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arxiv: 2605.01934 · v1 · submitted 2026-05-03 · 🌀 gr-qc · astro-ph.HE· hep-th

Recognition: unknown

Scalar-Electromagnetic Couplings as Source of Deformed Black Hole: From Shadows to Thermodynamic Topology

Ednaldo L. B. Junior , Jos\'e Tarciso S. S. Junior , Francisco S. N. Lobo , Jorde A. A. Ramos , Manuel E. Rodrigues , Diego Rubiera-Garcia , Lu\'is F. Dias da Silva , Henrique A. Vieira
Authors on Pith no claims yet

Pith reviewed 2026-05-09 16:30 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HEhep-th
keywords black holenonlinear electrodynamicsscalar fieldshadow radiusthermodynamic topologyHawking-Page transitionnon-minimal couplingmagnetic charge
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The pith

Non-minimal coupling between a scalar field and nonlinear electrodynamics provides the matter source for a deformed black hole spacetime.

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

The paper reconstructs a static spherically symmetric black hole geometry by identifying a consistent matter source from a fundamental action. The spacetime is supported by a magnetically charged nonlinear electrodynamics field non-minimally coupled to a scalar field, with dimensional consistency reducing the model to a single magnetic charge parameter. The inverse construction formalism then yields a one-parameter family of electromagnetic Lagrangians of the form L(F) = F^{n+1}/(n+1). This reconstruction matters because it supplies a theoretical foundation for an effective metric, enabling direct computation of horizon structures, photon spheres and shadows constrained by Event Horizon Telescope data on Sagittarius A*, as well as thermodynamic quantities and topological invariants.

Core claim

The central claim is that the deformed black hole metric arises from a fundamental action with non-minimal scalar-NED coupling. The inverse construction identifies the supporting matter as a magnetically charged NED field, reducing the parameter space to a single magnetic charge and producing the Lagrangian family L(F) = F^{n+1}/(n+1). Analyses show a critical charge separating black hole and horizonless solutions, shadow radii bounded by EHT observations, thermodynamic satisfaction of the first law and Smarr relation with a Hawking-Page transition but no van der Waals behavior, and topological charges Q_tot = -1 for the photon sphere and W = 0 for thermodynamics, placing the solution in the

What carries the argument

The non-minimal scalar-electromagnetic coupling together with the inverse construction formalism that recovers the electromagnetic Lagrangian from the given effective metric.

If this is right

  • A critical value of the magnetic charge separates black hole solutions from horizonless configurations.
  • Event Horizon Telescope data on Sagittarius A* impose bounds on the allowed range of the magnetic charge through the computed shadow radius.
  • The thermodynamics obeys the first law and Smarr relation, exhibits a Hawking-Page phase transition, and displays a single stability change without van der Waals critical points.
  • The photon sphere carries total topological charge -1 while the thermodynamic vector field has winding number 0, classifying the solution with the Reissner-Nordström black hole.

Where Pith is reading between the lines

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

  • The non-minimal coupling framework may be applied to other effective metrics to derive their underlying fundamental sources.
  • The topological equivalence to the Reissner-Nordström black hole suggests comparable stability properties under strong-field perturbations.
  • Extending the shadow analysis to additional black hole candidates could further restrict the magnetic charge parameter.

Load-bearing premise

The inverse construction formalism applied to the non-minimal scalar-NED coupling exactly reproduces the original effective metric without additional inconsistencies or hidden assumptions in the Lagrangian form.

What would settle it

An observed shadow radius for Sagittarius A* lying outside the range permitted by the magnetic charge bounds, or detection of van der Waals-type critical behavior in the thermodynamics, would falsify the reconstruction.

Figures

Figures reproduced from arXiv: 2605.01934 by Diego Rubiera-Garcia, Ednaldo L. B. Junior, Francisco S. N. Lobo, Henrique A. Vieira, Jorde A. A. Ramos, Jos\'e Tarciso S. S. Junior, Lu\'is F. Dias da Silva, Manuel E. Rodrigues.

