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arxiv: 2604.01521 · v2 · submitted 2026-04-02 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Probing Black Hole Thermodynamics and Microstructure via the Shadow of Sagittarius A*

Jose Miguel Ladino , Carlos E. Romero-Figueroa , Hernando Quevedo
Authors on Pith no claims yet

Pith reviewed 2026-05-13 21:26 UTC · model grok-4.3

classification 🌀 gr-qc
keywords black hole shadowthermodynamicsmicrostructureSagittarius A*phase transitionsGeometrothermodynamicsReissner-NordstromKerr black hole
0
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The pith

The shadow radius of Sagittarius A* encodes the same phase information as entropy for black hole thermodynamics.

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

This paper establishes a direct link between the observable shadow size of a black hole and its thermodynamic phase structure plus microstructure details. Selecting Geometrothermodynamic metrics derived from enthalpy and mass for Reissner-Nordström and Kerr black holes shows that these metrics match heat capacity singularities. The authors demonstrate that shadow radius carries equivalent phase information to entropy and introduce Shadow-Microstructure diagrams to read off stability and interaction types. Applying the method to Sagittarius A* uses observational shadow bounds to limit both macroscopic parameters and allowed microscopic thermodynamic phases.

Core claim

The shadow radius encodes the same phase information as entropy. Using the first Geometrothermodynamic metric from enthalpy and the second from mass, which correctly reproduce heat capacity singularities, Shadow-Microstructure diagrams are constructed to extract stability and microscopic interaction types, then applied to Sagittarius A* to constrain parameters and phases from observational bounds.

What carries the argument

Shadow-Microstructure diagrams that map shadow radius to thermodynamic stability and interaction types through selected Geometrothermodynamic metrics.

Load-bearing premise

That the chosen Geometrothermodynamic metrics correctly reproduce heat capacity singularities and that shadow radius maps directly to thermodynamic phase structure without additional model-dependent corrections.

What would settle it

A higher-precision measurement of the Sagittarius A* shadow radius that falls outside the ranges predicted for the allowed phases in the Shadow-Microstructure diagram.

Figures

Figures reproduced from arXiv: 2604.01521 by Carlos E. Romero-Figueroa, Hernando Quevedo, Jose Miguel Ladino.

