Physicists Use Topology to Classify the Stability of Singularity-Free Black Holes

Black holes are among the most extreme predictions of Einstein’s general relativity, and for more than a century they have carried an uncomfortable secret at their center: a singularity, a point where the equations of physics break down and quantities such as density and curvature diverge without limit. While observations from LIGO’s gravitational-wave detections and the Event Horizon Telescope’s images of supermassive black holes have confirmed that these objects exist, the singularity problem remains one of the deepest open wounds in classical gravitational theory. A new theoretical study published in The European Physical Journal C now offers a fresh way to interrogate a whole family of hypothetical black holes that cure this disease, using an unexpected tool: the mathematics of topology, the same branch of geometry that classifies objects by properties that survive stretching and twisting.

The research, carried out by A. A. M. Silva, M. H. Macêdo and R. R. Landim of the Federal University of Ceará in Brazil, focuses on the generalized Bardeen black hole, a two-parameter spacetime introduced by J. C. S. Neves and A. Saa that contains several celebrated singularity-free solutions as special cases. By tuning the parameters alpha and beta, the same metric reproduces the Bardeen black hole, the Hayward black hole, and a Simpson–Visser-type geometry, all of which replace the pathological central point of the Schwarzschild solution with a smooth, de Sitter-like core of finite curvature. The team’s central question was deceptively simple: do these different regular black holes belong to fundamentally distinct thermodynamic classes, or are they merely variations on a single topological theme?

To answer it, the authors employed a technique known as the off-shell generalized Helmholtz free energy method, a framework in which black hole thermodynamics is recast as the study of zeros of a carefully constructed vector field. In ordinary black hole thermodynamics, following the foundational work of Bekenstein and Hawking, the entropy of a black hole is proportional to the area of its event horizon and its temperature is proportional to its surface gravity. But local quantities like heat capacity can be cumbersome to work with, particularly for regular black holes where extra terms complicate the first law of thermodynamics. The topological approach sidesteps some of these difficulties by encoding stability information in winding numbers, integer-valued quantities that count how many times the vector field wraps around each of its zeros.

The machinery works as follows. The researchers defined a vector field whose first component is the derivative of the generalized Helmholtz free energy with respect to the horizon radius, evaluated at a Euclidean time period tau that acts as an inverse temperature. The zeros of this field correspond to thermodynamic equilibrium states of the black hole, and evaluating the on-shell condition tau equals one over T exactly recovers the Hawking temperature, a consistency check that the method passes for every geometry in the family. Each isolated, non-degenerate zero carries a winding number determined by the sign of the Jacobian of the mapping, which for this construction reduces to the sign of the second derivative of the free energy. Positive winding numbers mark locally stable branches with positive heat capacity; negative ones mark unstable branches.

The results are striking in their unity. Whenever two regular zeros exist simultaneously on the physical outer-horizon branch, they come as a pair: one smaller-radius defect with winding number plus one, corresponding to a locally stable configuration, and one larger-radius defect with winding number minus one, corresponding to an unstable one. Adding the charges gives a total topological number W equal to zero for the Bardeen case, the Hayward case, and the Simpson–Visser-type case alike. As the control parameter tau is lowered toward a critical value, the two defects slide toward each other along the inverse-temperature curve and coalesce at a critical radius, where the Jacobian vanishes and the black hole undergoes a phase transition signaled by a divergence and sign change of the heat capacity. Below the critical value, no physical zeros remain at all.

Crucially, the parameters alpha and beta, which encode how the singularity is regularized, do not change this global verdict. What they do control is everything local: the extremal bound below which no physical outer horizon exists, the exact location of the critical radius, the critical inverse temperature, the maximum Hawking temperature, and the width of the locally stable window. For the Bardeen case, with alpha equal to three and beta equal to two, the critical radius sits at roughly 2.697 times the length parameter a, while the Hayward case, with alpha and beta both equal to three, places it at approximately 2.168, and the Simpson–Visser-type case at exactly 1. The Simpson–Visser-type configuration also reaches the highest maximum temperature of the three, followed by Hayward and then Bardeen, demonstrating that the regularization mechanism directly shapes the thermal behavior even when the topological classification is unchanged.

The Schwarzschild limit tells a different story. As the length parameter a goes to zero, the regular core disappears, the singularity returns, and the inverse-temperature curve becomes monotonic, with no finite critical point. Only a single defect survives, carrying winding number minus one and yielding a total topological number W equal to minus one, in perfect agreement with the well-known thermodynamic instability of the Schwarzschild black hole, whose heat capacity is always negative. Placed within the recently developed universal thermodynamic-topological classification, the regular cases fall into the established W0-plus class, characterized by the stable-to-unstable ordering of their defects, while the Schwarzschild limit belongs to the W1-minus class. The authors emphasize that their contribution is not the discovery of a new topological class but a unified, family-wide analytical demonstration that varying the regularization parameters shifts local thermodynamic data without ever moving a configuration out of its global class.

Physically, the findings touch on one of the most tantalizing possibilities in black hole physics: remnants. Because the Hawking temperature of these regular geometries vanishes exactly at the extremal radius, evaporation halts at a finite mass, leaving behind a stable, non-radiating object. Such remnants have even been proposed as candidates for dark matter, and the new topological framework offers a systematic way to assess their thermodynamic viability across an entire family of models rather than case by case. The study also draws an intriguing comparison with recent work on regular black holes built from pure gravity through infinite towers of higher-curvature corrections, which, despite a completely different gravitational origin and entropy prescription, land in the very same W0-plus class, hinting at a universality that transcends the details of how regularity is achieved.

The authors are careful to note the limits of their analysis. The entire classification is performed in a fixed-parameter ensemble in which the length parameter a, along with alpha and beta, is held fixed during thermodynamic variations; promoting these quantities to thermodynamic variables would modify the first law and could, in principle, alter the topological structure. Future extensions to charged, rotating, or higher-dimensional generalizations remain open territory. Nevertheless, the work delivers a crisp analytical criterion separating parameter-dependent local stability from the global topological identity of regular black holes, and it reinforces a growing realization in gravitational physics: that the deepest questions about spacetime, from the fate of singularities to the stability of horizons, may be answered not by looking at any single solution, but by reading the topological signatures written into the thermodynamics of entire families of them.

Subject of Research: Topological thermodynamics and stability classification of regular black holes in the generalized Bardeen spacetime family

Article Title: Topological thermodynamics of generalized Bardeen black hole

Article References: Silva, A. A. M., Macêdo, M. H., & Landim, R. R. (2026). Topological thermodynamics of generalized Bardeen black hole. The European Physical Journal C, 86(9), Article 1086. https://doi.org/10.1140/epjc/s10052-026-16379-4

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16379-4

Keywords: black hole thermodynamics, topological classification, regular black holes, Bardeen black hole, Hayward black hole, Simpson–Visser spacetime, winding number, Helmholtz free energy, Hawking temperature, phase transition, heat capacity, black hole remnants