Extended thermodynamics and $P-v$ Criticality of Kalb-Ramond black hole coupled with nonlinear electrodynamics

Abstract: We present an exact black hole solution in anti-de Sitter (AdS) spacetime with a Kalb-Ramond field coupled to nonlinear electrodynamics (NLED), characterized by mass, magnetic monopole charge, and Lorentz-violating parameters. The geometry admits two horizons (inner and outer) that coalesce into a degenerate horizon at a critical monopole charge. Beyond this critical point, no black hole solutions exist. In the limit of vanishing Lorentz-violating parameters, the solution reduces to the modified Kalb-Ramond and Bardeen black holes, while suitable parameter choices reproduce the Reissner-Nordström-AdS and Schwarzschild-AdS geometries. We analyze the thermodynamics of the solution by computing the Hawking temperature, entropy, specific heat, and Gibbs free energy. The NLED source introduces nontrivial modifications: the Hawking temperature displays nonmonotonic behavior with possible local extrema, the entropy deviates from the standard area law, and the specific heat may assume negative values, signaling thermodynamic instabilities. The Gibbs free energy exhibits swallow-tail structures, indicative of first-order phase transitions. Furthermore, we derive the first law of black hole thermodynamics in the extended phase space, together with the Smarr relation, and confirm their validity for the Kalb-Ramond black holes with NLED sources. Our findings highlight the rich thermodynamic structure induced by Lorentz-violating effects and nonlinear electrodynamics in AdS black hole backgrounds.

• The paper demonstrates that coupling Kalb-Ramond fields with nonlinear electrodynamics regularizes singularities and alters black hole horizon structures.
• It employs extended phase space thermodynamics to reveal non-monotonic temperature profiles, entropy corrections, and clear indicators of first- and second-order phase transitions.
• The study establishes a black hole–fluid analogy via P–v criticality, linking Lorentz-violating effects to quantum corrections in gravity.

Extended Thermodynamics and $P$–$v$ Criticality of Kalb-Ramond Black Holes Coupled with Nonlinear Electrodynamics

Introduction and Motivation

The paper "Extended thermodynamics and $P-v$ Criticality of Kalb-Ramond black hole coupled with nonlinear electrodynamics" (2605.05281) presents an exact analysis of black holes in anti-de Sitter (AdS) backgrounds, incorporating both a Kalb-Ramond (KR) field and a nonlinear electrodynamics (NLED) sector. This combination provides a theoretical laboratory for probing the interplay between Lorentz symmetry violation, antisymmetric tensor fields from string theory, and quantum-corrected nonlinear field dynamics.

Fundamentally, the authors target two frontiers: (1) modifications to general relativity from higher-rank gauge fields that naturally emerge in string theories, and (2) regularization of spacetime singularities by NLED, following the historical Born-Infeld program. The interplay between these ingredients is analyzed through exact black hole solutions and their detailed thermodynamics, anchored in the extended phase space paradigm, with the cosmological constant interpreted as a pressure variable.

Exact Black Hole Solutions: Formal Development and Structure

The setup involves a four-dimensional AdS spacetime background, with the gravitational action coupled to a self-interacting Kalb-Ramond field (an antisymmetric rank-2 tensor), and a nonlinear electromagnetic field described by a specific Lagrangian. The general action includes mass $M$, magnetic monopole charge $q$, and Lorentz-violating couplings $(\gamma, \lambda)$ characterizing the strength and structure of deviation from Lorentz invariance due to the KR sector.

Solving the modified Einstein equations yields a metric with two essential non-standard features:

• The presence of both mass and a magnetic monopole charge in a regularized NLED configuration
• Hairy deformations governed by Lorentz-violating parameters

The metric admits inner and outer horizons, which coalesce to a degenerate horizon at a critical value of the monopole charge. Beyond this value, no event horizon exists—signaling the threshold for black hole formation in this framework. Several canonical solutions are subsumed as limiting cases, including the modified Kalb-Ramond, Bardeen, Reissner–Nordström–AdS, and Schwarzschild-AdS black holes. The energy conditions further constrain the Lorentz-violating couplings for physical admissibility of solutions.

