Lorentzian-Euclidean black holes and Lorentzian to Riemannian metric transitions

TL;DR

This paper explores signature-changing spacetimes, introducing Lorentzian-Euclidean black holes that satisfy Einstein's equations in a weak sense, and analyzes the geometric and causal properties of metric transitions.

Contribution

It presents a new concept of Lorentzian-Euclidean black holes with signature change and discusses the geometric structures and causal boundaries involved in metric signature transitions.

Findings

Proper time to the horizon remains finite.

The signature change hypersurface is spacelike and relates to causal boundaries.

Degeneracy of metrics corresponds to collapse of causal cones and structures.

Abstract

In recent papers on spacetimes with a signature-changing metric, the concept of a Lorentzian-Euclidean black hole and new elements for Lorentzian-Riemannian signature change have been introduced. A Lorentzian-Euclidean black hole is a signature-changing modification of the Schwarzschild spacetime satisfying the vacuum Einstein equations in a weak sense. Here the event horizon serves as a boundary beyond which time becomes imaginary. We demonstrate that the proper time needed to reach the horizon remains finite, consistently with the classical Schwarzschild solution. About Lorentzian to Riemannian metric transitions, we stress that the hypersurface where the metric signature changes is naturally a spacelike hypersurface which can be identified with the future or past causal boundary of the Lorentzian sector. Moreover, a number of geometric interpretations appear, as the degeneracy of the metric corresponds to the collapse of the causal cones into a line, the degeneracy of the dual metric corresponds to collapsing into a hyperplane, and additional geometric structures on the transition hypersurface (Galilean and dual Galilean) might be explored.