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arxiv: 2605.18371 · v1 · pith:3R2ZLSA7new · submitted 2026-05-18 · 🌀 gr-qc

Dyonic Black Holes in Lorentz-Violating Gravity with a Background Kalb--Ramond Field

Yu-Xuan Lin , Jia-Zhou Liu , Yu-Xiao Liu This is my paper

Pith reviewed 2026-05-20 09:14 UTC · model grok-4.3

classification 🌀 gr-qc
keywords dyonic black holeLorentz violationKalb-Ramond fieldblack hole shadowgeodesic motionthermodynamic phase transitionnonminimal couplingextended phase space
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The pith

A nonminimal coupling between the Kalb-Ramond and electromagnetic fields produces an exact dyonic black hole solution in Lorentz-violating gravity.

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

The paper constructs an exact static spherically symmetric dyonic black hole solution in Lorentz-violating gravity by introducing a nonminimal coupling between the Kalb-Ramond field and the electromagnetic field. This solution is then used to study null and timelike geodesics, showing how the Lorentz-violating parameter and dyonic charges change the photon sphere radius, shadow radius, and innermost stable circular orbit. Thermodynamic analysis in the extended phase space verifies the first law and Smarr relation while revealing a first-order phase transition between small and large black holes whose features depend on the same parameters. A sympathetic reader would care because the construction supplies concrete predictions for how Lorentz violation and extra fields could alter black hole observables such as shadows.

Core claim

By introducing a nonminimal coupling between the Kalb--Ramond field and the electromagnetic field, an exact four-dimensional static, spherically symmetric dyonic black hole solution is constructed in Lorentz-violating gravity with a background Kalb--Ramond field. The curvature invariants confirm a genuine singularity at r=0. Geodesic analysis of null and timelike particles yields the photon-sphere radius, shadow radius, and innermost stable circular orbit, all modified by the Lorentz-violating parameter and the dyonic charges. In the extended phase space the thermodynamic quantities satisfy the first law and the Smarr relation, and the system exhibits a first-order phase transition between小黑

What carries the argument

The nonminimal coupling term between the Kalb-Ramond field and the electromagnetic field, which permits the exact closed-form dyonic metric while preserving the static spherically symmetric ansatz.

If this is right

  • The shadow radius is modified by the Lorentz-violating parameter and the dyonic charges.
  • The domain of stable circular motion for timelike particles depends on the Lorentz-violating parameter and the dyonic charges.
  • The first law of black hole thermodynamics and the Smarr relation hold in the extended phase space.
  • A first-order phase transition occurs between small and large black holes and is influenced by the Lorentz-violating parameter and the dyonic charges.

Where Pith is reading between the lines

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

  • The explicit dependence of shadow size on the Lorentz-violating parameter offers a potential route to constrain such violations using future very-long-baseline interferometry observations.
  • The same nonminimal coupling strategy could be tested in other background-field models to generate additional exact solutions.
  • The parameter-dependent phase transition may link to questions of black-hole stability during early-universe evolution when similar fields are present.

Load-bearing premise

The nonminimal coupling between the Kalb-Ramond field and the electromagnetic field is assumed to permit an exact closed-form solution for the dyonic metric while preserving the static spherically symmetric ansatz.

What would settle it

Substituting the proposed metric, electromagnetic potential, and Kalb-Ramond background into the modified field equations and verifying whether they hold identically for the chosen nonminimal coupling.

read the original abstract

By introducing a nonminimal coupling between the Kalb--Ramond field and the electromagnetic field, we construct an exact four-dimensional static, spherically symmetric dyonic black hole solution in Lorentz-violating gravity with a background Kalb--Ramond field. The curvature invariants show that the spacetime retains a genuine curvature singularity at $r=0$. We then analyze the geodesic motion of null and timelike particles and obtain the photon-sphere radius, the shadow radius, and the innermost stable circular orbit, demonstrating that both the Lorentz-violating parameter and the dyonic charges can appreciably modify the shadow size and the domain of stable circular motion. In the extended phase space, we derive the thermodynamic quantities and verify the first law of black hole thermodynamics together with the Smarr relation. The system also exhibits a first-order phase transition between small and large black holes, and its phase structure is strongly influenced by the Lorentz-violating parameter and the dyonic charges.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a nonminimal coupling between the Kalb-Ramond field and the electromagnetic field to construct an exact four-dimensional static spherically symmetric dyonic black hole solution in Lorentz-violating gravity with a background Kalb-Ramond field. It verifies that curvature invariants exhibit a genuine singularity at r=0, computes geodesic quantities including the photon-sphere radius, shadow radius, and ISCO for null and timelike particles, and derives thermodynamic quantities in extended phase space to confirm the first law, Smarr relation, and a first-order phase transition between small and large black holes whose structure depends on the Lorentz-violating parameter and dyonic charges.

