Lorentzian-Euclidean black holes and Lorentzian to Riemannian metric transitions

In recent papers on spacetimes with a signature-changing metric, Capozziello et al. [Avoiding singularities in Lorentzian-Euclidean black holes: The role of atemporality, Phys. Rev. D 109, 104060 (2024).PRVDAQ2470-001010.1103/PhysRevD.109.104060] and Hasse and Rieger [Pseudo-Timelike loops in signature changing semi-Riemannian manifolds with a transverse radical, arXiv:2409.02403.] introduced, respectively, the concept of a Lorentzian-Euclidean black hole and new elements for Lorentzian-Riemannian signature change. In both cases the transition in the signature happens on a hypersurface H. The former is a signature-changing modification of the Schwarzschild spacetime satisfying the vacuum Einstein equations in a weak sense. Here H is the event horizon which serves as a boundary beyond which time becomes imaginary. We clarify an issue appearing in Capozziello et al. based on numerical computations which suggested that an observer in radial free fall would require an infinite amount of proper time to reach the event horizon. We demonstrate that the proper time needed to reach the horizon remains finite, consistently with the classical Schwarzschild solution, and suggesting that the model in Capozziello et al. should be revised. About the latter, we stress that H is naturally a spacelike hypersurface related to the future or past causal boundary of the Lorentzian sector. Moreover, a number of geometric interpretations appear, as the degeneracy of the metric g corresponds to the collapse of the causal cones into a line, the degeneracy of the dual metric g∗ corresponds to collapsing into a hyperplane, and additional geometric structures on H (Galilean and dual Galilean) might be explored.