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arxiv: 2605.26131 · v1 · pith:46BUC5FKnew · submitted 2026-05-20 · 🌀 gr-qc

ModMax black hole surrounded by perfect-fluid dark matter in Lorentz-violating Kalb-Ramond gravity

Fernando M. Belchior , Faizuddin Ahmed , Edilberto O. Silva This is my paper

Pith reviewed 2026-06-30 17:21 UTC · model grok-4.3

classification 🌀 gr-qc
keywords modmax electrodynamicsperfect fluid dark matterkalb-ramond gravitylorentz violationblack hole thermodynamicshawking temperatureheat capacityhorizon structure
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The pith

A static spherically symmetric black hole metric exists that merges ModMax nonlinear electrodynamics, Lorentz-violating Kalb-Ramond gravity, and perfect-fluid dark matter, producing altered horizons and thermodynamic quantities.

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

The paper derives the metric for a black hole that combines ModMax nonlinear electrodynamics, a Kalb-Ramond field inducing Lorentz violation, and surrounding perfect-fluid dark matter. It then computes the effects of the charge, ModMax parameter, Kalb-Ramond coupling, and dark matter parameter on the horizon radii and on thermodynamic quantities including Hawking temperature, entropy, heat capacity, and Helmholtz free energy. A sympathetic reader would care because these modifications can move the extremal limit, change regions of thermal stability, and produce new phase structures, while the dark matter term adds a logarithmic correction that matters at intermediate distances. The results indicate that ModMax can screen the electric field and that the Kalb-Ramond term amplifies geometric changes.

Core claim

We obtain the corresponding static and spherically symmetric black hole geometry and analyze how the charge, ModMax parameter, Kalb-Ramond coupling, and dark matter parameter affect the horizon structure and thermodynamic behavior. In particular, we study the Hawking temperature, entropy, heat capacity, and Helmholtz free energy, showing that the combined effects of nonlinear electrodynamics and Lorentz violation may shift the extremal configuration, modify the thermal stability regions, and generate nontrivial phase behavior. The perfect fluid dark matter contribution introduces an additional logarithmic correction to the geometry, becoming especially relevant at intermediate radial scales.

What carries the argument

The static spherically symmetric metric obtained by solving the field equations with ModMax electrodynamics, Kalb-Ramond Lorentz violation, and perfect-fluid dark matter contributions.

Load-bearing premise

A static and spherically symmetric solution exists for the combined system of ModMax electrodynamics, Kalb-Ramond gravity with Lorentz violation, and perfect fluid dark matter.

What would settle it

An explicit verification that the derived metric does not satisfy the Einstein equations for nonzero values of the ModMax, Kalb-Ramond, and dark matter parameters, or thermodynamic observations of a black hole showing no shift in stability regions when dark matter is present.

Figures

Figures reproduced from arXiv: 2605.26131 by Edilberto O. Silva, Faizuddin Ahmed, Fernando M. Belchior.

