Rotating black hole thermodynamics shaped by geometry and topology in Lorentz-violating gravity

In a result that is quietly reshaping how physicists think about the deep connection between gravity, heat, and geometry, a team of researchers has mapped the thermodynamic landscape of rotating black holes living in a universe where one of physics’ most sacred symmetries is allowed to break. The study, published in the journal General Relativity and Gravitation, dissects the thermal stability, phase transitions, and hidden topology of spinning black holes in Lorentz-violating gravity, a class of modified theories in which the speed and behavior of light need not be the same in every direction. The work, led by Aqsa Mehmood and M. Umair Shahzad of the University of Okara in Pakistan, together with A. Alkaoud and A. Eid of Imam Mohammad Ibn Saud Islamic University in Riyadh, suggests that the violent events occurring near a black hole’s event horizon can be understood as defects in an abstract thermodynamic space, much like vortices in a fluid or dislocations in a crystal.

Lorentz symmetry, the principle that the laws of physics are identical for all observers moving at constant velocity relative to one another, underlies both special relativity and the standard model of particle physics. Yet a growing number of theoretical frameworks, motivated by quantum gravity, loop quantum gravity, and string-inspired models such as bumblebee gravity, permit this symmetry to be violated at high energies or in strong gravitational fields. In these theories, a background field spontaneously selects a preferred direction in spacetime, subtly altering how black holes form, rotate, and radiate. Because rotating black holes are the most extreme laboratories of strong-field gravity available to theorists, understanding their thermodynamics in Lorentz-violating settings offers a potential window into physics beyond Einstein’s general relativity, and may eventually help constrain such models against observational data from gravitational-wave detectors and the Event Horizon Telescope.

The core of the new study is a careful thermodynamic audit of rotating black hole solutions in Lorentz-violating gravity. The researchers computed the Hawking temperature, the faint quantum radiation that Stephen Hawking showed all black holes must emit, as a function of the horizon radius, the characteristic size of the black hole’s boundary. They then derived the heat capacity, which measures how the black hole’s temperature responds to changes in its energy. When the heat capacity diverges or changes sign discontinuously, the system passes through a second-order phase transition, the thermodynamic equivalent of water boiling at a critical temperature. By scanning the full range of horizon radii, the team identified exactly where these transitions occur and cleanly separated the parameter space into thermodynamically stable regions, where the heat capacity is positive and the black hole can coexist peacefully with its radiation bath, and unstable regions, where the black hole will either evaporate away or grow without bound.

What elevates the analysis beyond a standard stability study is the authors’ systematic application of thermodynamic geometry. The idea, pioneered in the 1970s by Frank Weinhold and later developed by George Ruppeiner, is to treat the space of equilibrium thermodynamic states, coordinates such as temperature, entropy, and pressure, as a curved Riemannian manifold. The curvature of this manifold encodes statistical correlations among the microscopic degrees of freedom of the system: flat regions correspond to weakly interacting matter, while regions of strong curvature signal critical phenomena and phase transitions. In black hole physics, the scalar curvature of this thermodynamic geometry is expected to diverge precisely where the heat capacity vanishes or blows up, marking the boundaries between stable and unstable phases. If the geometry is constructed correctly, its singularities should act as a geometric echo of the black hole’s physical instabilities.

The team tested this expectation using not one but several competing geometric formalisms: the Ruppeiner metric rooted in thermodynamic fluctuation theory, the Weinhold energy metric, the HPEM metric introduced by Hendi, Panahiyan, Eslam Panah, and Momennia, and the geothermodynamic construction of Hernando Quevedo in two distinct formulations, labeled Case I and Case II. Each formalism builds its metric from different combinations of thermodynamic potentials, and they do not always agree. Remarkably, the study found that the scalar curvatures computed from the HPEM, Ruppeiner, and Quevedo metrics agree with each other and with the zeros of the heat capacity in excellent fashion, faithfully reproducing the full phase transition structure of the rotating Lorentz-violating black hole. The Weinhold metric, by contrast, proved less diagnostic, underscoring a long-standing puzzle in the field: why some thermodynamic geometries capture critical behavior while others fail, and what this hierarchy reveals about the microscopic origin of black hole entropy.

