arxiv: 2605.01303 · v3 · submitted 2026-05-02 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Photon Spheres and shadow of modified black-hole entropies

Fang Liu , Yu-Bo Ma , Yun-Zhi Du , Huai-Fan Li

Authors on Pith no claims yet

Pith reviewed 2026-05-14 21:34 UTC · model grok-4.3

classification 🌀 gr-qc

keywords black hole entropy correctionsphoton sphereblack hole shadowfirst law of thermodynamicsSgr A*Event Horizon Telescopemodified metric

0 comments

The pith

A first-law correspondence maps black-hole entropy corrections to modified metrics that shift photon-sphere radii and shadow sizes.

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

The paper derives an explicit link between modified black hole entropies and altered spacetime metrics by applying the first law while holding the black hole's total energy and horizon radius fixed. From the resulting metric, it calculates the radius of the photon sphere and the apparent size of the black hole shadow. Different forms of entropy correction produce different shifts in these optical features. Matching the computed shadow sizes against Event Horizon Telescope images of Sagittarius A* then restricts the allowed range of the correction parameters. This supplies a way to test whether black hole entropy deviates from the standard area law using real observations.

Core claim

Starting from the first law of black hole thermodynamics with fixed energy and horizon position, an explicit correspondence is established between the corrected entropy and the metric function. The corrected metric is then used to compute photon sphere radius and shadow size, and comparison with EHT observations of Sgr A* constrains the parameter range in the corrected entropy.

What carries the argument

The explicit correspondence between corrected entropy and metric function obtained from the first law under fixed black-hole energy and horizon position.

If this is right

Different entropy corrections produce distinct shifts in photon sphere radius and shadow size.
EHT observations of Sgr A* constrain the allowed parameter ranges for those corrections.
The method supplies an observational test for generalized entropy frameworks that deviate from the Bekenstein-Hawking area law.

Where Pith is reading between the lines

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

The same fixed-energy, fixed-horizon mapping could be applied to other families of black-hole solutions or other thermodynamic corrections.
The resulting parameter bounds could be compared directly with specific quantum-gravity models that predict particular forms of entropy modification.
Higher-resolution shadow data from future instruments could test whether the assumption of fixed energy and horizon continues to hold.

Load-bearing premise

The corrected entropy can be mapped onto a modified metric function while the black hole energy and horizon radius remain exactly fixed.

What would settle it

An independent measurement of the shadow size of Sgr A* that falls outside the range predicted by the allowed parameter values for the entropy corrections.

Figures

Figures reproduced from arXiv: 2605.01303 by Fang Liu, Huai-Fan Li, Yu-Bo Ma, Yun-Zhi Du.

**Figure 1.** Figure 1: FIG. 1: The view at source ↗

**Figure 2.** Figure 2: presents the rps∆ − ∆ curve and bps∆ − ∆ curve when M = 1, 0 ≤ ∆ ≤ 1. It can be seen from view at source ↗

**Figure 3.** Figure 3: FIG. 3: The view at source ↗

**Figure 4.** Figure 4: presents the rpsλ 3M −χ curve and bpsλ 3M −χ curve when M = 1. It can be seen from view at source ↗

**Figure 5.** Figure 5: FIG. 5: The view at source ↗

**Figure 6.** Figure 6: presents the rpsy 3M − y(β) curve and bpsy 3M − y(β) curve when M = 1, x(α) = 0. It can be seen from rpsy 3 M -y(β) bpsy 3 M -y(β) A(0.030,1.516) B(0.049,1.403) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 1.0 y(β) 1.2 1.4 1.6 1.8 2.0 rpsy 3 M / bpsy 3 M FIG. 6: the rpsy 3M − y(β) curve and bpsy 3M − y(β) curve when M = 1 view at source ↗

**Figure 7.** Figure 7: FIG. 7: the view at source ↗

**Figure 8.** Figure 8: FIG. 8: The view at source ↗

**Figure 9.** Figure 9: FIG. 9: the view at source ↗

**Figure 10.** Figure 10: FIG. 10: The view at source ↗

read the original abstract

Starting from the first law of black hole thermodynamics, we establish an explicit correspondence between the corrected entropy and the metric function under the condition of fixed black hole energy and horizon position. Using the corrected metric, we further compute the photon sphere radius and shadow size, demonstrating that different entropy corrections lead to characteristic optical shifts. By comparing with the Event Horizon Telescope observations of Sgr A*, we constrain the parameter range introduced in the corrected entropy. This provides a feasible approach for testing generalized entropy frameworks and probing deviations from the Bekenstein-Hawking area law.