Figure 1
Figure 1. Figure 1: (a) Black hole existence lies above the blue parabola S = πQ2 ; the orange parabola Sm = 3πQ2 is the critical curve of CQ. Local thermodynamic stability at fixed charge occurs between the two curves. (b) Temperature, (c) heat capacity at constant Q, and (d) mass, all expressed in terms of x ≡ S/πQ2 . 4 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Helmholtz free energy of the RN black hole; (b) parametrized entropy branches in terms of the reduced variable x ≡ S/πQ2 . The limits χ → 0 and χ → π correspond to x → 1 (T = 0) and x → 3 (T = Tmax), respectively. The Helmholtz free energy satisfies the thermodynamic identity  ∂F ∂T  Q = −S, (12) which remains finite for the RN black hole. However, when the free energy is expressed as a function of t… view at source ↗
Figure 3
Figure 3. Figure 3: (a) Black hole existence lies above the blue curve S = 2π|J|; the red curve S = Sm is the critical curve of CJ . Local thermodynamic stability at fixed J occurs between the two curves. (a) Temperature, (b) heat capacity at constant J, and (c) mass, all expressed in terms of x ≡ S/2π|J|. at which the system satisfies the universal relation p |J|/M = [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) Helmholtz free energy of the Kerr black hole; (b) entropy branches in terms of x ≡ S/2π|J| [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Reduced GTD scalars and heat capacity of the RN black hole. (a) RI computed from the potential H(S, U), (b) RII and (c) RIII computed from the potential M(S, Q). Gray regions indicate non-physical configurations where T < 0. All plots have been rescaled for clarity. summarized in Table I. The scalar curvatures RI and RII correctly capture the divergence associated with the heat capacity CQ. Moreover, RI pr… view at source ↗
Figure 6
Figure 6. Figure 6: GTD scalars of the RN black hole as functions of the temperature. (a) RI , (b) RII , and (c) comparison of both scalars in the Schwarzschild limit Q → 0. Here, R and A stand for repulsive and attractive interactions, respectively. instability. In the RN case, the problem is sufficiently simple to allow for an analytical treatment. Using the parametrization of the two thermodynamic branches given in Eq. (11… view at source ↗
Figure 7
Figure 7. Figure 7: Reduced GTD scalars and heat capacity of the Kerr black hole. (a) RI computed from the potential H(S, Ω), (b) RII and (c) RIII computed from the potential M(S, J). Gray regions indicate non-physical configurations where T < 0. All plots have been rescaled for clarity. R-Small R-Large 0.5 1.0 1.5 2.0 2.5 -40 -30 -20 -10 0 T/Tmax ℛI Kerr (a) Tmax A-Large A-Small 0.2 0.4 0.6 0.8 1.0 1.2 -5 0 5 10 15 20 25 30 … view at source ↗
Figure 8
Figure 8. Figure 8: GTD scalars of the Kerr black hole as functions of the temperature. (a) RI , (b) RII , and (c) comparison of both scalars in the Schwarzschild limit J → 0. Here, R and A stand for repulsive and attractive interactions, respectively. scaling ansatz [104] R(T) ∼ A (T − Tmax) ζ . (67) RI -L RI -S RII -L RII -S A -0.0018 -0.0020 0.0089 0.0087 ζ 1.0036 0.9960 0.9996 1.0013 TABLE II. Numerical scaling of the red… view at source ↗
Figure 9
Figure 9. Figure 9: Black hole shadows computed using the ray-tracing tool PyHole [141–143], revealing the shadow deformation in rotating spacetimes and the Einstein rings produced by gravitational lensing when the source, black hole, and observer are aligned. In (a), the celestial sphere modeling the ray-tracing background is shown for an equatorial observer (θ0 = π/2) from the interior. The black point indicates the region … view at source ↗
Figure 10
Figure 10. Figure 10: Shadow profiles of the RN black hole: (a) temperature T, (b) heat capacity CQ, (c) GTD scalar RI , and (d) GTD scalar RII . The orange dotted line indicates (a) T(Sm), (b) CQ → ∞, (c) RI → ∞, and (d) RII → ∞, highlighting their correspondence at the same shadow. We use βH = βM = 1, Q = M, and S ∈ [Sext, ∞). T, CQ, RI , and RII are expressed in units of M−1 , M2 , M−2 , and M−2 , respectively. indicates at… view at source ↗
Figure 11
Figure 11. Figure 11: Shadow profiles of the Kerr black hole: (a) temperature T, (b) heat capacity CJ , (c) GTD scalar RI , and (d) GTD scalar RII . The orange dotted line indicates (a) T(Sm), (b) CJ → ∞, (c) RI → ∞, and (d) RII → ∞, highlighting their correspondence at the same shadow. The cyan dotted line in (d) represents RII = 0. We use βH = βM = 1, θ0 = π/2, J = M2 , and S ∈ [Sext, +∞). T, CJ , RI , and RII are expressed … view at source ↗
Figure 12
Figure 12. Figure 12: Shadow–Microstructure diagrams for Sagittarius A* associated with the RN black hole. (a) Shadow–Microstructure diagram for {Rsh, Q, RI}. (b) Shadow–Microstructure diagram for {Rsh, Q, RII}. (c) Temperature T as a function of entropy S, highlighting the different interactions associated with RII and the characteristic points of S. The labels AS, RS, and RL denote the Attractive Small, Repulsive Small, and … view at source ↗
Figure 13
Figure 13. Figure 13: Shadow–Microstructure diagrams for Sagittarius A* associated with the Kerr black hole. (a) Shadow–Microstructure diagram for {Rsh, J, RI}. (b) Shadow–Microstructure diagram for {Rsh, J, RII}. (c) Temperature T as a function of entropy S, highlighting the different interactions associated with RII and the characteristic points of S. AS, RS, and RL denote the Attractive Small, Repulsive Small, and Repulsive… view at source ↗
read the original abstract

We explore the connection between black hole shadows, thermodynamic phase structure, and microstructure of charged and rotating black holes within General Relativity and Geometrothermodynamics. Focusing on Reissner-Nordstr\"om and Kerr solutions, we establish a criterion to select the most suitable Geometrothermodynamic metric for a system, revealing that the first metric from enthalpy and the second from mass correctly reproduce heat capacity singularities. We show that the shadow radius encodes the same phase information as entropy and introduce Shadow-Microstructure diagrams to extract insights into stability and microscopic interaction types directly from observational bounds. Applying this framework to Sagittarius A*, we constrain the macroscopic parameters and the allowed microscopic thermodynamic phases. Our findings indicate that shadow measurements offer a novel probe of thermodynamic and microscopic aspects of black holes, enabling tests of alternative theories of gravity and thermodynamic frameworks.

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 explores connections between black hole shadows, thermodynamic phase structure, and microstructure for Reissner-Nordström and Kerr black holes in GR and Geometrothermodynamics. It selects GTD metrics (from enthalpy and mass) that reproduce heat capacity singularities, claims the shadow radius encodes equivalent phase information to entropy, introduces Shadow-Microstructure diagrams to infer stability and interaction types from observations, and applies the framework to Sagittarius A* to constrain macroscopic parameters and allowed microscopic phases.

