechanism

“…The range of values of z is −∞ ≤ z ≤ +∞, whereas the ordinary radial variable r ≥ 0. The components of the Einstein tensor in an ortho-normal reference frame are [38] (and references therein) one arrives at a number consistent with the Dirac-Eddington-Weyl large number coincidences M o ∼ 10 61 m P lanck = 10 80 m proton since 10 80 is of the same order of magnitude as the square of the ratio (10 40 ) of the Hubble scale and classical electron radius and the square of the ratio ( 10 40 ) of the electrostatic force between an electron and a proton versus their corresponding gravitational force. In this case, one finds ( in natural units h = c = 1 ) that the temporal horizon is given by t h = 3 σ = 1.5 t Hubble such that the observed universe lies inside the temporal horizon because t Hubble < t h .…”

Kruskal Coordinates and Mass of Schwarzschild Black Holes: No Finite Mass Black Hole at All

“…Even the authors who, in a desperate bid to save the black hole paradigm, invent almost ridiculous definition of "free fall speed" by which an infalling particle has 0 V  even at the central singularity, were forced to conclude that [16] "The solutions that do away with the interior singularity and the event horizon, although interesting in themselves, sweep the inherent conceptual difficulties of black holes under the rug. In concluding, we note that the interior structure of realistic black holes have not been satisfactorily determined, and are still open to considerable debate."…”

“…Accordingly, they must obey the Buchdahl limit [20]. Indeed such quasi-Newtonian supermassive stars in principle may have a surface gravitational red-shift 2.0  z 1  [7,16]. In contrast, in principle there could be quasi-static extremely relativistic radiation pressure supported stars with (RRPSSs) and no lower mass limit as well [21][22][23].…”

Boundary sources in the Doran-Lobo-Crawford spacetime

“…Throughout the paper, we choose units in which the speed of light has the value = 1. Doran et al [7] have taken into consideration the case of a time-dependent ( ), when the metric (1) is no longer a solution of the vacuum Einstein equations. A timedependent mass inside the horizon is justified by the fact that it is equivalent to an -dependent mass ( ) outside the horizon [8], which is a common situation.…”