Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lorentzian-Euclidean black holes and Lorentzian to Riemannian metric transitions

Rossella Bartolo, Erasmo Caponio, Anna Valeria Germinario, Miguel Sánchez

TL;DR

This paper analyzes spacetimes with signature-changing metrics, focusing on Lorentzian-Euclidean black holes (LES) and Lorentzian-to-Riemannian transitions. The LES metric is examined with $g = - epsilon(r) (1 - 2m/r) dt^2 + (1 - 2m/r)^{-1} dr^2 + r^2 dOmega^2$, where epsilon(r) = sign(1 - 2m/r), and a horizon at $r=2m$ marks the signature change; a smooth regularization avoids divergences and facilitates geodesic analysis. Through Gullstrand-Painleve coordinates and an auxiliary smooth function, the paper shows that the proper time to reach the horizon is finite, contrary to some claims of divergence. It also discusses Lorentzian-Riemannian transitions, treating the hypersurface where the metric degenerates as a causal boundary and highlighting dual metric degenerations and potential Galilean structures, thereby motivating a revision of the LES model and inviting further study of global hyperbolicity notions in signature-changing spacetimes.

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.

Lorentzian-Euclidean black holes and Lorentzian to Riemannian metric transitions

TL;DR

This paper analyzes spacetimes with signature-changing metrics, focusing on Lorentzian-Euclidean black holes (LES) and Lorentzian-to-Riemannian transitions. The LES metric is examined with , where epsilon(r) = sign(1 - 2m/r), and a horizon at marks the signature change; a smooth regularization avoids divergences and facilitates geodesic analysis. Through Gullstrand-Painleve coordinates and an auxiliary smooth function, the paper shows that the proper time to reach the horizon is finite, contrary to some claims of divergence. It also discusses Lorentzian-Riemannian transitions, treating the hypersurface where the metric degenerates as a causal boundary and highlighting dual metric degenerations and potential Galilean structures, thereby motivating a revision of the LES model and inviting further study of global hyperbolicity notions in signature-changing spacetimes.

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.

Paper Structure

This paper contains 9 sections, 2 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Lorentzian-Euclidean black holes and finiteness of arrival proper time
  3. The signature-changing metric
  4. On the geodesics of the signature-changing metric
  5. Finiteness of the proper arrival time at the horizon in the Gullstrand-Painlevé coordinates
  6. Remarks on the Lorentzian-Riemannian signature change
  7. Conclusions
  8. The equation \ref{['dottr']} in Gullstrand-Painlevé coordinates
  9. Christoffel symbols

Key Result

Proposition 2.1

Let $\gamma (s) = (\mathscr T(s), r(s), \theta(s), \varphi(s))$, $s\in I$, be a causal geodesic with respect to the metric $g_{\tilde{\varepsilon}}$, such that $r(s) \in ( 2m, 3m)$, for all $s\in I$. Then function $r$ is strictly concave.

Theorems & Definitions (5)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof