Phase transition for a black hole with matter fields and the relation with the Lyapunov exponent

Abstract: We construct black hole geometries coexisting with anisotropic matter in (anti)-de Sitter spacetime. We specifically focus on the black hole phase transitions that occur in anti-de Sitter spacetime and analyze the effects of the incorporated matter fields. Its local stability is examined by evaluating the heat capacity, while global stability is investigated in greater detail through phase transition analysis. The black hole system coexisting with the matter field allows for a phase transition from a small black hole to a large black hole. This demonstrates that the constructed geometry with the matter field would resemble that of the Reissner-Nordström black hole. We examine null geodesics, particularly unstable homoclinic orbits, which allow us to obtain Lyapunov exponents that characterize sensitivity to initial conditions. Finally, we analyze the relationship between the different black hole phases and the behavior of these Lyapunov exponents.

• The paper develops novel black hole solutions in AdS spacetime featuring anisotropic matter fields and identifies a first-order small-to-large black hole phase transition.
• The analysis employs effective potential and free energy techniques to correlate thermodynamic stability with Lyapunov exponents from unstable photon orbits.
• Varying matter parameters tunes the phase structure, offering insights into gravitational thermodynamics and chaotic dynamics in holographic systems.

Phase Transitions and Lyapunov Exponents in Black Holes with Anisotropic Matter Fields

Introduction

This paper develops and analyzes black hole solutions in (anti)-de Sitter (AdS) spacetime incorporating an anisotropic matter field. The focus is on the AdS case and the interplay between thermodynamic phase transitions and dynamical properties as quantified by the Lyapunov exponent of unstable photon orbits. The study investigates local and global thermodynamic stability, conditions for phase transitions analogous to the van der Waals system, and the correlation between extrema in free energy and characteristic Lyapunov exponent behavior. It situates the results within the broader context of black hole thermodynamics, chaos, and the role of matter fields beyond the vacuum or Maxwell sector.

Construction of Black Holes with Anisotropic Matter

The paper generalizes previously known black hole solutions with anisotropic matter [Kim:2025sdj] by the inclusion of a cosmological constant, obtaining an explicit metric function for spherically, planar, and hyperbolic horizon topologies ($k = 1, 0, -1$). The matter sector is modeled effectively as a fluid described by a parameter $v_2$ (strength) with a characteristic decay scale $v_c$. For $v_2 = 0$ the solution reduces to Schwarzschild-AdS (SAdS), and for $v_c \rightarrow 0$ to Reissner-Nordström-AdS (RNAdS), establishing this family as interpolating between the vacuum and charged cases.

The horizon structure is governed by the effective potential, and the presence of the matter field allows for two black hole horizons, merging in an extremal case. Parameter space analysis demonstrates transitions between black-hole and naked-singularity configurations.

Thermodynamics and Phase Structure

Temperature, Heat Capacity, and Criticality

The Hawking temperature, derived from regular surface gravity arguments, depends sensitively on the parameters $v_2$ and $v_c$. Series expansions isolate three thermodynamic regimes: matter-dominated (small horizon radius), mass-dominated (intermediate scale), and AdS-dominated (large horizon radius). The small $r_H$ regime, dominated by the anisotropic matter, is locally stable ($C > 0$), in contrast to SAdS. The intermediate regime is locally unstable, while large AdS black holes regain stability. Importantly, tuning $v_2, v_c$ leads to merging of extrema, characteristic of critical phenomena, and a critical line in parameter space separates monotonic from multi-valued temperature regions.

Free Energy and Phase Transitions

The Helmholtz free energy $v_2$0 is used to diagnose global stability and phase structure, with the canonical ensemble held fixed ($v_2$1, $v_2$2 fixed, not the chemical potential). The analysis reveals a first-order phase transition between small and large black holes, closely analogous to the van der Waals liquid-gas system. The critical point marks the end of the coexistence line and is accompanied by the disappearance of the local instability (negative heat capacity) regime. The matter sector, through $v_2$3 and $v_2$4, allows detailed control over the onset and suppression of the phase transition.

Smarr Formula and Generalized First Law

A Smarr-type mass formula is derived using the explicit metric function and horizon condition, incorporating new "charge-like" contributions stemming from the matter field and its scale $v_2$5. The corresponding generalized first law features potentials conjugate to these new charges, along with pressure-volume terms. While the explicit first law structure is not entirely fixed (due to the phenomenological matter sector), the consistency with known charged and vacuum limits is established.

Lyapunov Exponent and Dynamical Stability

The analysis of geodesic motion focuses on the Lyapunov exponent $v_2$6 associated with unstable circular photon orbits. Despite the separability and integrability of the geodesic equations, the Lyapunov exponent provides a measure of exponential instability near the photon sphere. Using the effective potential approach, analytic expressions for $v_2$7 in terms of the black hole parameters are obtained.

Remarkably, a strong correlation is found between the phase structure inferred from thermodynamic quantities and the behavior of $v_2$8: the Lyapunov exponent exhibits distinct branches corresponding to small, intermediate, and large black hole phases. At criticality, branch structure disappears concomitant with single-valued temperature and free energy curves. Importantly, in regimes of phase coexistence, the large black hole phase always has a lower Lyapunov exponent than the small black hole phase at the same temperature, supporting the proposal that $v_2$9 can serve as a robust dynamical order parameter for black hole phase transitions, complementing traditional thermodynamic ones.

Implications and Future Directions

The results solidify the deep connection between black hole thermodynamics and dynamical instability indicators. The possibility of tuning the onset and order of phase transitions by varying matter field parameters $v_c$0 and $v_c$1 establishes this class of solutions as a versatile arena for studying critical phenomena, order parameters, and the interplay between gravitational, matter, and boundary effects in AdS. Furthermore, the efficacy of the Lyapunov exponent as a probe of phase structure suggests avenues for its use as an order parameter, both theoretically (in the context of holography, quantum chaos, and critical phenomena) and observationally (in the analysis of gravitational waveforms or black hole shadow properties).

The control afforded by the matter sector invites extension to higher-curvature corrections, rotation, and inclusion of further field content (e.g., non-linear electrodynamics, scalar hair). The connection to topological order parameters and explorations of phase transitions in grand canonical ensembles (Gibbs free energy analysis) are identified as valuable future research avenues.

Conclusion

This work constructs and analyzes black holes with anisotropic matter fields in AdS spacetime, identifying a tunable phase structure and a robust correlation between thermodynamic and dynamical stability properties. The black hole system exhibits a first-order small-to-large black hole phase transition, controlled by matter parameters, with corresponding behavior in the Lyapunov exponent of null geodesic instability. These insights reinforce the unity of gravitational thermodynamics and chaotic dynamics, and the detailed understanding of these interrelations serves as a platform for broad investigations into black hole microphysics and holography.