The second, and perhaps most striking, layer of the work ventures into topology. Building on a framework developed by Wei, Liu, and Mann in 2022, the researchers treated black hole solutions not merely as points in a parameter space but as topological defects embedded in a thermodynamic phase spanned by variables such as temperature and pressure. In this picture, the generalized free energy of the black hole defines a vector field on a two-dimensional parameter manifold, and equilibrium states correspond to the zeros of this field, points where the vector vanishes. Around each zero, the vector field winds a certain number of times, and that winding number, called the topological charge, is invariant under smooth deformations of the system. It cannot be changed by tweaking the black hole’s mass, spin, or the strength of the Lorentz-violating parameter; it can only jump through discrete events such as the creation or annihilation of a vortex-antivortex pair.

The winding number acts, in effect, as a fingerprint that classifies black hole solutions into distinct topological classes. The team computed these charges at both local and global levels and found that, for the rotating Lorentz-violating black holes they studied, the topological charge takes one of three discrete values: plus one, zero, or minus one. Physically, these values distinguish between different kinds of equilibrium points: locally stable states, unstable states, and bifurcation points where the character of the solution changes. Phase-space diagrams constructed by the authors expose these non-trivial structures directly, showing how stable branches of black holes emerge from, merge with, or annihilate against unstable ones as parameters vary. The methodology echoes topological techniques first introduced by Duan in 1984 in the study of defects in condensed matter systems, and it has recently been applied to Gauss-Bonnet gravity, Lovelock gravity, massive gravity, and Born-Infeld black holes. This study extends that program to Lorentz-violating spacetimes for the first time in the rotating sector.

The significance of the topological classification goes beyond mathematical elegance. Because the topological charge is quantized and robust, it provides a model-independent way to characterize black hole phase behavior, one that survives even when the underlying theory is modified or imperfectly known. In conventional thermodynamics, phase transitions are diagnosed by response functions like the heat capacity, which depend on the details of the theory. The topological approach, by contrast, extracts the essential skeleton of the phase structure from symmetry and continuity arguments alone. For Lorentz-violating gravity, where experimental guidance is scarce and theoretical predictions vary widely across models, such a robust diagnostic is especially valuable. The researchers suggest their findings could offer meaningful constraints on models that incorporate Lorentz symmetry violation, by identifying which thermodynamic features are generic and which are sensitive to the specific mechanism of symmetry breaking.

The timing of the work is notable. Observations of black hole shadows by the Event Horizon Telescope, analyses of rotating black holes in bumblebee gravity compared against those images, and detections of gravitational waves by LIGO and Virgo have all sharpened interest in testing whether Lorentz symmetry holds in the strong-field regime. Recent theoretical studies have shown that Lorentz violation can induce isospectrality breaking in the quasinormal mode spectra of rotating black holes, potentially observable signatures, and that exact rotating solutions exist in viable Lorentz-violating theories. The new thermodynamic and topological analysis complements these dynamical results: while quasinormal modes and shadows probe how black holes respond to perturbations and light, thermodynamic topology probes their equilibrium structure and stability, offering an independent and largely model-agnostic consistency check on any proposed modification of Einstein’s gravity.

The authors caution that their study is theoretical and involves no new observational data; the published work explicitly states that no datasets were generated or analyzed. Nevertheless, the convergence they demonstrate, in which three independent geometric formalisms, the heat capacity analysis, and the topological winding numbers all paint the same picture of stability, instability, and phase transition, lends considerable weight to the results. It suggests that the thermodynamic geometry of a black hole is not an artifact of a particular formalism but a genuine structural property of the underlying physics, one that persists even when the Lorentz symmetry of spacetime itself is allowed to fail.

The study also contributes to a rapidly growing literature connecting black hole thermodynamics to information geometry and quantum gravity. Recent years have seen thermodynamic topology applied to warped anti-de Sitter black holes through the lens of the AdS/CFT correspondence, to charged Gauss-Bonnet black holes, to black hole chemistry in massive gravity, and to regular black holes with zero-point length. Each application has refined the toolkit, and the present work extends it to one of the most theoretically active frontiers in gravitational physics. Whether Lorentz symmetry is exactly preserved in nature remains an open experimental question, but this research demonstrates that even a hypothetical violation leaves rich, quantifiable fingerprints in the thermal and topological anatomy of black holes, fingerprints that future observations may one day be able to read.