Thermodynamic Analysis: Temperature, Entropy, and Stability

The authors conduct a comprehensive thermodynamic analysis of the black hole solutions in the extended (or "black hole chemistry") phase space, defining the mass $M$ as enthalpy. Notable outcomes include:

• Hawking Temperature: The NLED-KR coupling introduces non-monotonic temperature profiles. Depending on $(\gamma, \lambda)$, $T_+$ can develop both local minima and maxima as a function of the horizon radius, yielding multivalued regions in the $T_+(r_+)$ curve and suggesting the presence of instability windows and critical behavior.
• Entropy: The entropy deviates from the Bekenstein-Hawking area law:

$v$0

The subleading term, proportional to $v$1, arises from the NLED source. This correction mandates a lower bound on $v$2 for positive entropy, directly connecting horizon structure and nonlinearity parameters.

• Specific Heat ($v$3): Regions of negative specific heat are generic in the phase diagram, indicating local thermodynamic instability. Transitions from negative to positive specific heat mark the onset of new stable phases, and divergences in $v$4 signal second-order phase transitions associated with the underlying microphysical structure.
• Gibbs Free Energy: Calculated as $v$5, the function exhibits swallowtail behavior typical of first-order phase transitions. Depending on parameter choices, $v$6 can possess multiple extrema, with negative branches indicating global thermodynamic preference.
• First Law and Smarr Relation: Despite highly nontrivial matter content, the standard form of the first law holds when incorporating the generalized potentials for $v$7, $v$8, and $v$9. The Smarr formula is explicitly verified.

$P-v$0–$P-v$1 Criticality and Phase Transitions

A central focus is placed on mapping the analogy between black hole thermodynamics and van der Waals fluids via $P-v$2–$P-v$3 (pressure–specific volume) criticality. The equation of state is extracted from the temperature expression, leading to the identification of inflection points ($P-v$4) corresponding to critical values $P-v$5.

Key results include:

• For $P-v$6, the critical ratio $P-v$7 exactly matches the van der Waals system, confirming the universality of the black hole–fluid analogy in this limit.
• For nonzero monopole charge, the critical radius and ratio grow with $P-v$8, while critical pressure and temperature decrease monotonically—demonstrating nontrivial modulation of the phase structure (and the possibility for tuning critical points with external parameters).
• The $P-v$9 diagrams reveal classic swallowtail structures for $M$0, confirming first-order phase transitions between small and large black hole phases. These join smoothly at $M$1, signaling a critical endpoint with universal scaling.

Implications and Outlook

This work extends the catalogue of regular black hole solutions with nontrivial matter couplings, systematically analyzing their thermodynamics and phase structure within the framework of Lorentz-violating theories and nonlinear electrodynamics. There are several implications:

• Theoretical: The findings reaffirm that higher-rank gauge fields from fundamental theory (e.g., string-motivated Kalb-Ramond fields) and quantum-corrected NLED sectors alter horizon structure and thermodynamics in nonperturbative ways. The deviation from the area law entropy serves as a testable imprint for quantum gravity effects in semiclassical gravity and AdS/CFT duals.
• Critical Phenomena: The detailed $M$2–$M$3 analysis connects the rich landscape of AdS black hole phase transitions to the thermodynamics of multi-component fluids, offering further insight into universality and the microphysical interpretation of black hole entropy and temperature.
• Future Directions: This class of solutions provides a fertile testing ground for further dynamical scenarios—such as stability against perturbations, scalarization effects, holographic conductivity in AdS/CFT, and the structure of quasinormal modes. Exploring higher-dimensional analogues, rotating solutions, or detailed dual field theory interpretations constitute relevant research avenues.

Conclusion

This paper provides a rigorous derivation and analysis of black holes with Kalb-Ramond fields coupled to nonlinear electrodynamics, focusing on the emergence and modification of thermodynamic and critical phenomena. The work highlights how field-theoretic and quantum-gravitational corrections shape black hole structure, thermodynamic stability, and phase transitions, thereby contributing to ongoing efforts probing the interface of Lorentz-violating field theory, regular black hole construction, and extended gravitating thermodynamic systems.