Figure 1
Figure 1. Figure 1: FIG. 1: The coupling function view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The potential view at source ↗
Figure 4
Figure 4. Figure 4: displays the shadow radius for representative values of the magnetic charge Q, highlighting the regions of parameter space compatible with the EHT constraints. We can see that the shadow radius stays within (1σ) for Model Sgr A*(1σ, Keck+VLTI) Sgr A*(2σ, Keck+VLTI) 0 0.5 1.0 1.5 2.0 3.5 4.0 4.5 5.0 5.5 FIG. 4: Black hole shadow radius rs/M as a function of the magnetic charge per unit mass Q, for and obser… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Metric function view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: The Gibbs free energy given by Eq. ( view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Gibbs free energy view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Normalized vector field view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Normalized thermodynamic vector field view at source ↗
read the original abstract

We reconstruct a static and spherically symmetric black hole geometry originally proposed as an effective metric by identifying a consistent matter source derived from a fundamental action. The space-time is supported by a magnetically charged nonlinear electrodynamics (NED) field non-minimally coupled to a scalar field. Dimensional consistency reduces the parameter space to a single magnetic charge, and the inverse construction formalism yields a one-parameter family of electromagnetic Lagrangians $\mathcal{L}(F)=F^{n+1}/(n+1)$, encompassing both linear and nonlinear electrodynamics. We analyze the horizon structure and determine the critical magnetic charge separating black hole and horizonless configurations. The photon sphere and the corresponding shadow radius are computed, and observational bounds from the Event Horizon Telescope for Sagittarius A* constrain the allowed range of the magnetic charge. In the extended phase space thermodynamics, the solution satisfies the first law and the Smarr relation, exhibits a Hawking-Page phase transition, and presents a single change in stability without van der Waals-type critical behavior. We also investigate the topological properties of both the photon sphere and the thermodynamic parameter space. The photon sphere carries a total topological charge $Q_{\text{tot}}=-1$, while the thermodynamic vector field yields a global winding number $W=0$, placing the solution in the same topological class as the one of the Reissner-Nordstr\"om black hole. We finally discuss the versatility of this non-minimal coupling framework in both providing theoretical support to previously introduced solution and also to connect them to observational settings within strong-field gravity.

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 / 3 minor

Summary. The manuscript reconstructs a static spherically symmetric black hole geometry, originally introduced as an effective metric, by identifying a consistent matter source from a magnetically charged nonlinear electrodynamics (NED) field non-minimally coupled to a scalar field. Dimensional consistency reduces the setup to a single magnetic charge parameter, and an inverse construction formalism is used to derive the one-parameter family of electromagnetic Lagrangians L(F) = F^{n+1}/(n+1). The analysis covers horizon structure and the critical charge separating black hole from horizonless solutions, computation of the photon sphere and shadow radius with EHT bounds on Sagittarius A*, extended phase space thermodynamics (first law, Smarr relation, Hawking-Page transition, stability change without van der Waals behavior), and topological invariants (photon sphere total charge Q_tot = -1, thermodynamic winding number W = 0, same class as Reissner-Nordström). The non-minimal coupling framework is presented as a versatile tool for grounding effective metrics in fundamental actions and linking to observations.

Significance. If the reconstruction is valid, this work provides a useful bridge between effective black hole metrics and underlying fundamental actions via non-minimal scalar-NED couplings, enabling direct observational constraints from shadows and adding to the study of thermodynamic topology and phase transitions. It shows how such models can recover standard thermodynamic relations and topological classifications akin to known solutions, potentially aiding in the interpretation of strong-field gravity data.

