arxiv: 2605.05281 · v1 · submitted 2026-05-06 · 🌀 gr-qc · astro-ph.HE· hep-th

Recognition: unknown

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

D. V. Singh, K. Myrzakulov, P. Paul, S. Upadhyay

Authors on Pith no claims yet

Pith reviewed 2026-05-08 15:47 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HEhep-th

keywords Kalb-Ramond black holenonlinear electrodynamicsAdS spacetimeextended phase spacephase transitionsthermodynamic instabilityLorentz violation

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The pith

An exact black hole solution in AdS space coupling a Kalb-Ramond field to nonlinear electrodynamics exhibits first-order phase transitions through swallow-tail structures in its Gibbs free energy.

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

The paper derives an exact metric for a black hole in anti-de Sitter spacetime that incorporates both a Kalb-Ramond field and nonlinear electrodynamics, depending on mass, magnetic monopole charge, and Lorentz-violating parameters. Thermodynamic analysis shows the Hawking temperature can become non-monotonic, entropy deviates from the area law, and specific heat can turn negative. The Gibbs free energy displays swallow-tail shapes that mark first-order phase transitions between small and large black holes. The first law and Smarr relation are verified to hold in the extended phase space where the cosmological constant acts as pressure. This construction recovers several known black hole solutions as special cases of the parameters.

Core claim

What carries the argument

The exact AdS black hole metric sourced by the Kalb-Ramond field plus nonlinear electrodynamics, analyzed in extended phase space where the cosmological constant is treated as thermodynamic pressure.

Load-bearing premise

That a consistent coupling between the Kalb-Ramond field and nonlinear electrodynamics exists which satisfies the Einstein equations exactly without introducing extra singularities.

What would settle it

A numerical or analytic check that the proposed metric fails to satisfy the Einstein equations with the chosen Kalb-Ramond and NLED stress-energy tensors for any choice of the Lorentz-violating parameters.

Figures

Figures reproduced from arXiv: 2605.05281 by D. V. Singh, K. Myrzakulov, P. Paul, S. Upadhyay.

**Figure 1.** Figure 1: FIG. 1: Metric function view at source ↗

**Figure 2.** Figure 2: FIG. 2: Metric function view at source ↗

**Figure 3.** Figure 3: FIG. 3: The plot of temperature view at source ↗

**Figure 4.** Figure 4: FIG. 4: The plot of heat capacity ( view at source ↗

**Figure 5.** Figure 5: FIG. 5: The plot of Gibbs free energy ( view at source ↗

**Figure 6.** Figure 6: FIG. 6: The plot view at source ↗

read the original 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-Nordstr\"om-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.

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

read the letter

The paper gives a new exact AdS black hole metric coupling Kalb-Ramond field to nonlinear electrodynamics with Lorentz violation, then maps its extended thermodynamics including non-monotonic temperature and swallow-tail phase transitions. They derive the metric explicitly, show it has inner and outer horizons that merge at a critical charge, and recover several known limits such as Bardeen, modified Kalb-Ramond, RN-AdS, and Schwarzschild-AdS when parameters are set to zero or special values. The thermodynamic quantities follow: Hawking temperature with possible local extrema from the NLED source, entropy that deviates from the area law, specific heat that can go negative, and Gibbs free energy plots with clear swallow tails indicating first-order transitions. They also state the first law in extended phase space and verify the Smarr relation holds independently. This is a concrete addition to the catalog of solutions in modified gravity with exotic matter. The calculations are standard but applied cleanly to the new coupling, and the reduction to prior cases gives some built-in consistency checks. The main soft spot is the exactness claim. The stress-test note is right to flag that the Einstein equations with the summed NLED and Kalb-Ramond stress-energy tensors need transparent verification; if the paper only shows the final metric and limits without component-level cancellation or code, a referee will want that spelled out. The entropy modification is handled but would benefit from a short derivation of how it enters the first law. The Lorentz-violating parameters remain free knobs with limited physical motivation, which is common but worth noting. This is for researchers tracking black hole thermodynamics in AdS and modified theories, especially those collecting examples of phase transitions. A reader wanting explicit new solutions and plots will get value from it. It deserves a serious referee because the central claims are specific and checkable rather than vague. I would send it to peer review with the expectation that the solution verification gets the main attention.

Referee Report

3 major / 2 minor

Summary. The manuscript claims to derive an exact black hole solution in AdS spacetime sourced by a Kalb-Ramond field coupled to nonlinear electrodynamics with Lorentz-violating parameters and magnetic monopole charge. It computes modified thermodynamic quantities (non-monotonic Hawking temperature, entropy deviating from the area law, specific heat), shows swallow-tail structures in the Gibbs free energy indicating first-order phase transitions, and states that the first law in extended phase space together with the Smarr relation hold for this system. The solution is asserted to reduce to known limits such as Reissner-Nordström-AdS and Schwarzschild-AdS.

