Hořava-Lifshitz gravity is not a minor variation on Einstein’s framework. Proposed originally as a candidate for quantum gravity, it abandons full Lorentz invariance—the deep equivalence of space and time—at short distances while restoring it at large scales. In doing so, it introduces a preferred cosmic time foliation and opens the door to physics that would be impossible in general relativity. Black holes in this theory possess a modified metric and correspondingly modified thermodynamics, raising a natural question that has occupied theorists for over a decade: do the famous phase transitions of anti-de Sitter black holes, first catalogued by Hawking and Page in 1983 and later recast in the language of van der Waals chemistry, survive in this Lorentz-violating setting? And if they do, can they be detected by something other than standard thermodynamic quantities like heat capacity?

The answer, according to Mozib Bin Awal and Prabwal Phukon of Dibrugarh University in Assam, India, is a resounding yes, and their instrument of choice is the Lyapunov exponent. Conceptually, the idea is elegant. When a particle orbits a black hole on an unstable circular trajectory—a so-called light ring for photons, or its massive-particle analogue—it sits balanced on a knife’s edge. The slightest perturbation sends it spiraling either into the horizon or off to infinity. The Lyapunov exponent λ measures the exponential rate of this divergence: large values mean violent, rapidly amplifying instability; small values mean gentler departure. For particles skimming the horizon, Hashimoto and Tanahashi showed in 2017 that this exponent is universal, scaling with the horizon temperature as λ = 2πT. That universal ratio, when it cannot exceed one, becomes the celebrated Maldacena-Shenker-Stanford chaos bound, a conjectured ceiling on how fast information can scramble in any quantum system with a gravitational dual.

The Dibrugarh team computed these exponents for both massless and massive test particles moving in the equatorial plane of the four-dimensional Hořava-Lifshitz black hole, tracking how λ varies as a function of the black hole’s temperature across a range of theory parameters. What emerged was striking. In parameter regimes where the black hole is known, from thermodynamic analysis, to undergo a first-order phase transition—analogous to the liquid-gas transition of a van der Waals fluid—the Lyapunov exponent does not trace a single smooth curve against temperature. Instead, it becomes multivalued: at one and the same temperature, distinct branches of the exponent coexist, corresponding to the small, intermediate, and large black hole phases that the standard thermodynamic treatment identifies. The chaos quantity, in other words, remembers which phase it belongs to.

This multivaluedness is not a numerical artifact. The authors show that it is a direct geometric consequence of the black hole’s phase structure. Where the free energy landscape supports several competing extrema—several locally stable black hole configurations at the same temperature—the unstable circular orbits associated with each configuration yield distinct exponents. The number of branches and their arrangement encode the small-large coexistence region, the spinodal curves where metastable phases lose stability, and the characteristic swallowtail structure familiar from the thermodynamics of first-order transitions. Then, as the system parameters approach the critical point—the unique point where the first-order line terminates and the distinction between small and large black holes dissolves, just as liquid and gas merge at the critical point of water—the multivalued behavior smoothly disappears. At criticality, the branches merge into a single continuous curve, mirroring the mean-field picture in which the order parameter vanishes exactly at the critical temperature.

Perhaps the most quantitative result of the study concerns what happens just below that critical point. The team demonstrates that the discontinuity in the Lyapunov exponent—the jump between the branches of small and large black hole solutions—behaves as an effective order parameter for the transition. Its scaling with the reduced temperature follows a critical exponent of exactly δ = 1/2, the hallmark of mean-field universality class shared by van der Waals fluids, superconductors described by Landau theory, and charged AdS black holes in Einstein’s gravity. This is a remarkable statement about universality: a quantity defined entirely through the instability of particle orbits, calculated in a Lorentz-violating theory of gravity, reproduces the same critical scaling as everyday condensed matter systems. Phase transition physics, it seems, cares little for the fine details of the underlying gravitational dynamics and everything for the topology of the thermodynamic potential.

The second half of the paper turns to the chaos bound itself, and here the news is more provocative. Below a threshold horizon radius, the Hořava-Lifshitz black hole generically violates the bound: the ratio of the Lyapunov exponent to the temperature exceeds its conjectured maximum, implying faster-than-allowed scrambling if the usual holographic interpretation holds. Crucially, the authors find that this violation is not tied to thermodynamic instability. It occurs entirely within the region where the black hole is thermodynamically stable—where the heat capacity is positive and the phase is locally safe—and it persists even in parameter regimes where no phase transition takes place at all. The bound violation is thus decoupled from criticality, a structural feature of small Hořava-Lifshitz black holes rather than a symptom of phase change.

This decoupling carries real interpretive weight. In earlier studies of charged and rotating black holes in Einstein gravity, chaos bound violations were often linked to charged probes, electromagnetic coupling, or specific extremal limits. Here, the violation emerges from the modified near-horizon geometry that Hořava-Lifshitz gravity enforces, and it raises questions about whether the chaos bound, which was formulated within holographic frameworks assuming Lorentz invariance, should be expected to hold universally in theories with a preferred time direction. The authors’ results suggest that Lorentz violation provides a natural and persistent mechanism for super-fast scrambling, at least at the level of classical probe dynamics, and that any future resolution must account for theories beyond general relativity rather than treating the bound as sacred.

Methodologically, the study adds Hořava-Lifshitz black holes to a rapidly growing list of systems where Lyapunov exponents have proven to be faithful thermodynamic probes. In recent years, researchers have used the technique to diagnose van der Waals-like transitions in charged AdS black holes, Born-Infeld black holes, Gauss-Bonnet gravity, Hayward regular black holes, massive gravity, and quintessence-surrounded spacetimes. The consistent message across these analyses is that the instability of orbits and the stability of phases are two faces of the same underlying potential. What the new work establishes is the robustness of this correspondence even when Lorentz symmetry—the assumption underlying nearly all prior analyses—is abandoned. The Lyapunov-based toolkit, in other words, is not an accident of Einstein’s theory but a genuinely universal diagnostic of black hole thermodynamics.

The broader implications reach toward quantum gravity itself. Hořava-Lifshitz theory was designed to be power-counting renormalizable, offering a window into physics at energies where quantum effects should dominate, and its black holes therefore serve as laboratories for testing how quantum-gravity candidates behave thermodynamically. If chaos measures can serve as order parameters in these settings, they may eventually help discriminate between competing approaches to quantum gravity—flagging which theories support standard mean-field criticality and which allow the chaos bound to fail. For now, the Dibrugarh results stand as a vivid demonstration that the line between order and chaos at the edge of a black hole is drawn precisely where thermodynamics says it should be, even when Einstein’s symmetry principles are no longer in force. The Universe, it appears, encodes its phase diagrams in the mathematics of instability, and researchers are learning to read them one diverging trajectory at a time.

Subject of Research: Thermodynamic phase transitions and chaos bound violations in four-dimensional Hořava-Lifshitz black holes probed via Lyapunov exponents

Article Title: Phase transitions and chaos bound in Horava Lifshitz black holes using Lyapunov exponents

Article References: Awal, M. B., & Phukon, P. (2026). Phase transitions and chaos bound in Horava Lifshitz black holes using Lyapunov exponents. General Relativity and Gravitation, 58(9), Article 107. https://doi.org/10.1007/s10714-026-03611-5

Image Credits: AI Generated

DOI: 10.1007/s10714-026-03611-5

Keywords: black holes, Hořava-Lifshitz gravity, Lyapunov exponent, phase transitions, chaos bound, black hole thermodynamics, critical exponent, quantum gravity, unstable circular orbits, mean-field universality, anti-de Sitter space, Lorentz violation