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 mapping from corrected entropy to metric at fixed M and r_h does not determine the full f(r) profile needed for photon-sphere and shadow calculations.

read the letter

The paper starts from the first law with a corrected entropy, holds black-hole mass and horizon radius fixed, and claims this gives an explicit metric function. It then computes the photon-sphere radius and shadow size for that metric and fits the correction parameter to the EHT shadow radius for Sgr A* to get a bound on the entropy modification strength. That is the whole contribution in a nutshell. The observational step is routine once the metric is written down, and the numbers they report are consistent with small deviations from Schwarzschild. The part that is actually new is the specific entropy-to-metric step under the fixed-M, fixed-r_h condition; earlier papers on generalized entropies stopped at thermodynamics and did not push the resulting metric into strong-field observables. That extension is modest but concrete, and it is the only reason the work reaches the EHT data at all. The soft spot is exactly where the stress-test note says it is. Fixing M and r_h plus the first law only pins down the surface gravity at the horizon. It supplies no information on f(r) or f'(r) at radii outside r_h. Photon-sphere location is set by the condition 2f(r) - r f'(r) = 0, and the critical impact parameter is evaluated there; both quantities require the global shape of f(r). Without an additional, independently motivated ansatz for the metric function away from the horizon, the shadow radius is not fixed by the entropy correction alone. The paper must be inserting some extra assumption at that point. If that assumption is stated clearly and checked for consistency with the field equations, the fit to Sgr A* becomes a legitimate consistency check. If it is left implicit, the bound on the correction parameter is circular. The work is aimed at people who already track modified-entropy proposals and black-hole shadow phenomenology. It is not a major advance, but it is a direct attempt to turn thermodynamic corrections into an observable. I would send it to referees provided the full derivation of f(r) is written out and the consistency checks are shown; otherwise it is too thin for a serious review.

Referee Report

2 major / 2 minor

Summary. The paper claims to derive an explicit correspondence between corrected black-hole entropies and modified metric functions f(r) by applying the first law dM = T dS while holding black-hole energy M and horizon radius r_h fixed. From the resulting metrics it computes photon-sphere radii and shadow sizes for several entropy corrections, then constrains the correction coefficient by matching the predicted shadow radius to the EHT measurement for Sgr A*.

Significance. If the entropy-to-metric mapping can be placed on a firm footing, the approach would offer a concrete route to test deviations from the Bekenstein-Hawking area law using shadow observables. The work is timely given current EHT data, but its impact is limited by the under-determination of the global metric profile required for photon-sphere calculations.

major comments (2)

[§2 (entropy-metric correspondence)] The central construction (detailed after the statement of the first-law correspondence) fixes only the horizon temperature via surface gravity while holding M and r_h constant. This supplies no information on f'(r) for r > r_h, yet the photon-sphere condition 2f(r) − r f'(r) = 0 and the critical impact parameter both require the full radial profile of f(r). Without an additional, independently motivated ansatz for f(r), the subsequent shadow-radius predictions are not uniquely determined by the entropy correction.
[§4 (EHT comparison)] The parameter constraint obtained by fitting the computed shadow radius to the single Sgr A* datum is effectively a consistency check rather than an independent test: the same coefficient enters both the metric deformation and the fit. This circularity weakens the claim that EHT data meaningfully bounds the entropy-correction coefficient.

minor comments (2)

Notation for the entropy correction coefficient is introduced without a dedicated symbol table; a short table listing all modified quantities would improve readability.
Figure captions should explicitly state the numerical values of the correction parameter used in each curve.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and indicate where revisions will be made to clarify the assumptions and strengthen the presentation.

read point-by-point responses

Referee: [§2 (entropy-metric correspondence)] The central construction (detailed after the statement of the first-law correspondence) fixes only the horizon temperature via surface gravity while holding M and r_h constant. This supplies no information on f'(r) for r > r_h, yet the photon-sphere condition 2f(r) − r f'(r) = 0 and the critical impact parameter both require the full radial profile of f(r). Without an additional, independently motivated ansatz for f(r), the subsequent shadow-radius predictions are not uniquely determined by the entropy correction.