Significance. If the mapping from shadow radius to thermodynamic critical points is explicitly validated, the work offers a novel observational route to probe black hole thermodynamics and microstructure using shadow data, with potential to test alternative gravity theories and thermodynamic frameworks against Sgr A* bounds.

major comments (1)
  1. §3 (RN analysis) and §4 (Shadow-Microstructure diagrams): The claim that shadow radius encodes the same phase information as entropy requires an explicit demonstration that heat-capacity singularities are preserved when thermodynamic quantities are re-expressed in terms of r_sh. For RN, the photon-sphere condition r_ph^2 - 3M r_ph + 2Q^2 = 0 is algebraically independent of the horizon r_+ = M + sqrt(M^2 - Q^2), so the map (M,Q) → r_sh does not automatically preserve the loci of C divergences; without this check the diagrams risk mislocating stability boundaries when applied to observational bounds on Sgr A*.
minor comments (2)
  1. Abstract and §2: The precise forms of the two selected GTD metrics (first from enthalpy, second from mass) are referenced but not written explicitly; including the metric tensors or line elements would aid reproducibility.
  2. Figure captions: Several diagrams lack error bands or uncertainty propagation from the Sgr A* shadow radius measurement, which would strengthen the constraint statements.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive comments on our manuscript. We have carefully considered the major concern raised and provide our response below, along with revisions to the manuscript.

read point-by-point responses

  1. Referee: §3 (RN analysis) and §4 (Shadow-Microstructure diagrams): The claim that shadow radius encodes the same phase information as entropy requires an explicit demonstration that heat-capacity singularities are preserved when thermodynamic quantities are re-expressed in terms of r_sh. For RN, the photon-sphere condition r_ph^2 - 3M r_ph + 2Q^2 = 0 is algebraically independent of the horizon r_+ = M + sqrt(M^2 - Q^2), so the map (M,Q) → r_sh does not automatically preserve the loci of C divergences; without this check the diagrams risk mislocating stability boundaries when applied to observational bounds on Sgr A*.

    Authors: We agree that an explicit check is valuable to confirm the preservation of thermodynamic features under the shadow radius mapping. Although the defining equations for the photon sphere and the horizon are independent, the shadow radius r_sh depends on the combination of M and Q in a way that, when inverting the relations to express C(M,Q) as C(r_sh), the singularities remain at the corresponding critical values. In the revised manuscript, we have added an explicit demonstration in Section 3 for the RN case: we derive the expression for the heat capacity in terms of r_sh and verify that the points of divergence coincide with those from the standard thermodynamic analysis. This is achieved by solving for the parameter space and plotting C versus r_sh, confirming no mislocation of stability boundaries. Consequently, the Shadow-Microstructure diagrams in Section 4 remain valid for application to Sgr A* bounds. revision: yes

Circularity Check

0 steps flagged

No circularity: shadow-to-thermodynamics mapping is algebraically independent

full rationale

The derivation computes the shadow radius from the independent photon-sphere equation (r_ph^2 - 3M r_ph + 2Q^2 = 0 for RN) while entropy follows from the horizon area; these are distinct functions of the same parameters, so re-expressing phase structure in terms of r_sh is a genuine change of variables rather than a tautology. GTD metric selection is performed by explicit verification that the chosen metrics reproduce C singularities, not by definition or prior self-citation. No fitted parameters are relabeled as predictions, no uniqueness theorem is imported from the authors' own work to force the result, and the Shadow-Microstructure diagrams are constructed from the observable r_sh bounds without reducing to the input entropy by construction. The chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields limited visibility into parameters and axioms; the work relies on standard general relativity and Geometrothermodynamics assumptions whose independence from the target result cannot be verified here.

axioms (1)
  • domain assumption Selected Geometrothermodynamic metrics reproduce heat capacity singularities for RN and Kerr black holes
    Used as the criterion to choose the enthalpy-based and mass-based metrics.

pith-pipeline@v0.9.0 · 5444 in / 1254 out tokens · 38946 ms · 2026-05-13T21:26:30.317561+00:00 · methodology

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Lean theorems connected to this paper

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  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
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    Relation between the paper passage and the cited Recognition theorem.

    We show that the shadow radius encodes the same phase information as entropy and introduce Shadow-Microstructure diagrams... gI constructed from H(S,I1) and gII from M(S,E1) to correctly reproduce the phase structure encoded in the heat capacity singularities.

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Reference graph

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