Significance. If the exact solution and thermodynamic derivations are rigorously verified, the work adds to the literature on extended black hole thermodynamics with nonlinear sources and Lorentz violation by exhibiting rich phase structure, including possible instabilities and criticality. The recovery of standard geometries in parameter limits is a strength that supports broader applicability.

major comments (3)

[Metric solution and field equations] The load-bearing claim is that the proposed metric is an exact solution satisfying G_{μν} − Λ g_{μν} = 8π (T_{μν}^{NLED} + T_{μν}^{KR}) identically. The abstract and summary assert this without visible component-by-component substitution, derivation steps, or error checks, leaving open whether the identity holds only asymptotically, for restricted parameters, or after unstated constraints. This directly affects all subsequent thermodynamic results.
[Thermodynamic quantities and first law] The entropy is stated to deviate from the area law due to the NLED source, yet the explicit form and its consistency with the first law (dM = T dS + Φ dQ + ... in extended phase space) are not shown in detail. Without this, the asserted validity of the first law and Smarr relation cannot be independently confirmed and risks circularity with the metric ansatz.
[Gibbs free energy and phase transitions] The Gibbs free energy is claimed to exhibit swallow-tail structures for first-order phase transitions, but the definition of the extended phase space pressure P, volume V, and critical points (including dependence on Lorentz-violating parameters) requires explicit derivation to support the P-v criticality analysis and the non-existence of black holes beyond the critical monopole charge.

minor comments (2)

[Notation and limits] Clarify the notation for the Lorentz-violating parameters and magnetic charge in equations and figures to avoid ambiguity when taking limits.
[Introduction and references] Include additional references to prior Kalb-Ramond and NLED black hole solutions in AdS to better situate the novelty of the coupled system.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments have prompted us to strengthen the presentation of the derivations. We address each major comment below and have revised the manuscript to incorporate explicit calculations and additional explanations.

read point-by-point responses

Referee: [Metric solution and field equations] The load-bearing claim is that the proposed metric is an exact solution satisfying G_{μν} − Λ g_{μν} = 8π (T_{μν}^{NLED} + T_{μν}^{KR}) identically. The abstract and summary assert this without visible component-by-component substitution, derivation steps, or error checks, leaving open whether the identity holds only asymptotically, for restricted parameters, or after unstated constraints. This directly affects all subsequent thermodynamic results.

Authors: We appreciate the referee pointing out the need for explicit verification. The metric was obtained by direct substitution into the Einstein equations with the given sources, and the identity holds for the full parameter range as confirmed by our internal calculations. To make this transparent, we have added Appendix A containing the explicit non-zero components of G_{μν}, T^{NLED}_{μν}, and T^{KR}_{μν}, demonstrating that the equations are satisfied identically without extra constraints or asymptotic approximations. revision: yes
Referee: [Thermodynamic quantities and first law] The entropy is stated to deviate from the area law due to the NLED source, yet the explicit form and its consistency with the first law (dM = T dS + Φ dQ + ... in extended phase space) are not shown in detail. Without this, the asserted validity of the first law and Smarr relation cannot be independently confirmed and risks circularity with the metric ansatz.

Authors: We agree that the explicit steps should be shown. The entropy follows from the Wald formula applied to the NLED-modified action, yielding S = π r_+² plus correction terms proportional to the nonlinear parameters. In the revised Section III we now display the full expression for S, compute T = (∂M/∂S)_{Q,P} directly, and verify dM = T dS + Φ dQ + V dP by explicit differentiation of the mass function. The Smarr relation is obtained via Euler homogeneity and is checked numerically for sample parameter values. These additions remove any potential circularity. revision: yes
Referee: [Gibbs free energy and phase transitions] The Gibbs free energy is claimed to exhibit swallow-tail structures for first-order phase transitions, but the definition of the extended phase space pressure P, volume V, and critical points (including dependence on Lorentz-violating parameters) requires explicit derivation to support the P-v criticality analysis and the non-existence of black holes beyond the critical monopole charge.

Authors: We have expanded Section IV with the required derivations. We define P = −Λ/8π and V = (4/3)π r_+³ (standard for the extended phase space in this geometry). Critical points are located by solving ∂T/∂r_+ = 0 and ∂²T/∂r_+² = 0, with the resulting expressions for T_c, r_c, and P_c shown explicitly as functions of the Lorentz-violating parameters. The critical monopole charge Q_crit is obtained from the discriminant of the horizon equation; for Q > Q_crit the cubic has no positive real roots, confirming the absence of black-hole solutions. Additional P-v isotherms and Gibbs free-energy plots illustrating the swallow-tail behavior have been included. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper constructs an exact metric solution by direct substitution into the Einstein equations sourced by NLED and Kalb-Ramond stress-energy tensors, then derives thermodynamic quantities (temperature, entropy, Gibbs free energy) and verifies the extended first law plus Smarr relation by explicit differentiation and integration from the metric. These steps are independent of any fitted parameters or self-citations; the solution reduces to known limits by parameter choice but does not presuppose the thermodynamic results. No self-definitional loops, renamed empirical patterns, or load-bearing self-citations appear in the derivation chain.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard general relativity axioms plus the specific matter content and the extended thermodynamics framework.

free parameters (2)

Lorentz-violating parameters
Parameters introduced to describe the Kalb-Ramond field effects, their values are chosen to recover known limits.
magnetic monopole charge
Parameter in the solution, with a critical value beyond which no black hole exists.

axioms (2)

domain assumption The Einstein equations hold with the energy-momentum tensor from the Kalb-Ramond and NLED fields
Fundamental assumption for finding the solution.
domain assumption The extended phase space thermodynamics applies, treating Lambda as pressure
Used for P-v criticality and first law.

invented entities (1)

Kalb-Ramond field coupled to NLED no independent evidence
purpose: To source the black hole geometry with Lorentz violation
Postulated in the model, no independent evidence provided beyond the solution itself.

pith-pipeline@v0.9.0 · 5579 in / 1353 out tokens · 47648 ms · 2026-05-08T15:47:13.915474+00:00 · methodology

discussion (0)

Reference graph

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