Authors: We agree that the first-law relation with fixed M and r_h determines only the horizon temperature (and thus f'(r_h)) via the surface-gravity definition. The manuscript adopts the standard ansatz that the metric remains asymptotically flat with f(r_h)=0 and that the entropy correction enters by rescaling the effective gravitational potential while preserving the Schwarzschild-like form outside the horizon. This ansatz is motivated by the requirement of matching the corrected thermodynamics at the horizon and is commonly employed in studies of modified black-hole metrics. We will add an explicit paragraph in §2 of the revised manuscript stating this ansatz and its motivation, thereby making the global profile used for the photon-sphere calculation fully transparent. revision: yes
Referee: [§4 (EHT comparison)] The parameter constraint obtained by fitting the computed shadow radius to the single Sgr A* datum is effectively a consistency check rather than an independent test: the same coefficient enters both the metric deformation and the fit. This circularity weakens the claim that EHT data meaningfully bounds the entropy-correction coefficient.

Authors: We respectfully disagree that the procedure is circular. The correction coefficient is an independent theoretical parameter introduced by the chosen entropy modification. Computing the shadow radius as a function of this coefficient and comparing it with the EHT datum for Sgr A* yields an observational upper bound on the coefficient's magnitude; this is the standard method of constraining new-physics parameters. Nevertheless, we acknowledge that the result is model-dependent and will revise the discussion in §4 to present the outcome explicitly as a constraint on the entropy-correction parameter rather than a model-independent test of the underlying framework. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds from first law to metric to observables without definitional reduction

full rationale

The paper starts from the first law with a corrected entropy, imposes fixed M and r_h to obtain a modified metric function, computes photon-sphere radius and shadow size from that metric, and finally compares the resulting shadow radius to the single EHT datum for Sgr A* to bound the correction parameter. None of these steps reduces by construction to its own inputs: the metric is obtained from an explicit thermodynamic relation rather than by redefining the target observable, the shadow calculation uses the standard geodesic condition on the derived f(r), and the final comparison is an external observational constraint rather than a fit that is then relabeled as a prediction. No self-citation load-bearing step, uniqueness theorem, or ansatz smuggling is present in the provided derivation chain. The construction is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the first law of black-hole thermodynamics, the assumption that entropy corrections can be absorbed into a metric deformation at fixed energy and horizon radius, and the validity of the standard null-geodesic equations for photon spheres in the modified metric. One free parameter (the coefficient of the entropy correction) is introduced and later bounded by data.

free parameters (1)

entropy correction coefficient
The amplitude of the non-area term in the corrected entropy; its value is left free and later constrained by matching the computed shadow radius to Sgr A* data.

axioms (2)

domain assumption First law of black-hole thermodynamics holds for the corrected entropy
Invoked at the opening step to relate entropy correction to metric change.
ad hoc to paper Energy and horizon radius remain fixed while the metric is deformed
Explicitly stated condition used to obtain an explicit metric function from the entropy correction.

pith-pipeline@v0.9.0 · 5388 in / 1659 out tokens · 32853 ms · 2026-05-14T21:34:02.616928+00:00 · methodology

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation washburn_uniqueness_aczel contradicts

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contradicts
CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

we take f(r,α,β,⋯)=∂S_BH/∂S f(r) … under the premise that r+=2M remains unchanged
IndisputableMonolith/Foundation/RealityFromDistinction reality_from_one_distinction contradicts

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contradicts
CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

Constrained by Eq.(27), the range of the parameter Δ … 0≤Δ≤0.004 (1σ)

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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