arxiv: 2604.13140 · v1 · submitted 2026-04-14 · 🌀 gr-qc

Recognition: unknown

Black holes in general relativity coupled with NEDs surrounded by PFDM: thermodynamics, epicyclic oscillations, QPOs, and shadow

Faizuddin Ahmed , Sardor Murodov , Bekzod Rahmatov

Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:18 UTC · model grok-4.3

classification 🌀 gr-qc

keywords black holesnonlinear electrodynamicsperfect fluid dark matterquasiperiodic oscillationsepicyclic frequenciesthermodynamicsblack hole shadowMarkov Chain Monte Carlo

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The pith

Quasiperiodic oscillation data from four X-ray sources constrains the mass, magnetic charge, and dark matter parameters of regular black holes in general relativity coupled to nonlinear electrodynamics and perfect fluid dark matter.

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

The paper examines regular black hole solutions in general relativity that incorporate nonlinear electrodynamics and are embedded in a perfect fluid dark matter environment. It first computes the horizon structure and thermodynamic quantities such as Hawking temperature and specific heat capacity to determine stability. It then derives the effective potential for neutral test particles using the Hamiltonian formalism and extracts the epicyclic frequencies that govern quasiperiodic oscillations. By fitting these frequencies via Markov Chain Monte Carlo to observational QPO data from XTE J1550-564, GRO J1655-40, GRS 1915+105, and M82 X-1, the work obtains joint constraints on black hole mass, the nonlinear electrodynamics magnetic charge, the perfect fluid dark matter parameter, and the orbital radius. It closes by showing how the same parameters shape the apparent shadow. A reader cares because the analysis supplies a concrete route to test the viability of these modified black hole models against real astrophysical timing data rather than leaving them as purely theoretical constructions.

Core claim

The spacetime metric of a regular black hole in general relativity coupled with nonlinear electrodynamics and surrounded by perfect fluid dark matter yields epicyclic frequencies for neutral test particles whose values can be matched to observed quasiperiodic oscillations, permitting a Markov Chain Monte Carlo analysis that constrains the black hole mass, magnetic charge parameter, perfect fluid dark matter parameter, and characteristic orbital radius for the sources XTE J1550-564, GRO J1655-40, GRS 1915+105, and M82 X-1.

What carries the argument

Epicyclic frequencies obtained from the effective potential of neutral test particles in the equatorial plane, inserted into a Markov Chain Monte Carlo fit against observed QPO frequencies.

Load-bearing premise

The observed quasiperiodic oscillations in the listed sources arise from the epicyclic motion of neutral test particles orbiting in the equatorial plane of this regular black hole spacetime.

What would settle it

A set of QPO frequency measurements from any one of the four sources whose values lie outside the range reproducible by the epicyclic frequency expressions for all physically allowed combinations of mass, magnetic charge, perfect fluid dark matter parameter, and radius.

Figures

Figures reproduced from arXiv: 2604.13140 by Bekzod Rahmatov, Faizuddin Ahmed, Sardor Murodov.

**Figure 2.** Figure 2: Hawking temperature TH versus the event horizon radius rh for the Schwarzschild black hole and for different values of q/M and λ/M. The Schwarzschild case decreases monotonically, while the modified solutions show a peak and lower temperatures [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: Heat capacity C as a function of the event horizon radius rh for the Schwarzschild black hole and for the modified black hole solutions with different values of q/M and λ/M. The Schwarzschild case remains negative over the whole range, whereas the modified solutions exhibit divergences at critical radii and a sign change of C, indicating a thermodynamic phase transition. The horizontal line C = 0 separates… view at source ↗

**Figure 4.** Figure 4: Effective potential Ueff as a function of r/M for different values of q and λ with fixed angular momentum L = 20. The plot shows that the magnetic charge and PFDM parameters noticeably affect the orbital structure of test particles. tum Lz. In our case at hand, we find these are pt m = −f(r) dt dτ = −E, (29) pϕ m = r 2 sin2 θ dϕ dτ = Lz, (30) pθ m = r 2 dθ dτ = Lθ, (31) where E = E/m and Lz = Lz/m represen… view at source ↗

**Figure 5.** Figure 5: Effective radial force M2F as a function of r/M for different values of q/M and λ/M, with fixed angular momentum L/M = √ 15. shape of the potential curve and shifts its minimum toward larger radii. This behavior indicates that stable circular orbits are formed farther from the black hole. At the same time, the reduction in the depth of the potential well suggests that test particles become less strongly bo… view at source ↗

