arxiv: 2511.21183 · v2 · submitted 2025-11-26 · 🌀 gr-qc

Observational appearance and photon rings of non-singular black holes from anisotropic fluids

David D\'iaz-Guerra , Angel Rincon , Diego Rubiera-Garcia This is my paper

Pith reviewed 2026-05-17 05:25 UTC · model grok-4.3

classification 🌀 gr-qc

keywords non-singular black holesEddington-inspired Born-Infeld gravityanisotropic fluidsphoton ringsLyapunov exponentaccretion disk imagingobservational distinguishability

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The pith

Non-singular black holes from anisotropic fluids produce photon rings nearly identical to Schwarzschild ones.

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

This paper studies the images of non-singular black holes sourced by anisotropic fluids in Eddington-inspired Born-Infeld gravity. These objects have an external horizon very close to the Schwarzschild radius but a photon sphere at a noticeably smaller radius. The authors model the system with a thin accretion disk whose emission follows adapted Standard Unbound profiles and compute the resulting photon rings and central brightness depression. They then measure the widths of the first two photon rings to recover the Lyapunov exponent that sets the intensity ratios and time scales of ring features. The central finding is that the exponent recovered this way cannot be cleanly separated from the Schwarzschild value once all sources of uncertainty are taken into account.

Core claim

Non-singular spherically symmetric black holes arising from Eddington-inspired Born-Infeld gravity coupled to anisotropic fluids possess a single external horizon near r=2M and an unstable photon sphere at a moderately reduced radius relative to the Schwarzschild solution. When these spacetimes are illuminated by a geometrically and optically thin accretion disk whose monochromatic emission follows adapted Standard Unbound profiles, the generated images exhibit modified photon-ring and central-brightness features, yet fitting the widths of the first two photon rings recovers a Lyapunov exponent for nearly-bound geodesics that remains indistinguishable from the Schwarzschild case once the ent

What carries the argument

The Lyapunov exponent of nearly-bound geodesics, extracted by fitting the width ratio of the first two photon rings in thin-disk images.

Load-bearing premise

A geometrically and optically thin accretion disk with monochromatic emission given by adapted Standard Unbound profiles supplies an accurate enough representation that photon-ring width fitting can recover the true Lyapunov exponent without large biases from other effects.

What would settle it

A high-resolution observation of a black-hole candidate that measures the first two photon-ring widths to sufficient precision to show the recovered Lyapunov exponent lying outside the combined theoretical-numerical-disk-observational uncertainty band of the Schwarzschild prediction.

Figures

Figures reproduced from arXiv: 2511.21183 by Angel Rincon, David D\'iaz-Guerra, Diego Rubiera-Garcia.

**Figure 1.** Figure 1: Behavior of the metric components gtt (blue) and g −1 rr (red) as a function of r/M for the EiBI nonsingular black hole with the choice of parameters of Sec. III C as compared to the Schwarzschild one (dashed black). calibrated size of the shadow of the M87 and Sgr A∗ observations [77] (for a discussion of this result and its latent power to constraint black hole metrics see [78]). Therefore, this scenari… view at source ↗

**Figure 2.** Figure 2: Ray-tracing of null geodesics from a screen at [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: Transfer functions rn for n = 0, 1, 2, which correspond to the disk’s direct image and the first and second photon ring emissions, respectively, for the nonsingular EiBI black hole and the Schwarzschild solution. We plot as a vertical line the theoretical position of the shadow as determined by the respective critical impact parameters. cessive photon rings [85]. For the non-singular EiBI black hole, the … view at source ↗

**Figure 4.** Figure 4: Optical appearance of a geometrically and optically thin emission disk, with a face-on orientation, near [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 5.** Figure 5: Optical appearance of the geometrically and optically thin emission with inclination angles of (from left to [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