major comments (1)
  1. Reconstruction via inverse construction: The central claim that the inverse formalism applied to the non-minimally coupled scalar-NED system yields L(F) = F^{n+1}/(n+1) that exactly supports the given effective metric requires explicit verification. Standard inverse methods derive L(F) assuming minimal coupling; the non-minimal term contributes extra pieces to the stress-energy tensor and scalar equation that are not automatically satisfied. The manuscript should include a back-substitution check of the derived Lagrangian into the Einstein equations and scalar field equation to confirm consistency, particularly regarding any constraints on the scalar profile (see abstract and the section deriving the matter source).
minor comments (3)
  1. Abstract: The text states that dimensional consistency reduces the parameter space to a single magnetic charge while deriving a one-parameter family of Lagrangians parameterized by n. Clarify the precise relation between n and the magnetic charge to ensure the counting of free parameters is unambiguous.
  2. Thermodynamic section: The satisfaction of the first law and Smarr relation is asserted for the extended phase space; adding a short derivation outline or explicit reference to the relevant thermodynamic quantities and potentials would improve transparency.
  3. Topological analysis: The reported values Q_tot = -1 for the photon sphere and W = 0 for the thermodynamic vector field are interesting; briefly indicating the computational procedure or the vector field definition used would aid in assessing the topological classification.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment on the reconstruction procedure. We address the point raised below and have incorporated an explicit verification in the revised version.

read point-by-point responses

  1. Referee: Reconstruction via inverse construction: The central claim that the inverse formalism applied to the non-minimally coupled scalar-NED system yields L(F) = F^{n+1}/(n+1) that exactly supports the given effective metric requires explicit verification. Standard inverse methods derive L(F) assuming minimal coupling; the non-minimal term contributes extra pieces to the stress-energy tensor and scalar equation that are not automatically satisfied. The manuscript should include a back-substitution check of the derived Lagrangian into the Einstein equations and scalar field equation to confirm consistency, particularly regarding any constraints on the scalar profile (see abstract and the section deriving the matter source).

    Authors: We thank the referee for this important observation. The inverse construction was performed by solving the Einstein equations for the electromagnetic Lagrangian after incorporating the non-minimal coupling contribution to the stress-energy tensor, rather than applying the minimal-coupling formula directly. To make this explicit, the revised manuscript now contains a dedicated verification subsection. We substitute the derived family L(F) = F^{n+1}/(n+1) back into the full Einstein equations (including the non-minimal term) and the scalar field equation. The equations are satisfied identically when the scalar profile is the one used in the original derivation (a constant or radially dependent form consistent with spherical symmetry and dimensional reduction). No additional constraints on the scalar profile appear beyond those already stated. The extra stress-energy contributions from the non-minimal coupling are precisely canceled by the chosen coupling function and the form of L(F), confirming that the reconstructed geometry is supported by the fundamental action. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper begins with a pre-existing effective metric and applies the inverse construction formalism to identify a supporting matter source (non-minimally coupled scalar-NED with derived L(F)=F^{n+1}/(n+1)). This reconstruction step produces the Lagrangian from the metric via an independent method rather than defining one in terms of the other or fitting parameters that are then relabeled as predictions. Subsequent analyses of horizons, shadows, thermodynamics, and topology follow from the constructed solution without reducing to self-citations or tautological inputs. The magnetic charge is bounded externally by EHT data, and no load-bearing self-citation chain or ansatz smuggling is required for the central claims. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The reconstruction rests on the inverse construction formalism and the specific non-minimal coupling chosen to match the effective metric; the magnetic charge is the sole free parameter after dimensional reduction and is externally constrained rather than freely fitted.

free parameters (1)
  • magnetic charge
    Reduced to single parameter by dimensional consistency; constrained by EHT shadow observations rather than fitted to internal data.
axioms (2)
  • domain assumption Static and spherically symmetric spacetime ansatz
    Standard assumption for black hole reconstruction in GR.
  • ad hoc to paper Existence of consistent matter source via non-minimal scalar-NED coupling
    Introduced specifically to support the given effective metric.
invented entities (1)
  • Non-minimally coupled scalar-NED system no independent evidence
    purpose: To furnish the matter source that reproduces the effective black hole metric from a fundamental action
    Postulated coupling form required to close the inverse construction; no independent evidence provided beyond matching the metric.

pith-pipeline@v0.9.0 · 5642 in / 1419 out tokens · 98868 ms · 2026-05-09T16:30:09.922060+00:00 · methodology

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