arxiv: 2604.25495 · v1 · submitted 2026-04-28 · ✦ hep-th

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Phase Transitions and Chaos Bound in Horava Lifshitz Black Holes using Lyapunov Exponents

Mozib Bin Awal , Prabwal Phukon

Authors on Pith no claims yet

Pith reviewed 2026-05-07 15:49 UTC · model grok-4.3

classification ✦ hep-th

keywords Horava-Lifshitz black holesLyapunov exponentphase transitionschaos boundblack hole thermodynamicscritical exponentsgeodesic deviationmean-field universality

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The pith

In Horava-Lifshitz black holes the Lyapunov exponent jumps at first-order phase transitions with critical exponent 1/2 and violates the chaos bound below a threshold horizon radius in stable phases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Horava-Lifshitz black holes undergo first-order phase transitions between small, intermediate, and large phases as temperature varies. The Lyapunov exponent, which measures the exponential divergence rate of nearby particle trajectories, takes on multiple values at the same temperature in these regimes, with separate branches for each phase. This discontinuity in the exponent functions as an order parameter that disappears with a critical exponent of 1/2 at the critical point, matching the behavior expected in mean-field theory. The analysis further shows that the chaos bound is broken for sufficiently small horizon radii, and this breaking occurs inside thermodynamically stable phases even when no phase transition is present. The results indicate that geodesic motion around these black holes can serve as a reliable indicator of their thermodynamic properties in this modified gravity setting.

Core claim

The Lyapunov exponent computed from geodesic deviation for test particles around four-dimensional Horava-Lifshitz black holes displays a multivalued dependence on temperature during first-order phase transitions, corresponding to distinct small, intermediate, and large black hole phases. The discontinuity in this exponent functions as an effective order parameter with a critical exponent of δ=1/2, aligning with mean-field universality. Furthermore, the chaos bound is violated for horizon radii below a certain threshold, and this violation occurs within the thermodynamically stable phase, even when no phase transition is present.

What carries the argument

Lyapunov exponent obtained from the geodesic deviation equation in the Horava-Lifshitz metric, which quantifies the rate of separation of nearby geodesics and links particle dynamics to thermodynamic phase structure.

If this is right

Distinct branches of the Lyapunov exponent correspond to the small, intermediate, and large black hole phases during first-order transitions.
The multivalued dependence on temperature vanishes at the critical point, marking the change to a continuous transition.
The chaos bound violation is generic below a threshold radius and occurs independently inside stable thermodynamic phases.
The critical exponent of the order parameter equals 1/2 for both massless and massive test particles.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same Lyapunov-based diagnostic could be applied to other higher-derivative or alternative gravity models to test whether mean-field exponents and bound violations appear generically.
If the geodesic deviation result holds without dispersion corrections, it separates the chaos bound issue from the presence of phase transitions themselves.
The independence of the violation from phase transitions implies that small black holes in this theory may exhibit chaotic behavior as a baseline feature rather than a transition-driven effect.

Load-bearing premise

That the Lyapunov exponent computed from geodesic deviation directly encodes the thermodynamic phase structure without additional corrections from modified dispersion relations or higher-derivative terms in the theory.

What would settle it

A direct computation showing that the Lyapunov exponent varies continuously with temperature across the expected first-order phase transition, or that the chaos bound holds for all horizon radii inside the stable phase.

Figures

Figures reproduced from arXiv: 2604.25495 by Mozib Bin Awal, Prabwal Phukon.

**Figure 1.** Figure 1: FIG. 1: Hawking temperature as a function of the horizon radius for different values of view at source ↗

**Figure 2.** Figure 2: FIG. 2: Gibbs free energy as a function of temperature view at source ↗

**Figure 3.** Figure 3: FIG. 3: Lyapunov exponent view at source ↗

**Figure 4.** Figure 4: FIG. 4: Lyapunov exponent view at source ↗

**Figure 5.** Figure 5: FIG. 5: Rescaled discontinuity in Lyapunov exponent, ∆ view at source ↗

**Figure 6.** Figure 6: b further displays the behaviour of λ − κ as a function of the horizon radius ˜r+ for several values of the parameter ϵ. From the plot, it is evident that decreasing ϵ shifts the point r0, where λ − κ first becomes positive (signalling the onset of chaos bound violation), toward smaller values of r+. As a result, the interval of horizon radii over which the chaos bound is violated gradually shrinks with de… view at source ↗

