arxiv: 2604.05646 · v1 · submitted 2026-04-07 · 🌀 gr-qc · hep-th

Recognition: 2 theorem links

· Lean Theorem

Thermodynamics, Phase Transitions, and Geodesic Structure of F(R)-Phantom BTZ Black Holes

Behzad Eslam Panah , Bilel Hamil , Manuel E. Rodrigues

Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:58 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords F(R) gravityphantom BTZ black holesphase transitionEhrenfest relationsgeodesicsnonlinear electrodynamicsthermodynamicsstability

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The pith

F(R) phantom BTZ black holes confirm second-order phase transitions via Ehrenfest relations and support stable orbits in phantom regimes.

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

This paper aims to demonstrate that phantom BTZ black holes in F(R) gravity with a conformally invariant Maxwell source undergo a second-order phase transition at their critical point. The authors first construct the solutions and verify that their thermodynamic quantities obey the first law of thermodynamics. They then analyze stability through heat capacity and Gibbs free energy, and rigorously check that the Ehrenfest relations hold at the critical point. The geodesic analysis further shows that stable circular orbits for massive particles and photons are possible specifically in the phantom field case for backgrounds with negative curvature.

Core claim

The core discovery is that these F(R)-phantom BTZ black holes satisfy both Ehrenfest equations at the critical point, confirming a second-order phase transition in the thermodynamic system. The thermodynamic potentials are derived from the metric and shown to be consistent. For the orbital dynamics, the effective potential for timelike and null geodesics reveals that stable circular orbits for test particles exist only in the phantom regime when the curvature is negative, and the phantom configuration also supports stable photon circular orbits.

What carries the argument

The key machinery consists of the F(R)-modified action coupled to phantom power-Maxwell electrodynamics, which determines the black hole metric used for both thermodynamic calculations and geodesic effective potentials.

If this is right

The system exhibits a second-order phase transition concurrent with the critical point.
Stable timelike circular orbits exist only in the phantom regime for negative curvature backgrounds.
The phantom configuration allows for stable circular photon orbits.
Stability regions in canonical and grand canonical ensembles are affected by the phantom field and scalar curvature.

Where Pith is reading between the lines

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

The phase transition verification could extend to other modified gravity models with exotic matter sources.
Stable orbits in the phantom case might influence the dynamics of matter accretion and the resulting observational signatures.
Further study of the geodesic structure could connect to the black hole shadow or lensing effects in such spacetimes.

Load-bearing premise

The particular choice of the F(R) function together with the phantom sign in the nonlinear electrodynamics must generate solutions that fulfill the first law and Ehrenfest relations without introducing unaccounted instabilities.

What would settle it

A direct computation at the critical point that shows one of the Ehrenfest relations is violated, or an effective potential analysis that finds no stable circular orbit for phantom parameters in negative curvature.

Figures

Figures reproduced from arXiv: 2604.05646 by Behzad Eslam Panah, Bilel Hamil, Manuel E. Rodrigues.

**Figure 2.** Figure 2: FIG. 2: The total mass [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: The heat capacity [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: The Gibbs potential [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5: Effective potential [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6: Effective potential curve of timelike geodesic and trajecto [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7: Effective potential curve of null geodesic and trajectorie [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗

read the original abstract

This paper investigates phantom BTZ black holes within the high-curvature gravity theory framework, specifically using a special case of power-Maxwell theory, which functions as a nonlinear electrodynamics source called $F(R)-$conformally invariant Maxwell gravity. We examine how the phantom or anti-Maxwell field affects the structure of these black holes and how the theory's parameters influence their horizon structure. Additionally, we derive the conserved and thermodynamic potentials associated with these black holes, thereby establishing their conformance to the foundational first law of thermodynamics. Next, the stability characteristics of BTZ black holes endowed with phantom and Maxwell fields are explored under canonical and grand canonical ensemble conditions by inspecting their heat capacity and Gibbs free energy profiles. This assessment reveals how the phantom field and scalar curvature affect these stability regions. We then perform a rigorous analytical verification of the Ehrenfest equations to determine whether the critical behavior of the phantom BTZ black hole corresponds to a second-order phase transition. Our results demonstrate adherence to both Ehrenfest relations, thereby confirming the occurrence of a second-order phase transition within the black hole system concurrent with the critical point. Furthermore, we explore the geodesic structure of the obtained solutions to analyze the motion of massive and massless test particles in the $F(R)$-phantom BTZ spacetime. The analysis demonstrates that stable timelike circular orbits exist only in the phantom regime for negative curvature backgrounds, while the phantom configuration also allows for stable circular photon orbits. These results underscore the significant influence of the phantom field and the $F(R)$ correction on the spacetime geometry and orbital dynamics.

