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Consistent Regularization of Signature-Changing BTZ Black Holes

Farzad Milani

TL;DR

This work establishes a mathematically rigorous framework for signature-changing black holes by resolving a key inconsistency in prior regularization schemes and demonstrating a consistent, vacuum BTZ geometry with a Euclidean interior. The authors introduce a modified Hadamard regularization that properly handles distributional curvature terms, yielding geodesically complete, linearly stable spacetimes with unitary quantum-field propagation across the change surface. The central singularity is replaced by a topological boundary, achieved through atemporality where infalling observers require infinite proper time to reach the horizon, while curvature remains finite. External thermodynamics remain identical to the standard BTZ black hole, supporting the physical viability of signature-change as a classical mechanism for singularity avoidance with potential implications for quantum gravity and black hole physics.

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.

Consistent Regularization of Signature-Changing BTZ Black Holes

TL;DR

This work establishes a mathematically rigorous framework for signature-changing black holes by resolving a key inconsistency in prior regularization schemes and demonstrating a consistent, vacuum BTZ geometry with a Euclidean interior. The authors introduce a modified Hadamard regularization that properly handles distributional curvature terms, yielding geodesically complete, linearly stable spacetimes with unitary quantum-field propagation across the change surface. The central singularity is replaced by a topological boundary, achieved through atemporality where infalling observers require infinite proper time to reach the horizon, while curvature remains finite. External thermodynamics remain identical to the standard BTZ black hole, supporting the physical viability of signature-change as a classical mechanism for singularity avoidance with potential implications for quantum gravity and black hole physics.

Abstract

Spacetime singularities represent a fundamental challenge in gravitational physics. We present a mathematically consistent framework for signature-changing black holes based on the -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 , 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 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.

Paper Structure

This paper contains 63 sections, 5 theorems, 81 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The Signature-Changing BTZ Geometry and its Consistent Regularization
  3. The Lorentzian-Euclidean BTZ Metric
  4. Distributional Curvature and Regularization Scheme
  5. Modified Hadamard Regularization Scheme
  6. Application to BTZ Curvature
  7. Physical Analysis: Curvature and Singularity Avoidance
  8. Regularized Curvature and Energy Conditions
  9. Ricci Tensor and Scalar Curvature
  10. Weyl Tensor and Tidal Forces
  11. Energy Conditions and Effective Matter
  12. Singularity Avoidance through Atemporality
  13. Geodesic Motion and Proper Time Asymptotics
  14. Accelerated Observers and Universal Horizon Asymptotics
  15. Physical Interpretation and Observational Implications
  16. ...and 48 more sections

Key Result

Theorem 1

The signature-changing BTZ geometry is linearly stable against gravitational perturbations. Comprehensive analytical and numerical analyses confirm that no exponentially growing modes exist. All perturbations remain bounded in time, and the quasi-normal mode spectrum exhibits characteristic exponent

Figures (7)

  • Figure 1: Regularized signature function $\varepsilon(r)$ for varying $\rho$ ($\kappa=1$ fixed). Larger $\rho$ (red) yields smoother transitions; smaller $\rho$ (green) approaches the discontinuous limit.
  • Figure 2: Regularized $\varepsilon(r)$ for varying $\kappa$ ($\rho=0.1 r_h^2$ fixed). Lower $\kappa$ (red) gives sharper transitions; higher $\kappa$ (green) produces smoother transitions.
  • Figure 3: (a) Real proper time $\sigma(\eta)$ and (b) complete proper time for radial geodesic motion. Divergence at $\eta = \eta_H$ demonstrates infinite proper time to reach horizon.
  • Figure 4: (a) Coordinate time $t(\eta)$ in Lorentzian region. (b) Complete behavior showing Lorentzian (real) and Euclidean (imaginary) parts.
  • Figure 5: (a) Comparison of $\sigma(\eta)$ and $t(\eta)$ in Lorentzian exterior. (b) Full comparison across both regions, showing imaginary proper time in Euclidean interior.
  • ...and 2 more figures

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • proof