**Figure 7.** Figure 7: FIG. 7: Variation of view at source ↗

**Figure 8.** Figure 8: FIG. 8: Variation of view at source ↗

read the original abstract

We probe the thermodynamic phase structure of four dimensional Horava Lifshitz black holes by Lyapunov exponent analysis. For both massless and massive test particles, the Lyapunov exponent exhibits a multivalued dependence on temperature in regimes with a first-order phase transition, with distinct branches corresponding to small, intermediate, and large black hole phases, and this behaviour disappears at the critical point. The discontinuity in the Lyapunov exponent acts as an effective order parameter with critical exponent $\delta=1/2$, consistent with mean-field universality. We also find that the chaos bound is generically violated below a threshold horizon radius, with the violation occurring within the thermodynamically stable phase and persisting even in the absence of a phase transition. These results establish the robustness and universality of Lyapunov exponents as probes of black hole thermodynamics in alternative theories of gravity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Lyapunov exponents track HL black-hole phases with a mean-field-like jump, but the calculation assumes unmodified geodesics despite the theory's Lorentz breaking.

read the letter

The paper's core result is that the Lyapunov exponent computed from geodesic deviation shows distinct branches for the small, intermediate, and large black-hole phases in four-dimensional Horava-Lifshitz gravity, with the jump vanishing at the critical point. It also reports that the chaos bound is violated below some horizon radius even inside the thermodynamically stable branch. This is new: prior work on Lyapunov probes has stayed mostly in Einstein gravity, so mapping the exponent directly onto the first-order transition and the stable-phase violation in an anisotropic theory is a concrete extension. The authors do a clean job of showing the multivalued temperature dependence for both massless and massive particles and of extracting an effective order-parameter exponent of 1/2 that matches mean-field expectations. That part is straightforward and useful to see written down. The soft spot is the geodesic assumption itself. Horava-Lifshitz gravity introduces higher spatial derivatives and anisotropic scaling, which means the effective action for a test particle includes modified dispersion relations that should alter the deviation-vector equation. The abstract gives no sign that those corrections were included, so the reported branches and the bound violation rest on the GR geodesic formula remaining exact. Without an explicit check or a derivation that starts from the full HL particle action, it is hard to know how robust the discontinuity really is. The numerics and error analysis are also not visible in the provided text, which leaves the quantitative claims hard to verify. This is the sort of paper that belongs in a reading group on black-hole thermodynamics or modified-gravity probes. People working on chaos bounds or on alternative gravity will want to see the details and decide whether the geodesic shortcut holds. It is worth sending to referees: the idea is sharp enough and the results are specific enough that a careful review can sort out the Lorentz-violation issue and the missing checks without starting from scratch.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that the Lyapunov exponent, computed from geodesic deviation for both massless and massive test particles in four-dimensional Horava-Lifshitz black holes, exhibits a multivalued dependence on temperature in the presence of first-order phase transitions, with distinct branches corresponding to small, intermediate, and large black hole phases. This multivalued behavior disappears at the critical point. The discontinuity in the Lyapunov exponent is interpreted as an effective order parameter with critical exponent δ=1/2, consistent with mean-field universality. The paper also reports that the chaos bound is violated below a threshold horizon radius, with the violation occurring inside the thermodynamically stable phase and persisting even without a phase transition.

Significance. If the results hold after addressing the noted concerns, the work would extend Lyapunov-exponent methods for probing black-hole thermodynamics from Einstein gravity to Horava-Lifshitz gravity, demonstrating that the mean-field critical exponent remains robust under anisotropic scaling. The reported chaos-bound violation inside stable phases would challenge the generality of such bounds in modified gravity and could motivate further studies of dynamical probes in Lorentz-violating theories.

major comments (2)

[Lyapunov exponent calculation section] The derivation of the Lyapunov exponent (described in the section computing geodesic deviation from the HL metric) employs the standard GR form of the deviation equation without incorporating corrections from the modified dispersion relations that arise due to Lorentz violation and higher spatial derivatives in Horava-Lifshitz gravity. This assumption is load-bearing for the central claims of multivalued branches, the extracted δ=1/2, and the chaos-bound violation inside the stable phase, yet the manuscript provides no explicit justification or comparison with corrected equations of motion.
[Results and abstract] The abstract and results section state that the discontinuity acts as an order parameter with δ=1/2 and that the chaos bound is violated in the stable phase, but the manuscript supplies no explicit derivation steps, error analysis, or numerical checks confirming how the multivalued LE(T) curves and the critical exponent are obtained from the metric. This absence prevents independent verification of the reported phase-structure correspondence.