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

This paper verifies Ehrenfest relations for second-order transitions in F(R)-phantom BTZ black holes and maps geodesics, but the entropy handling needs checking.

read the letter

The key takeaway is that the authors analytically confirm both Ehrenfest equations hold at the critical point for these F(R)-phantom BTZ solutions, which they tie to a second-order phase transition, and they show stable timelike orbits only in the phantom regime with negative curvature while photon orbits can stabilize under phantom conditions. They also derive thermodynamic potentials that satisfy the first law and examine stability via heat capacity and Gibbs free energy in canonical and grand canonical ensembles. This builds directly on earlier BTZ and phantom field work by adding the F(R) piece and the explicit Ehrenfest check plus orbit analysis. The derivations appear to be done by hand from the metric, which is a plus for transparency in this setting. The geodesic section gives concrete conditions on parameters for stable orbits, which is useful for anyone modeling particle motion in these spacetimes. The main soft spot is the entropy. F(R) gravity modifies the Wald entropy by an F'(R) factor at the horizon, so any derivatives entering the Ehrenfest ratios or heat capacity should incorporate that. The abstract claims analytic satisfaction of the relations, but it is not clear from the summary whether they used the corrected entropy or fell back on the standard area law. If the latter, the match could be an artifact rather than a property of the modified theory. The parameter choices for F(R) and the phantom strength also look tuned to produce the desired horizons and stability windows, which is common but limits how general the conclusions are. This work is aimed at researchers in 3D modified gravity and black hole thermodynamics who already follow the BTZ literature. A reader looking for concrete examples of phase transitions or orbit stability in phantom setups will find usable results, though the scope stays narrow. It is solid enough on its own terms to warrant a serious referee, even if the entropy step requires clarification or correction in revision.

Referee Report

2 major / 2 minor

Summary. The paper derives solutions for phantom BTZ black holes in F(R) gravity with power-Maxwell nonlinear electrodynamics, obtains thermodynamic potentials and verifies the first law, analyzes stability via heat capacity and Gibbs free energy in canonical and grand canonical ensembles, analytically confirms both Ehrenfest relations at the critical point to establish a second-order phase transition, and examines geodesic motion to identify regimes with stable timelike circular orbits (only in phantom regime for negative curvature) and stable photon orbits.

Significance. If the thermodynamic derivations are correct, the work extends black hole thermodynamics in modified gravity by providing an explicit check of Ehrenfest relations for phase transitions and mapping geodesic stability to the phantom and curvature parameters. The analytical (rather than numerical) verification of the two Ehrenfest equations and the orbit classification are potentially useful contributions if the entropy is the Wald entropy appropriate to F(R).

major comments (2)

[Thermodynamic potentials and Ehrenfest verification] Thermodynamic section (first law and Ehrenfest verification): In F(R) gravity the Wald entropy is S = (A/4G) F'(R) evaluated at the horizon, so the derivatives entering the Ehrenfest relations (∂S/∂T, ∂²S/∂T², etc.) differ from the pure area-law expressions. The manuscript asserts analytic satisfaction of both Ehrenfest equations at the heat-capacity divergence point, yet gives no indication that the F'(R) factor was inserted before differentiation. If the standard BTZ area entropy was used instead, the reported equality of the two ratios would be an artifact rather than a property of the F(R)-phantom solution.
[Stability analysis and critical behavior] Stability and critical-point analysis: The F(R) coupling and phantom-field strength are free parameters that appear to have been chosen to produce the reported horizon structure, heat-capacity divergence, and stability regions. The paper should state the explicit ranges of these parameters for which the solutions satisfy the null energy condition and the first law with the correct Wald entropy; without such constraints the phase-transition claim risks being parameter-tuned rather than generic.

minor comments (2)

[Geodesic structure] Geodesic section: The effective potential for timelike and null geodesics should be written explicitly (including the metric function f(r) and the conserved quantities E and L) so that the conditions for stable circular orbits can be reproduced from the given expressions.
[Metric solution] Notation: The distinction between the phantom and Maxwell cases is sometimes blurred in the text; consistently label the sign flip of the Maxwell term and its effect on the metric function.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major point below and will incorporate clarifications and additions in the revised version.

read point-by-point responses

Referee: Thermodynamic section (first law and Ehrenfest verification): In F(R) gravity the Wald entropy is S = (A/4G) F'(R) evaluated at the horizon, so the derivatives entering the Ehrenfest relations (∂S/∂T, ∂²S/∂T², etc.) differ from the pure area-law expressions. The manuscript asserts analytic satisfaction of both Ehrenfest equations at the heat-capacity divergence point, yet gives no indication that the F'(R) factor was inserted before differentiation. If the standard BTZ area entropy was used instead, the reported equality of the two ratios would be an artifact rather than a property of the F(R)-phantom solution.

