Consistent Regularization of Signature-Changing BTZ Black Holes Enables Finite Curvature and Resolves Distributional Inconsistencies

The nature of spacetime singularities represents a long-standing problem in theoretical physics, and now Farzad Milani from Technical and Vocational University, along with colleagues, presents a mathematically consistent approach to understanding these enigmatic points in spacetime. The team investigates signature-changing black holes, focusing on a modified version of the well-known BTZ metric, where spacetime transitions between Lorentzian and Euclidean characteristics at the event horizon. This research addresses a fundamental inconsistency in previous attempts to regularise these black holes, introducing a novel method that ensures a smooth and physically realistic solution, free from problematic surface layers or impulsive gravitational waves. By demonstrating linear stability and consistent propagation of scalar fields, the team establishes the viability of this approach, effectively reinterpreting the singularity not as a breakdown of physics, but as a removable boundary, and solidifying the concept of atemporality as a robust mechanism for resolving singularities.

The core idea is that black holes aren’t necessarily defined by a singularity at their center, but by a boundary where spacetime transitions into a fundamentally different, atemporal, Euclidean domain. The work focuses on a three-dimensional black hole model, chosen for its mathematical simplicity, and utilizes established techniques to analyze the spacetime around the black hole. The research proposes that singularities can be avoided by allowing spacetime to transition into a Euclidean domain at the black hole’s center, where time as we understand it ceases to exist.

This transition isn’t a sharp boundary, but a gradual shift in the nature of spacetime, and is not merely a mathematical trick, but a potentially physical description of what happens at the heart of a black hole. The mathematical framework successfully avoids the formation of a singularity, and all causal geodesics remain complete, a crucial requirement for a physically realistic spacetime. A Penrose diagram reveals a novel global structure with a clear transition between Lorentzian and Euclidean regions, and the Euclidean region is demonstrably atemporal. The research suggests that this atemporal region isn’t just a mathematical artifact, but a potentially physical description of the black hole’s interior.

The transition between spacetime types introduces complex mathematical terms, which are carefully regularized to ensure a consistent solution. This research offers an alternative to the standard singularity theorems of general relativity, and provides a new perspective on the nature of black hole interiors, suggesting they may not be characterized by singularities but by fundamentally different spacetime structures. The atemporal region could potentially be related to concepts in quantum gravity, where time may not be a fundamental concept. While not explicitly addressed, the avoidance of a singularity could have implications for resolving the black hole information paradox. In essence, this paper proposes a radical but mathematically consistent solution to the singularity problem in black holes by allowing for a transition into an atemporal, Euclidean domain, offering a new and potentially fruitful avenue for exploring the nature of black holes and the fundamental structure of spacetime.

Signature-Changing Black Holes, Hadamard Regularization Achieved

Scientists have achieved a mathematically consistent framework for describing black holes with a changing spacetime signature, resolving a long-standing challenge in classical gravity. The work introduces a modified Hadamard regularization scheme to address inconsistencies in previous approaches, definitively eliminating spurious surface layers and impulsive gravitational waves, yielding a genuine vacuum solution that accurately represents the geometry of these exotic objects. The team constructed the model in both standard and Painlevé-Gullstrand coordinates, establishing a rigorous regularization scheme for its distributional curvature. This provides a consistent geometric framework for analyzing singularity resolution, demonstrating a distributionally well-defined vacuum geometry with no problematic energy contributions.

Detailed analysis of the curvature tensors reveals bounded curvature invariants, confirming the absence of both surface layers and impulsive gravitational waves, and demonstrating a mechanism for singularity avoidance where radially infalling observers require infinite proper time to reach the horizon. Crucially, the research extends beyond classical consistency to establish physical robustness. Scientists proved the linear stability of the geometry against gravitational perturbations through both analytical and numerical methods, and demonstrated that quantum scalar field propagation remains well-defined, unitary, and singularity-free across the horizon. The team established the comprehensive physical consistency of the solution by demonstrating that the r = 0 singularity is resolved into a removable topological boundary, interpreting the Euclidean interior as a frozen quantum gravitational phase, and proving that standard black hole thermodynamics remains unchanged. The resulting Ricci tensor satisfies the vacuum equations of general relativity everywhere, including in a distributional sense, demonstrating a consistent and physically viable resolution of the black hole singularity problem. This work challenges the necessity of quantum gravity for resolving fundamental spacetime pathologies and opens new avenues for exploring the interface between classical and quantum gravity.

Horizon Transition Resolves Black Hole Singularities

This work presents a mathematically rigorous framework for understanding black holes with horizons that transition from Lorentzian to Euclidean signature, effectively resolving the classical problem of spacetime singularities. By revisiting a well-established model, researchers developed a modified Hadamard regularization scheme to address inconsistencies in previous approaches to defining vacuum solutions, successfully eliminating surface layers and impulsive waves. The resulting geometry demonstrates finite curvature throughout and requires infinite proper time for infalling observers to reach the horizon, indicating a physically plausible resolution of the singularity. Crucially, the team established the stability of this solution through analysis of gravitational perturbations and propagation of quantum scalar fields, confirming its robustness against small disturbances and consistent behaviour with quantum mechanics.

This analysis revealed that the singularity at the black hole’s centre is effectively reinterpreted as a removable topological boundary, suggesting a fundamentally different structure for these objects than previously understood. Future research directions include exploring the observational consequences of this geometry, such as potential alterations to black hole silhouettes and accretion disk structures, as well as investigating its connection to quantum gravity theories. The elimination of the central singularity offers a promising avenue for probing the interior of black holes and potentially resolving long-standing puzzles in theoretical physics, opening new possibilities for understanding these enigmatic objects and the nature of spacetime itself.