Stellar gravitational collapse, singularity formation and theory breakdown

We review three definitions (missing point(s) unsteadiness, infinite quadratic curvature invariant, and geodesic incompleteness) of what a gravitational singularity is, and argue that prediction of a gravitational singularity is problematic for General Relativity (GR), indicating breakdown of Lorentzian geometry, only insofar as it concerns the infinite curvature singularity characterization. In contrast, the geodesic incompleteness characterization is GR's innovating hallmark, which is not meaningfully available in Newtonian gravity formulations (locally infinite density field, and locally infinite gravitational force) of what a gravitational singularity is. It is the continuous, non-quantized, nature of Lorentzian geometry which admits gravitational contraction be continued indefinitely. The Oppenheimer-Snyder 1939 analytical solution derives formation of a locally infinite curvature singularity and of incomplete geodesics, while Penrose's 1965 theorem concerns formation of incomplete (null) geodesics only. We critically examine the main physical arguments against gravitational singularity formation in stellar collapse, with scope restriction to decades spanning in between Schwarzschild's 1916 solution and Penrose's 1965 singularity theorem. As the most robust curvature singularity formation counterargument, we assess Markov's derivation of an upper bound on the quadratic curvature invariant R mu nu lambda delta R mu nu lambda delta <= 1/l(P)(4) from a ratio of natural constants (h) over bar, c and G, in connection with Wheeler's grounding of the premise that the Planck scale l(P) is ultimate.