Smooth Signature Change as a Mechanism for Singularity Avoidance in BTZ Black Holes

Abstract

Spacetime singularities represent a fundamental challenge in classical general relativity, prompting investigations into mechanisms that could resolve or avoid them. The paradigm of \emph{signature change}, where the metric transitions from Lorentzian to Euclidean signature across the horizon, offers a geometric approach to singularity resolution. However, previous implementations based on the discontinuous sign function $\varepsilon(r)$ encounter mathematical inconsistencies in distributional curvature and lead to complex-valued metrics in regular coordinate systems. In this work, we introduce a novel, mathematically rigorous framework for signature-changing black holes by replacing $\varepsilon(r)$ with a smooth, real transition function $\mathcal{S}_\delta(r) = \tanh[(r-r_h)/\delta]$. We develop this framework within the analytically tractable $(2+1)$-dimensional Ba\~nados-Teitelboim-Zanelli (BTZ) geometry. The resulting metric is globally smooth and real for any $\delta > 0$. We prove it satisfies $R_{\mu\nu}=0$ identically, confirming it as a vacuum solution without surface layers. Curvature invariants remain finite everywhere. Geodesic analysis reveals that radially infalling observers require infinite proper time to reach the horizon, implementing the \emph{atemporality} mechanism quantitatively. We further establish the physical robustness of the solution by demonstrating its linear stability against gravitational perturbations, well-defined propagation of quantum scalar fields, and preservation of standard BTZ thermodynamics for external observers. Our smooth-transition framework resolves the foundational issues of prior distributional approaches and provides a consistent, computationally tractable model for signature change as a mechanism for classical singularity avoidance.