Significance. If the exact solution is shown to satisfy the full set of modified field equations, the work supplies a concrete example of how Lorentz violation combined with a Kalb-Ramond background and dyonic charges alters observable features such as shadow size and the domain of stable orbits, as well as the thermodynamic phase structure. The explicit verification of the first law and Smarr relation in the presence of the new parameters strengthens the thermodynamic analysis.

major comments (2)
  1. [§2] §2 (action and field equations): The nonminimal coupling term between the Kalb-Ramond field and the electromagnetic field is introduced to permit an exact dyonic solution, yet the manuscript does not display the component-by-component substitution of the assumed metric, dyonic vector potential, and background Kalb-Ramond vev into the modified Einstein equations. Without this explicit check, it remains unclear whether the coupling cancels all angular or radial source terms that would otherwise violate the static spherically symmetric ansatz when both electric and magnetic charges are nonzero.
  2. [§3] §3 (metric and curvature invariants): The curvature invariants are stated to confirm a genuine singularity at r=0, but the explicit expressions for the metric functions (including the dependence on the Lorentz-violating parameter and the two charges) are not accompanied by the algebraic steps that solve the modified field equations. This verification is load-bearing for the central claim of an exact closed-form solution.
minor comments (2)
  1. [Abstract] The abstract and introduction refer to the nonminimal coupling without specifying its precise tensorial form (e.g., the contraction involving B_{μν}F^{μν}); adding the explicit Lagrangian term would improve readability.
  2. Figure captions for the shadow and phase-transition plots should explicitly state the fixed values of the Lorentz-violating parameter and the two charges used in each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and indicate the revisions we plan to implement.

read point-by-point responses

  1. Referee: [§2] §2 (action and field equations): The nonminimal coupling term between the Kalb-Ramond field and the electromagnetic field is introduced to permit an exact dyonic solution, yet the manuscript does not display the component-by-component substitution of the assumed metric, dyonic vector potential, and background Kalb-Ramond vev into the modified Einstein equations. Without this explicit check, it remains unclear whether the coupling cancels all angular or radial source terms that would otherwise violate the static spherically symmetric ansatz when both electric and magnetic charges are nonzero.

    Authors: We agree that an explicit component-by-component verification would improve clarity. In the revised manuscript we will add an appendix that substitutes the static spherically symmetric metric, the dyonic electromagnetic potential, and the background Kalb-Ramond vacuum expectation value into the modified Einstein equations, showing term-by-term cancellation of all non-spherically-symmetric source terms due to the nonminimal coupling. revision: yes

  2. Referee: [§3] §3 (metric and curvature invariants): The curvature invariants are stated to confirm a genuine singularity at r=0, but the explicit expressions for the metric functions (including the dependence on the Lorentz-violating parameter and the two charges) are not accompanied by the algebraic steps that solve the modified field equations. This verification is load-bearing for the central claim of an exact closed-form solution.

    Authors: We acknowledge that the algebraic steps leading to the closed-form metric functions should be shown explicitly. We will insert these intermediate steps into §3 (or a dedicated subsection) of the revised manuscript so that the dependence on the Lorentz-violating parameter and the dyonic charges is derived transparently from the field equations. revision: yes

Circularity Check

0 steps flagged

No significant circularity: exact solution constructed via ansatz and modified field equations

full rationale

The paper introduces a nonminimal coupling in the gravitational action specifically to admit an exact static spherically symmetric dyonic solution with a fixed background Kalb-Ramond field. The metric, electromagnetic potential, and KR field are posited as ansatzes, the modified Einstein and matter field equations are solved for the metric functions, and the resulting spacetime is then used to compute geodesics, shadow radius, and thermodynamic quantities. The first law and Smarr relation are verified as consistency checks on the obtained solution rather than as independent predictions. No parameter is fitted to data and then relabeled as a prediction, no uniqueness theorem is imported from self-citation to forbid alternatives, and no ansatz is smuggled through prior work. The derivation chain is self-contained against the chosen action and ansatz.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The central claim depends on a nonminimal coupling chosen to allow exact solvability, a static spherical symmetry ansatz, and the standard framework of extended black-hole thermodynamics; these are introduced without independent derivation in the abstract.

free parameters (2)
  • Lorentz-violating parameter
    Controls the strength of Lorentz symmetry breaking; appears as a free parameter that modifies geodesics and thermodynamics.
  • Nonminimal coupling constant
    Introduced by hand to couple Kalb-Ramond and electromagnetic fields so that an exact dyonic solution exists.
axioms (2)
  • domain assumption The metric ansatz is static and spherically symmetric in four dimensions
    Standard assumption invoked to reduce the field equations to ordinary differential equations.
  • domain assumption Extended phase space thermodynamics applies with cosmological constant as pressure
    Used to derive first law, Smarr relation and phase transition without additional justification in abstract.
invented entities (1)
  • Background Kalb-Ramond field no independent evidence
    purpose: Provides Lorentz violation and participates in nonminimal coupling to electromagnetic field
    Postulated as a fixed background tensor field whose independent existence is not demonstrated within the paper.

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