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

Central to this cutting-edge research is the examination of rotating regular black holes, a theoretical construct that deviates from the singularity-ridden classical black hole models. Unlike their singular counterparts, regular black holes possess a smooth structure at their core, avoiding the infinite densities and curvatures that plague traditional descriptions. This crucial distinction allows for a more nuanced understanding of black hole physics, particularly concerning phenomena close to their event horizons. The rotation of these black holes adds another layer of complexity, introducing frame-dragging effects and altering the dynamics of particles and radiation in their vicinity. The interplay between the regular nature of the core and the rotational dynamics presents a fertile ground for exploring novel gravitational phenomena that might not be observable in simpler black hole scenarios, potentially leading to new observational signatures.

The study meticulously investigates scalar quasinormal modes, which are essentially the characteristic frequencies at which a black hole oscillates when subjected to external disturbances. Imagine dropping a pebble into a pond; ripples spread outwards, and the pond’s surface oscillates at specific frequencies. Similarly, when matter or radiation interacts with a black hole, it induces these quasinormal modes, which then decay over time as the black hole settles back to equilibrium. The frequencies and damping rates of these modes are intrinsically linked to the black hole’s properties, such as its mass and spin. By calculating these scalar quasinormal modes for rotating regular black holes, the researchers are able to characterize their dynamical response to perturbations, providing valuable insights into their fundamental nature.

Moreover, the research introduces the concept of Lyapunov exponents into the study of black holes, a measure of the rate at which nearby trajectories in a dynamical system diverge. In the context of black holes, a positive Lyapunov exponent signifies chaotic behavior, indicating that even infinitesimally small differences in initial conditions can lead to vastly different outcomes over time. This has profound implications for understanding the predictability and information scrambling properties of black holes. The presence and magnitude of Lyapunov exponents for particles orbiting or falling into rotating regular black holes can reveal the extent of chaotic mixing within their gravitational influence, potentially shedding light on the black hole information paradox.

A significant aspect of the investigation involves the analysis of null geodesics, which represent the paths of light rays in spacetime. The curvature of spacetime around a black hole dictates the trajectories of these null geodesics. The study examines the radii of these paths to understand how light propagates in the vicinity of rotating regular black holes. This includes exploring phenomena such as light bending and the formation of photon spheres, regions where photons can orbit the black hole. By analyzing the properties of these orbits, the researchers can infer crucial information about the geometry of spacetime around these exotic objects and how gravity distorts the paths of light.

The mathematical framework employed in this research is both sophisticated and rigorous, drawing upon advanced concepts in general relativity and differential geometry. The team has developed theoretical models that allow for the precise calculation of scalar quasinormal modes and Lyapunov exponents for a range of parameters characterizing rotating regular black holes. This involves solving complex differential equations that describe the propagation of scalar fields in the curved spacetime around these objects. The precision of these calculations is paramount in obtaining reliable results that can be compared with potential future observational data. The theoretical advancements made here are a testament to the ongoing evolution of astrophysical and cosmological modeling.

The implications of this study extend far beyond theoretical physics, potentially paving the way for new observational strategies. While directly observing the quasinormal modes of black holes is currently beyond our technological capabilities, this research provides a theoretical blueprint for what to look for. Future generations of gravitational wave detectors and advanced telescopes might be able to detect subtle imprints of these modes, offering direct evidence for the existence and properties of rotating regular black holes. Such observations would be revolutionary, providing empirical validation for these theoretical predictions and opening up a new window into the universe.

The concept of regular black holes itself has significant theoretical appeal. The resolution of singularities, points of infinite density and curvature where the laws of physics as we know them break down, is a long-standing challenge in general relativity. Regular black holes offer a potential solution by proposing an alternative structure that avoids these problematic infinities. This research, by exploring the dynamics of rotating versions of these regular black holes, further solidifies their importance as theoretical laboratories for probing the limits of our current understanding of gravity and quantum mechanics.

The behavior of particles close to the event horizon of a black hole is a deeply fascinating area of study. The intense gravitational fields can lead to extreme relativistic effects, and the presence of rotation further complicates these dynamics. By analyzing Lyapunov exponents, the researchers can determine whether the motion of particles in these regions is predictable or exhibits chaotic characteristics. This is crucial for understanding how information is processed and potentially lost within black holes, a key aspect of the long-standing black hole information paradox, which questions whether information that falls into a black hole is truly destroyed or somehow preserved.

The study’s focus on null geodesics is also critical for understanding how black holes interact with light. The bending of light around massive objects, as predicted by Einstein’s theory, is a well-established phenomenon. However, around black holes, this bending can be so extreme that light can be trapped in orbits. The analysis of null geodesics helps to delineate the regions where such phenomena occur and how they are affected by the black hole’s rotation and its regular internal structure. This has direct relevance to observations of gravitational lensing and the appearance of objects around black holes, such as accretion disks.

Understanding the stability of black hole solutions is a cornerstone of theoretical astrophysics. Quasinormal modes provide a powerful tool for assessing this stability. If these modes exhibit rapid damping, it suggests that the black hole is stable and will return to its equilibrium state after a disturbance. Conversely, modes that grow over time would indicate an unstable configuration. The research presented here provides crucial insights into the stability landscape of rotating regular black holes, confirming their robustness as theoretical entities and bolstering confidence in their potential importance.

The integration of scalar quasinormal modes and Lyapunov exponents represents a significant analytical advancement. By considering both the oscillatory behavior and the chaotic dynamics, the researchers gain a more comprehensive picture of the complex interactions occurring in the vicinity of rotating regular black holes. This multi-faceted approach allows for a deeper probing of the physical processes at play, moving beyond single-aspect analyses to a more holistic understanding of these extreme environments. It is this kind of integrated approach that often yields the most profound discoveries in physics.

The theoretical predictions stemming from this research hold the promise of guiding future observational efforts. As our astronomical instruments become more sensitive and sophisticated, the ability to test these intricate theoretical models will increase. The specific signatures predicted for scalar quasinormal modes and the chaotic behavior associated with Lyapunov exponents could become the fingerprints that allow us to identify and study rotating regular black holes, if they exist, in the distant cosmos. This study, therefore, serves as a vital bridge between theoretical exploration and potential empirical verification.

In conclusion, this research on rotating regular black holes represents a significant leap forward in our quest to understand the universe. By employing sophisticated theoretical tools like scalar quasinormal modes and Lyapunov exponents, and by analyzing the paths of light, scientists are unraveling some of the deepest mysteries of gravity and spacetime. The findings not only deepen our theoretical understanding but also offer tantalizing possibilities for future observational discoveries, potentially revolutionizing our cosmology and our place within it. The universe, it seems, continues to whisper its secrets, and with every new discovery like this, we learn to listen a little better.

Subject of Research: The dynamical behavior, stability, and spacetime properties of rotating regular black holes.

Article Title: Scalar quasinormal modes, Lyapunov exponents and radii of null geodesics of rotating regular black holes.

Article References: Peng, Y., Huang, JH. Scalar quasinormal modes, Lyapunov exponents and radii of null geodesics of rotating regular black holes.
Eur. Phys. J. C 85, 1312 (2025). https://doi.org/10.1140/epjc/s10052-025-14999-w

Image Credits: AI Generated

DOI: https://doi.org/10.1140/epjc/s10052-025-14999-w

Keywords: Black Holes, General Relativity, Quasinormal Modes, Lyapunov Exponents, Null Geodesics, Regular Black Holes, Gravitational Physics, Theoretical Astrophysics.