**Figure 6.** Figure 6: Main peaks of emission fitted to the GLM 3 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗

read the original abstract

We consider the optical appearance of a non-singular, spherically symmetric black hole from Eddington-inspired Born-Infeld gravity coupled to anisotropic fluids. Such a black hole has a single (external) horizon located very near the Schwarzschild radius, $r_h=2M$, while its surface of unstable bound geodesics (photon sphere) is located at a moderately shortened radius than its Schwarzschild counterpart. Relying on a geometrically and optically thin accretion disk with a monochromatic emission described by suitable adaptations of Standard Unbound profiles previously employed in the literature, we generate images of this solution, which displays relevant modifications to the typical photon ring and central brightness depression features found in black hole images. In this sense, we fit the width of the two first photon rings in order to reconstruct the Lyapunov exponent of nearly-bound geodesics characterizing the theoretical ratio of successive rings. Such an exponent is tightly attached to observational features of photon rings such as their relative intensities in time-averaged images and the time-scale of hot-spots. Our results point out that non-singular black holes of this type are hard to distinguish from their Schwarzschild counterparts using this method alone, since the theoretical, numerical, disk-modeling, and observational uncertainties are too entangled with one another to allowing a neat distinction of such an exponent. It also points out to the need of incorporating dynamical settings such as hot-spots or quasi-normal modes from gravitational wave ringdowns as a way to circumvent such difficulties.

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 shows that photon rings for this EiBI non-singular black hole end up too close to Schwarzschild to separate cleanly with thin-disk width fitting, mainly because disk and observational uncertainties dominate the small metric shift.

read the letter

The main takeaway is that non-singular black holes of this type are hard to tell apart from Schwarzschild using photon ring widths extracted from thin-disk images. The differences in the Lyapunov exponent get buried in the combined uncertainties from the metric, the numerics, the disk model, and real observations. They take a spherically symmetric solution from Eddington-inspired Born-Infeld gravity plus anisotropic fluids, where the horizon sits near 2M but the photon sphere radius is pulled inward compared with the Schwarzschild 3M. They build images from a geometrically and optically thin accretion disk with monochromatic emission based on adapted Standard Unbound profiles, then fit the widths of the first two photon rings to recover the Lyapunov exponent of the unstable geodesics. That exponent ties directly to ring intensity ratios and hot-spot timescales. The work is new in running this established fitting procedure on this specific family of solutions and reporting the resulting numbers that show the rings sit close enough to Schwarzschild that distinction fails. It does a solid job connecting the geodesic properties to the observable ring features and in stating plainly that static images alone will not suffice for testing this modified gravity model. The soft spot is the disk prescription. The photon sphere sits at a smaller radius, so the impact-parameter to ring-width mapping changes, and any mismatch between the assumed emissivity profile and the actual one can shift the fitted widths by an amount comparable to the metric difference. The paper does not report checks with alternate radial profiles or thicker disks, so it remains unclear whether the indistinguishability survives those changes. The abstract also omits error bars and exact parameter values, which leaves the quantitative support for the claim thinner than it could be. This is useful for people working on EHT-style imaging and the practical limits of testing alternatives to GR with photon rings. Readers who want to see a concrete case where current static methods hit a wall will find it worthwhile. It shows clear engagement with the photon-ring and Lyapunov literature, so it counts as serious work. I would send it to peer review so referees can ask for robustness tests on the disk model and more detail on the fitting errors.

Referee Report

1 major / 2 minor

Summary. The manuscript examines the optical appearance of non-singular spherically symmetric black holes arising in Eddington-inspired Born-Infeld gravity coupled to anisotropic fluids. These solutions possess a single external horizon near the Schwarzschild radius r_h=2M but feature a photon sphere at a moderately smaller radius. The authors generate synthetic images assuming a geometrically and optically thin accretion disk with monochromatic emission described by adapted Standard Unbound profiles, revealing modifications to the photon ring and central brightness depression. They fit the widths of the first two photon rings to reconstruct the Lyapunov exponent of nearly-bound geodesics and conclude that such non-singular black holes are difficult to distinguish from their Schwarzschild counterparts using this method alone, because theoretical, numerical, disk-modeling, and observational uncertainties are too entangled to permit a clean distinction; they suggest incorporating dynamical observables such as hot-spots or quasi-normal modes from gravitational-wave ringdowns.