**Figure 6.** Figure 6: Energy E and angular momentum L/M for circular motion around the black hole for different values of q and λ. The left and middle panels show their radial profiles, while the right panel presents the E–L/M relation. The black solid curve corresponds to the Schwarzschild case. At larger distances, however, all curves gradually approach zero, showing that the effects of these parameters become weaker far from… view at source ↗

**Figure 7.** Figure 7: Three-dimensional plot of ISCO radius for various [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗

**Figure 8.** Figure 8: Particle trajectory in the (X, Y ) plane corresponding to L = 1, and M = 1. The above second-order non-linear differential equation (45) represents the test particles trajectory in the considered black hole gravitational field. We observe that the geometric parameters λ and q together with the conserved angular momentum L alter the particles trajectory in the field. In Figures 8, we illustrate the particle… view at source ↗

**Figure 9.** Figure 9: Left panel: radial epicyclic frequency Ω [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗

**Figure 10.** Figure 10: νU versus νL in the RP model for different q and λ. The black curve shows the Schwarzschild case, while the dashed curves represent the PFDM black hole. Larger q and λ shift the curves upward. for several values of the magnetic charge parameter q and the PFDM parameter λ. The Schwarzschild case is shown by the black solid curve, whereas the remaining curves correspond to the regular black hole surrounded … view at source ↗

**Figure 11.** Figure 11: Posterior distributions of M, q, λ, and r obtained from the MCMC analysis in the RP model for XTE J1550– 564, GRO J1655–40, GRS 1915+105, and M82 X-1. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗

**Figure 12.** Figure 12: Shadow profiles in the celestial plane ( [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗

**Figure 13.** Figure 13: Three-dimensional behavior of the photon sphere radius and the shadow radius for different values of [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗

read the original abstract

In this work, we investigate the thermodynamics and motion of neutral test particles around a regular black hole immersed in a perfect fluid dark matter environment. We begin by examining the horizon structure and key thermodynamic properties, with particular emphasis on quantities such as the Hawking temperature and the specific heat capacity. These aspects provide important insight into the stability and physical behavior of the black hole system. We then proceed to analyze the dynamics of neutral test particles using the Hamiltonian formalism, through which we derive the effective potential governing particle motion. Using the effective potential, we further study quasiperiodic oscillations by determining the associated epicyclic frequencies and comparing them with available observational data. Using the observed QPO data of XTE J1550-564, GRO J1655-40, GRS 1915+105, and M82 X-1, we perform a Markov Chain Monte Carlo analysis to constrain the black hole mass, the magnetic charge parameter, the PFDM parameter, and the characteristic orbital radius. Finally, we investigate the black hole shadow and demonstrate how various geometric parameters influence its optical appearance. This analysis highlights the potential observational signatures of such black holes and their surrounding dark matter environment.

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

The paper fits a NED+PFDM black hole to QPO data from four sources via MCMC but the frequency-to-epicyclic mapping is the untested step.

read the letter

The paper combines nonlinear electrodynamics with perfect fluid dark matter in a regular black hole spacetime. It works out the thermodynamics, the motion of neutral test particles, the epicyclic frequencies, and the shadow. Then it uses MCMC to match the model to QPO data from XTE J1550-564, GRO J1655-40, GRS 1915+105, and M82 X-1, constraining mass, magnetic charge, PFDM parameter, and orbital radius. The specific pairing of NED and PFDM with this QPO analysis is the targeted extension here. The calculations follow the usual route from the metric to the effective potential and frequencies. The shadow part shows how the parameters change the appearance. The derivations seem standard and should be checkable. The MCMC is applied directly to the data. The issue is that the fit assumes the QPOs come exactly from the radial and vertical epicyclic frequencies of neutral geodesics. The paper does not test alternative QPO models or see how the credible intervals change if that mapping is not exact. That makes the constraints dependent on the choice of mechanism. This kind of paper is for specialists in black hole solutions with additional matter fields or in fitting observations to modified metrics. It supplies numbers for this model but does not tackle larger problems. I would discuss it in a reading group on observational black hole physics. I would not cite it in my own research. It should go to peer review so the details of the derivations and the robustness of the fit can be examined.

Referee Report

1 major / 2 minor

Summary. The manuscript examines the thermodynamic properties and geodesic motion of neutral test particles around a regular black hole in general relativity coupled with nonlinear electrodynamics (NED) and surrounded by perfect fluid dark matter (PFDM). It derives the Hawking temperature and specific heat, computes the effective potential using the Hamiltonian formalism, calculates epicyclic frequencies for QPOs, performs MCMC analysis to fit parameters (mass, magnetic charge, PFDM parameter, orbital radius) to observational QPO data from XTE J1550-564, GRO J1655-40, GRS 1915+105, and M82 X-1, and studies the black hole shadow.

