Consistent Regularization of Signature-Changing BTZ Black Holes

TL;DR

This paper introduces a consistent mathematical framework for signature-changing BTZ black holes, resolving previous regularization issues, and demonstrating that such models can avoid singularities while remaining physically stable and well-defined.

Contribution

It develops a modified Hadamard regularization for signature-changing black holes, ensuring a surface-layer-free vacuum solution and establishing singularity avoidance through atemporality.

Findings

Regularization scheme rectifies previous inconsistencies.

Radial infall requires infinite proper time, preventing access to singularity.

Geometry remains stable and quantum fields propagate unitarily across signature change.

Abstract

Spacetime singularities represent a fundamental challenge in gravitational physics. We present a mathematically consistent framework for signature-changing black holes based on the $(2+1)$-dimensional BTZ metric, where the signature transitions from Lorentzian $(-,+,+)$ to Euclidean $(+,+,+)$ at the horizon. We identify and rectify a critical inconsistency in previous regularization schemes concerning second-order distributional terms $\varepsilon''(r)$, introducing a \emph{modified Hadamard regularization} that respects distribution theory. This produces a vacuum solution free of surface layers and impulsive gravitational waves. Geodesic analysis reveals that radially infalling observers require infinite proper time to reach the horizon, effectively preventing access to the would-be singularity while maintaining finite curvature invariants throughout the spacetime. We further establish the physical robustness of the geometry by demonstrating linear stability against gravitational perturbations, showing that quantum scalar field propagation remains unitary and well-defined across the signature change, and reinterpreting the $r=0$ region as a topological boundary rather than a curvature singularity. Our work establishes atemporality via signature change as a mathematically rigorous mechanism for \emph{singularity avoidance} in black hole spacetimes.