Topological Thermodynamics of Generalized Bardeen Black Hole

Abstract: Neves and Saa introduced a two parameter spacetime that includes the Hayward, Bardeen, and Simpson-Visser geometries as particular cases. In this work, we employ the generalized off-shell Helmholtz free energy method to investigate the thermodynamic properties of the generalized Bardeen black hole within a topological framework. We construct the associated vector field and analyze its zeros, whose winding numbers allow us to classify the thermodynamic branches and identify critical points associated with phase transitions. The regular black hole configurations exhibit two topological defects with opposite winding numbers, resulting in a vanishing total topological charge, while the Schwarzschild case contains a single unstable branch. Our results demonstrate how the regularization parameters affect the thermodynamic stability and phase structure of the spacetime.

• The paper introduces a unified analysis combining thermodynamics and topology to study regular black holes using a generalized Bardeen metric.
• It employs an off-shell Helmholtz free energy framework and winding number analysis to detect critical points and phase transitions, linking changes in heat capacity with stability.
• Numerical results for Bardeen, Hayward, and Simpson–Visser cases reveal distinct critical values, suggesting implications for black hole remnants and dark matter.

Topological Thermodynamics of Generalized Bardeen Black Hole: A Technical Review

Background and Motivation

The paper conducts a rigorous analysis of the thermodynamic and topological properties of the generalized Bardeen black hole spacetime, which subsumes a variety of singularity-free metrics (including Hayward and Simpson-Visser) as special cases through a two-parameter family. The motivation stems from the need to regularize central singularities present in classical black hole solutions (notably Schwarzschild) and to elucidate the impact of regularization parameters on thermodynamic stability and phase structure. The study leverages the generalized off-shell Helmholtz free energy framework, providing a geometric-topological interpretation of black hole thermodynamics via the construction and classification of critical points using winding numbers.

Generalized Metric and Thermodynamic Quantities

The generalized Bardeen metric is parametrized by mass $M$, length scale $a$, and two positive real parameters $\alpha$ and $\beta$, yielding a line element that includes the Bardeen ($\alpha=3,\ \beta=2$), Hayward ($\alpha=3,\ \beta=3$), and Simpson–Visser ($\alpha=1,\ \beta=2$) metrics as special cases. For $r\gg a$, it asymptotically reduces to Schwarzschild. The regularization parameter $a$ ensures the curvature (Kretschmann scalar) remains finite everywhere, eliminating singularities by adopting a de Sitter core structure for small $r$.

Thermodynamic quantities are computed as follows:

• The Hawking temperature $a$0 is parameter-dependent, vanishing at the extremal radius and recovering the Schwarzschild (for large $a$1) and de Sitter (for small $a$2, $a$3) limits.
• The heat capacity $a$4 is explicitly calculated and exhibits a single critical point, corresponding to the maximum Hawking temperature and indicating a phase transition.
• Entropy $a$5 is derived via an integral with hypergeometric functions, and regular black holes may break standard area laws.

Notably, the black hole remnant scenario is realized: at certain parameter choices, the Hawking temperature vanishes while the remnant mass is finite, implying non-evaporating configurations potentially relevant for dark matter.

Topological Thermodynamics Framework

Adopting the topological thermodynamic formalism, the paper defines a vector field from the off-shell Helmholtz free energy. The analysis of zeros of this vector field (where both components vanish) reveals topological defects whose winding numbers classify the thermodynamic branches:

• Winding number $a$6 corresponds to a locally stable phase (positive heat capacity).
• Winding number $a$7 corresponds to a locally unstable phase (negative heat capacity).
• The total topological charge in regular black holes is zero, as branches of opposing stability coexist.

The critical value $a$8 (dependent on $a$9 and $\alpha$0) is analytically derived from quadratic polynomials, delineating the phase transition point where the heat capacity diverges and changes sign. This is mathematically formalized via the sign of the winding number and its correspondence with the sign of the heat capacity.

Case Analyses

• Bardeen ($\alpha$1): Critical value $\alpha$2. The system exhibits one stable and one unstable branch separated by a divergent heat capacity.
• Hayward ($\alpha$3): Critical value $\alpha$4, same qualitative structure as Bardeen.
• Simpson-Visser ($\alpha$5): Critical value $\alpha$6, again showing two branches.
• Schwarzschild (asymptotic): No finite critical point; only an unstable branch with always negative heat capacity.

These regular solutions display two topological defects (opposite winding numbers), while the Schwarzschild case shows only one (winding number $\alpha$7).

Numerical Results and Critical Values

Critical values are tabulated for $\alpha$8 and $\alpha$9 for each regular spacetime, with Simpson-Visser exhibiting the highest maximum temperature. The position of the critical point is a direct function of the regularization parameters, altering thermodynamic stability domains. Strong numerical claims are supported by explicit calculation of critical radii and temperatures for each case.

The vector field analysis visually confirms the correspondence between winding number and heat capacity sign, further validating the topological classification.

Practical and Theoretical Implications

The unified approach clarifies how regularization parameters control thermodynamic stability and the presence of black hole remnants, with ramifications for quantum gravity and black hole evaporation scenarios. The topological method enables direct identification of phase transitions from geometric objects (defects), offering new analytical tools for black hole thermodynamics. The methodology’s implications extend to the stability analysis of generic regular black holes, suggesting a systematic avenue for comparing different regularization mechanisms.

Potential future directions include generalization to charged, rotating, or higher-dimensional black holes and exploration of how additional thermodynamic variables restructure the topological phase space. The framework’s ability to distinguish stable relics—potential dark matter candidates—from unstable evaporating solutions is of particular relevance in cosmological contexts.

Conclusion

The paper establishes a comprehensive topological thermodynamic analysis for the generalized Bardeen metric, situating it as a unifying case for multiple regular black hole solutions. The explicit link between winding numbers and thermodynamic stability provides a robust geometric interpretation of black hole phase structure. Numerical precision in critical values substantiates claims regarding thermodynamic transitions. The findings underscore the importance of the regularization parameters in controlling phase domains and stability, with future developments poised to extend the topological methodology to broader classes of black hole spacetimes and to refine the understanding of black hole thermodynamics at the intersection with quantum gravity and astrophysics.

(2605.22686)