Figure 1
Figure 1. Figure 1: Behavior of the metric function f(r) under variations of the model parameters. Panel (a) displays the effect of the KR/LV parameter, panel (b) the effect of the PFDM parameter, panel (c) the effect of the electric charge, and panel (d) the effect of the ModMax parameter. In the graphical legends, γ ≡ α and λ ≡ λMM. and the internal graphical labels should be read with the identifications γ ≡ α and λ ≡ λMM.… view at source ↗
Figure 2
Figure 2. Figure 2: Behavior of the timelike effective potential [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Behavior of the null effective potential [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Apparent shadow profiles in the celestial plane ( [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Behavior of the mass function M(rh) under variations of the model parameters. Panel (a) displays the effect of the KR/LV parameter, panel (b) the effect of the PFDM parameter, panel (c) the effect of the electric charge, and panel (d) the effect of the ModMax parameter. In the graphical legends, γ ≡ α and λ ≡ λMM. The temperature curves show that the black hole does not behave as a simple Schwarzschild obj… view at source ↗
Figure 6
Figure 6. Figure 6: Behavior of the Hawking temperature TH under variations of the model parameters. Panel (a) displays the effect of the KR/LV parameter, panel (b) the effect of the PFDM parameter, panel (c) the effect of the electric charge, and panel (d) the effect of the ModMax parameter. In the graphical legends, γ ≡ α and λ ≡ λMM. second-order phase transitions, since the temperature has an extremum there and the respon… view at source ↗
Figure 7
Figure 7. Figure 7: Behavior of the heat capacity CQ under variations of the model parameters. Panel (a) displays the effect of the KR/LV parameter, panel (b) the effect of the PFDM parameter, panel (c) the effect of the electric charge, and panel (d) the effect of the ModMax parameter. In the graphical legends, γ ≡ α and λ ≡ λMM. where the charge term is strongest. Larger α generally increases the thermodynamic cost of a bla… view at source ↗
Figure 8
Figure 8. Figure 8: Behavior of the Helmholtz free energy F under variations of the model parameters. Panel (a) displays the effect of the KR/LV parameter, panel (b) the effect of the PFDM parameter, panel (c) the effect of the electric charge, and panel (d) the effect of the ModMax parameter. In the graphical legends, γ ≡ α and λ ≡ λMM. Then, we write the Hawking temperature as follows TH = Θh 4π √ δ rh . (89) The condition … view at source ↗
Figure 9
Figure 9. Figure 9: Behavior of the scalar effective potential [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Behavior of the scalar transmission factor, namely the greybody lower bound [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Behavior of the partial scalar absorption cross section [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
read the original abstract

We investigate a ModMax black hole surrounded by perfect-fluid dark matter within the framework of Lorentz-violating Kalb-Ramond gravity. The model combines three physically distinct contributions: nonlinear electrodynamic corrections from the ModMax sector, Lorentz-symmetry-breaking effects induced by the background Kalb-Ramond field, and environmental modifications associated with the surrounding dark matter fluid. We obtain the corresponding static and spherically symmetric black hole geometry and analyze how the charge, ModMax parameter, Kalb-Ramond coupling, and dark matter parameter affect the horizon structure and thermodynamic behavior. In particular, we study the Hawking temperature, entropy, heat capacity, and Helmholtz free energy, showing that the combined effects of nonlinear electrodynamics and Lorentz violation may shift the extremal configuration, modify the thermal stability regions, and generate nontrivial phase behavior. The perfect fluid dark matter contribution introduces an additional logarithmic correction to the geometry, becoming especially relevant at intermediate radial scales. Our results indicate that ModMax electrodynamics can effectively screen the electric sector, while the Kalb-Ramond parameter amplifies the geometric deformation and changes the thermodynamic response of the system. These features suggest that black holes in Lorentz-violating backgrounds surrounded by dark matter provide a useful arena for probing deviations from standard charged black-hole thermodynamics.

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

1 major / 2 minor

Summary. The paper investigates a ModMax black hole surrounded by perfect-fluid dark matter in Lorentz-violating Kalb-Ramond gravity. It claims to obtain the corresponding static spherically symmetric black hole geometry incorporating nonlinear electrodynamic corrections, Lorentz-symmetry-breaking effects from the Kalb-Ramond field, and dark matter modifications, then analyzes the effects of the charge, ModMax parameter, Kalb-Ramond coupling, and dark matter parameter on horizon structure and thermodynamic quantities (Hawking temperature, entropy, heat capacity, Helmholtz free energy), concluding that these can shift extremal configurations, modify thermal stability regions, generate nontrivial phase behavior, and that dark matter introduces a logarithmic correction to the geometry.

