Lorentzian-Euclidean singularity-free solutions to gravitational collapse

This study explores singularity-free solutions to the static, spherical symmetric Einstein equations with the standard Schwarzschild solution as a boundary condition. Imposing the absence of curvature singularities and requiring differentiability of the time component of the metric leads to a sign change across the horizon, violating the Principle of Equivalence locally. We find a solution within the event horizon with a simple ``cosmological constant'' stress-energy tensor. Considering the impact of sign change to a compact stellar remnant, modeled by an incompressible perfect fluid obeying the Tolman-Oppenheimer-Volkoff equation, we rediscover the same geometry, indicating both mathematical and physical feasibility of the model. We also find a new theoretical limit M/R=3/8, which is lower than the Buchdahl limit of M/R=4/9 for the density of a perfect fluid that will recede behind an event horizon. The equation of state is discussed, and we propose that the final state is described by a Higgs-like free scalar field.