Authors: We agree that explicit use of the Wald entropy is essential in F(R) gravity. The derivations in the manuscript employed S = (A/4) F'(R) (with G=1) for the entropy, and the analytic verification of both Ehrenfest relations was performed with this expression, including the F'(R) dependence in all temperature derivatives. However, the intermediate differentiation steps were not displayed in sufficient detail, which may have obscured this. In the revised manuscript we will insert the explicit steps showing how F'(R) enters ∂S/∂T and the second derivatives, confirming that the reported satisfaction of the Ehrenfest equations is not an artifact. revision: yes
Referee: Stability and critical-point analysis: The F(R) coupling and phantom-field strength are free parameters that appear to have been chosen to produce the reported horizon structure, heat-capacity divergence, and stability regions. The paper should state the explicit ranges of these parameters for which the solutions satisfy the null energy condition and the first law with the correct Wald entropy; without such constraints the phase-transition claim risks being parameter-tuned rather than generic.

Authors: We accept this criticism. The manuscript presents results for specific choices of the F(R) coupling and phantom strength that yield the reported thermodynamics and geodesics, but does not delineate the full allowed intervals. In the revised version we will add a dedicated paragraph (or subsection) specifying the ranges of these parameters that simultaneously satisfy the null energy condition, ensure a positive Hawking temperature, and permit the first law to hold with the Wald entropy. This will clarify the domain in which the second-order phase transition and stability features are physically realized rather than tuned. revision: yes

Circularity Check

0 steps flagged

No significant circularity; thermodynamic and geodesic derivations are independent of inputs

full rationale

The paper selects a specific F(R) form and phantom Maxwell source, solves the field equations for the metric, then computes thermodynamic quantities (mass, temperature, entropy) directly from that metric to verify the first law. Heat capacity and Gibbs free energy are obtained from these potentials, the critical point is located via heat-capacity divergence, and Ehrenfest relations are checked analytically at that point. Geodesic equations follow from the same metric. None of these steps reduce by construction to the input ansatz or to a fitted parameter; the Ehrenfest verification is a non-trivial consistency test on the derived potentials rather than an identity. No self-citations are load-bearing, no uniqueness theorems are invoked, and no renaming of known results occurs. The chain is therefore self-contained against the metric solution.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The central claim rests on the standard 3D Einstein-Hilbert action modified by an F(R) term and a power-Maxwell nonlinear electrodynamics source with a phantom sign flip; these extensions introduce free parameters whose values control horizon structure and stability without independent external calibration.

free parameters (2)

F(R) coupling constant
Controls the strength of the higher-order curvature correction in the gravitational action.
Phantom field strength parameter
Determines the magnitude and sign flip of the anti-Maxwell contribution to the energy-momentum tensor.

axioms (2)

domain assumption The spacetime is asymptotically AdS with BTZ-like topology in three dimensions.
Standard background assumption for BTZ black hole solutions.
domain assumption The first law of black hole thermodynamics holds for the derived mass, entropy, and potentials.
Invoked and claimed to be verified but treated as foundational for the thermodynamic analysis.

invented entities (1)

Phantom field no independent evidence
purpose: Provides a source with negative kinetic energy to modify black hole horizons, thermodynamics, and geodesics.
Introduced via the nonlinear electrodynamics term with a sign flip relative to standard Maxwell fields.

pith-pipeline@v0.9.0 · 5596 in / 1873 out tokens · 101024 ms · 2026-05-10T19:58:29.250055+00:00 · methodology

discussion (0)

Reference graph

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(13), can inc lude an event horizon when R0 is negative

Our analysis shows that the solutions obtained in Eq. (13), can inc lude an event horizon when R0 is negative. This leads to two distinct behaviors for BTZ black holes depending on w hether R0 is negative or positive: 1- For R0 > 0: The solutions yield a real root that does not correspond to the eve nt horizon. Therefore, for R0 > 0, no BTZ black hole sol...
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