Significance. If the central assumption that the observed QPOs correspond to epicyclic frequencies holds, the work provides new constraints on the model parameters and illustrates the impact of NED and PFDM on black hole observables such as QPOs and shadows. This contributes to the growing literature on testing modified gravity and dark matter models with astrophysical data. The inclusion of MCMC fitting and shadow analysis adds quantitative and visual elements to the study.

major comments (1)

[QPO and MCMC analysis section] The MCMC constraints on the black hole mass, magnetic charge parameter, PFDM parameter, and characteristic orbital radius are obtained by directly equating the observed upper and lower QPO frequencies to the radial (ν_r) and vertical (ν_θ) epicyclic frequencies derived from the effective potential. This identification is load-bearing for the central claim but lacks justification against alternative QPO models (e.g., disk precession or resonance mechanisms), and the uncertainty in the frequency assignment is not propagated into the posterior distributions. A sensitivity analysis or comparison with other models would strengthen the results.

minor comments (2)

[Thermodynamics section] The discussion of specific heat capacity could benefit from clearer plots or additional analysis of phase transitions.
[Shadow section] The figures showing the shadow could include more quantitative measures, such as the shadow radius as a function of parameters.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment on the QPO and MCMC analysis. We address the point below and indicate the revisions we will make.

read point-by-point responses

Referee: The MCMC constraints on the black hole mass, magnetic charge parameter, PFDM parameter, and characteristic orbital radius are obtained by directly equating the observed upper and lower QPO frequencies to the radial (ν_r) and vertical (ν_θ) epicyclic frequencies derived from the effective potential. This identification is load-bearing for the central claim but lacks justification against alternative QPO models (e.g., disk precession or resonance mechanisms), and the uncertainty in the frequency assignment is not propagated into the posterior distributions. A sensitivity analysis or comparison with other models would strengthen the results.

Authors: We agree that the direct identification of the observed upper and lower QPO frequencies with the vertical and radial epicyclic frequencies is a central assumption of the MCMC analysis. This choice follows the standard relativistic precession model commonly employed in the literature for twin-peak QPOs in black-hole X-ray binaries. While alternative mechanisms such as disk precession or resonance models exist, our work is focused on the implications of the NED+PFDM spacetime within the epicyclic framework. To address the referee's concern, we will revise the manuscript by (i) adding an explicit discussion of the adopted identification together with references to both supporting and alternative QPO models, and (ii) performing a limited sensitivity analysis in which the frequency assignment is swapped and the resulting shifts in the posterior distributions are reported. The MCMC already incorporates the reported observational uncertainties on the QPO frequencies; the revised text will clarify this point and note the additional model-choice uncertainty. These changes will be presented as an expanded subsection without altering the numerical results or conclusions of the original analysis. revision: partial

Circularity Check

0 steps flagged

No significant circularity: model derivation independent of QPO fitting

full rationale

The paper constructs the NED+PFDM metric from the Einstein equations, derives the effective potential and epicyclic frequencies (ν_r, ν_θ) from the geodesic Hamiltonian, and computes thermodynamic quantities directly from the metric functions. It then performs MCMC fitting of the four parameters to external QPO frequency data from four named sources by matching those derived frequencies to observations. This is standard parameter estimation against independent data; the fitted values are outputs of the likelihood, not inputs that force the frequency expressions by construction. No self-definitional equations, renamed predictions, or load-bearing self-citations that reduce the central results to tautology are exhibited in the abstract or described chain. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central results rest on an assumed exact black hole metric whose derivation is not shown in the abstract, plus the standard assumption that PFDM can be treated as a perfect fluid background. The MCMC step introduces fitted parameters whose values are determined by the data rather than derived from first principles.

free parameters (3)

magnetic charge parameter
Introduced via the NED coupling and directly constrained by the MCMC fit to QPO data.
PFDM parameter
Density parameter of the perfect fluid dark matter halo, fitted via MCMC.
characteristic orbital radius
Orbital radius at which QPOs are assumed to occur, also fitted.

axioms (2)

domain assumption The background spacetime is a regular black hole solution of GR coupled to NEDs and immersed in PFDM.
Invoked as the starting metric for all thermodynamic, geodesic, and shadow calculations.
domain assumption Quasi-periodic oscillations in the listed X-ray sources are produced by epicyclic motion of neutral test particles.
Required to link the derived frequencies to the observational data used in MCMC.

pith-pipeline@v0.9.0 · 5523 in / 1647 out tokens · 55303 ms · 2026-05-10T15:18:21.519968+00:00 · methodology

discussion (0)

Reference graph

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