Significance. If the metric is a valid solution to the combined field equations, the work could illustrate how multiple extensions (ModMax NED, KR Lorentz violation, and perfect-fluid DM) interact to alter black hole thermodynamics beyond standard charged solutions. The explicit inclusion of a logarithmic DM term and parameter screening effects offers a concrete arena for probing deviations, though the absence of limit checks or comparisons to known cases reduces immediate impact.

major comments (1)
  1. [Geometry section (following abstract claim of obtaining the solution)] The central claim that a static spherically symmetric geometry exists for the combined ModMax + KR + perfect-fluid DM system is asserted in the abstract and introduction but not supported by an explicit metric function f(r), the full action, or variation steps confirming that the three stress-energy contributions are simultaneously satisfied. This is load-bearing because the subsequent analysis of horizons, temperature, heat capacity, and phase behavior all presuppose this metric; without the derivation, the thermodynamic results cannot be verified as following from the model.
minor comments (2)
  1. The abstract refers to 'nontrivial phase behavior' without specifying whether this involves swallowtail structures in free energy, critical points in heat capacity, or other diagnostics; the main text should include explicit calculations or figures for at least one parameter set.
  2. Notation for the free parameters (ModMax parameter, Kalb-Ramond coupling, dark matter parameter) should be introduced with clear definitions and ranges early in the text to avoid ambiguity in later thermodynamic expressions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for explicit verification of the metric derivation. We address the single major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The central claim that a static spherically symmetric geometry exists for the combined ModMax + KR + perfect-fluid DM system is asserted in the abstract and introduction but not supported by an explicit metric function f(r), the full action, or variation steps confirming that the three stress-energy contributions are simultaneously satisfied. This is load-bearing because the subsequent analysis of horizons, temperature, heat capacity, and phase behavior all presuppose this metric; without the derivation, the thermodynamic results cannot be verified as following from the model.

    Authors: We agree that the original manuscript did not provide sufficient detail on the derivation. In the revised version we will include: (i) the complete action combining the ModMax nonlinear electrodynamics, the Lorentz-violating Kalb-Ramond term, and the perfect-fluid dark-matter stress-energy; (ii) the explicit static spherically symmetric metric function f(r) that incorporates the logarithmic dark-matter correction; and (iii) the step-by-step variation of the action showing that the three stress-energy tensors are simultaneously satisfied by this f(r). These additions will allow direct verification that the subsequent thermodynamic quantities follow from the field equations. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states that the static spherically symmetric geometry is obtained from the combined field equations of ModMax NED, Lorentz-violating KR gravity, and perfect-fluid DM, after which Hawking temperature, entropy, heat capacity, and free energy are computed directly from the resulting metric function. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the thermodynamic analysis follows from the derived line element without renaming or circular re-use of inputs. The derivation chain is therefore self-contained against the Einstein equations and external to any fitted data.

Axiom & Free-Parameter Ledger

4 free parameters · 2 axioms · 2 invented entities

Ledger inferred from abstract only; full action and field equations unavailable.

free parameters (4)
  • ModMax parameter
    Controls strength of nonlinear electrodynamic corrections; introduced without derivation from more fundamental theory.
  • Kalb-Ramond coupling
    Sets scale of Lorentz violation; chosen as free parameter in the model.
  • dark matter parameter
    Sets amplitude of perfect-fluid dark matter contribution; fitted or chosen to produce logarithmic term.
  • electric charge
    Standard charge parameter of the black hole.
axioms (2)
  • domain assumption Static and spherically symmetric metric ansatz
    Invoked to reduce the system to an integrable form for the metric function.
  • domain assumption Perfect fluid equation of state for surrounding dark matter
    Assumed to produce the logarithmic correction at intermediate radii.
invented entities (2)
  • ModMax nonlinear electromagnetic field no independent evidence
    purpose: Provide nonlinear corrections to Maxwell sector
    Postulated as part of the model; no independent evidence supplied.
  • Background Kalb-Ramond field no independent evidence
    purpose: Induce Lorentz symmetry breaking
    Introduced as a fixed background; no independent evidence supplied.

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