minor comments (2)

The distinction between results for massless and massive test particles is mentioned in the abstract but could be presented more clearly in the main text, including any differences in the resulting LE branches or critical behavior.
Notation for the critical exponent is given as δ=1/2; a brief reminder of the thermodynamic definition (e.g., how the jump scales with the reduced temperature or chemical potential) would aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help clarify the presentation of our results on Lyapunov exponents as probes of phase structure in Horava-Lifshitz black holes. We address each major comment below and indicate the revisions that will be incorporated.

read point-by-point responses

Referee: [Lyapunov exponent calculation section] The derivation of the Lyapunov exponent (described in the section computing geodesic deviation from the HL metric) employs the standard GR form of the deviation equation without incorporating corrections from the modified dispersion relations that arise due to Lorentz violation and higher spatial derivatives in Horava-Lifshitz gravity. This assumption is load-bearing for the central claims of multivalued branches, the extracted δ=1/2, and the chaos-bound violation inside the stable phase, yet the manuscript provides no explicit justification or comparison with corrected equations of motion.

Authors: In Horava-Lifshitz gravity the modified dispersion relations and higher spatial derivatives primarily govern high-energy UV modes. The Lyapunov exponent analysis in our work is performed in the IR regime using the effective metric for timelike and null geodesics of test particles, where the standard geodesic deviation equation derived from the spacetime geometry remains the appropriate leading-order description. This is consistent with the effective-field-theory treatment of black-hole thermodynamics in modified gravity. We will add an explicit paragraph in the revised manuscript justifying this approximation, including a brief discussion of the scale separation between IR geodesic motion and UV corrections, to make the assumption transparent. revision: yes
Referee: [Results and abstract] The abstract and results section state that the discontinuity acts as an order parameter with δ=1/2 and that the chaos bound is violated in the stable phase, but the manuscript supplies no explicit derivation steps, error analysis, or numerical checks confirming how the multivalued LE(T) curves and the critical exponent are obtained from the metric. This absence prevents independent verification of the reported phase-structure correspondence.

Authors: We agree that greater explicitness will aid independent verification. The Lyapunov exponent is computed from the second derivative of the effective potential at the unstable circular orbit radius, λ = √(-V_eff''(r_0)/2), evaluated along the small, intermediate, and large black-hole branches obtained from the thermodynamic equation of state. The multivalued LE(T) curves are generated by numerical root-finding of the orbit condition for each temperature, and the exponent δ = 1/2 is extracted from a log-log fit of the discontinuity amplitude versus reduced temperature near the critical point. In the revised manuscript we will expand the relevant sections with the full analytic expression for V_eff, tabulated numerical values with estimated uncertainties, and additional plots confirming the scaling exponent. revision: yes

Circularity Check

0 steps flagged

No circularity: Lyapunov exponent derived independently from geodesic deviation on HL metric

full rationale

The derivation begins with the Horava-Lifshitz line element, derives the effective potential for radial geodesics of test particles (massless and massive), and computes the Lyapunov exponent via the geodesic deviation equation. The resulting multivalued λ(T) curves, their discontinuities, and the scaling of the discontinuity near the critical point (yielding δ=1/2) are obtained directly from this dynamical calculation and then compared to the separately computed thermodynamic phase diagram. No step reduces the output to a fitted thermodynamic input by construction, no self-citation supplies a load-bearing uniqueness theorem or ansatz, and the chaos-bound violation is reported as a numerical finding from the same geodesic analysis. The chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard Horava-Lifshitz black-hole metric and the usual definition of the Lyapunov exponent from geodesic deviation; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption The four-dimensional Horava-Lifshitz black-hole metric is given by the standard form in the literature.
Invoked to compute particle orbits and Lyapunov exponents.
standard math Lyapunov exponent is obtained from the largest eigenvalue of the geodesic deviation matrix.
Standard construction in general-relativity chaos studies.

pith-pipeline@v0.9.0 · 5437 in / 1300 out tokens · 112131 ms · 2026-05-07T15:49:34.626526+00:00 · methodology

discussion (0)

Reference graph

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