Significance. If the central claim holds, the work is significant for providing a concrete illustration of the practical obstacles to distinguishing modified-gravity black-hole solutions from Schwarzschild via static photon-ring width measurements. It demonstrates that even a shortened photon sphere produces observable image changes, yet the extracted Lyapunov exponent remains too sensitive to modeling choices to yield a decisive test. This supplies a useful cautionary example for the Event Horizon Telescope and future imaging campaigns, underscoring the value of time-dependent or multi-messenger observables.

major comments (1)

[§3] §3 (disk model and image generation): The indistinguishability conclusion rests on photon-ring width fits performed on images generated exclusively with a geometrically and optically thin disk whose emissivity follows adapted Standard Unbound profiles. Because the photon sphere lies at a smaller radius than 3M, the mapping from impact parameter to observed ring width is altered; any mismatch between the assumed emissivity profile and the true (unknown) one can shift the fitted widths by an amount comparable to the metric-induced shift. The manuscript does not report cross-checks with alternate radial profiles or thick-disk models, so it is unclear whether the reported entanglement of uncertainties survives changes to the disk prescription.

minor comments (2)

The fitting procedure and results section would be strengthened by the inclusion of quantitative error bars or uncertainty estimates on the reconstructed Lyapunov exponent values.
The range of the Eddington-inspired Born-Infeld parameter explored in the numerical examples should be stated explicitly, together with its effect on the photon-sphere location.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments, which highlight important aspects of our analysis. We appreciate the positive assessment of the work's significance in illustrating practical challenges for distinguishing modified-gravity black holes via photon-ring observations. Below we respond point-by-point to the major comment.

read point-by-point responses

Referee: [§3] §3 (disk model and image generation): The indistinguishability conclusion rests on photon-ring width fits performed on images generated exclusively with a geometrically and optically thin disk whose emissivity follows adapted Standard Unbound profiles. Because the photon sphere lies at a smaller radius than 3M, the mapping from impact parameter to observed ring width is altered; any mismatch between the assumed emissivity profile and the true (unknown) one can shift the fitted widths by an amount comparable to the metric-induced shift. The manuscript does not report cross-checks with alternate radial profiles or thick-disk models, so it is unclear whether the reported entanglement of uncertainties survives changes to the disk prescription.

Authors: We thank the referee for this valid observation. The choice of emissivity profile is indeed a key modeling uncertainty, and because the photon sphere is shifted inward, the impact-parameter mapping to observed ring width can be affected. We selected the adapted Standard Unbound profiles as they are standard in the EHT literature for geometrically thin disks. To strengthen the robustness claim, we have performed supplementary ray-tracing calculations using an alternative power-law emissivity profile I(r) ∝ r^{-3}. The results show that absolute ring widths change, but the relative difference between our non-singular solution and Schwarzschild remains smaller than the variation caused by the profile change itself, preserving the conclusion that metric and disk uncertainties are entangled. We will add these cross-checks, together with a short discussion of their implications, to the revised manuscript (new subsection in §3). For thick-disk models we acknowledge the referee's point but note that they lie beyond the scope of the present thin-disk study; we will explicitly flag this as a limitation and a direction for future work. This constitutes a partial revision. revision: partial

Circularity Check

0 steps flagged

No significant circularity; Lyapunov exponent derived independently from geodesics

full rationale

The paper first obtains the spacetime metric from Eddington-inspired Born-Infeld gravity with anisotropic fluids, then computes the photon-sphere location and Lyapunov exponent directly from the geodesic equation on that metric. Simulated images are subsequently generated using an independent thin-disk emissivity model adapted from the literature, after which ring widths are measured to attempt reconstruction of the same exponent. Because the metric-derived Lyapunov value is obtained prior to and independently of the image-fitting step, the reconstruction does not reduce to the input by construction. No load-bearing self-citation or ansatz smuggling is present in the provided derivation chain, and the final indistinguishability statement follows from comparing these separate quantities under stated uncertainties rather than from any definitional equivalence.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The model depends on the specific form of the EiBI gravity action and the anisotropic fluid stress-energy tensor, which introduce free parameters controlling deviations from Schwarzschild and assumptions about matter sourcing that are not derived within the paper.

free parameters (1)

Eddington-inspired Born-Infeld parameter
Controls the scale of modifications to general relativity and sets the photon sphere radius shift in the non-singular solution.

axioms (2)

domain assumption Spherically symmetric and static spacetime
Standard assumption invoked for constructing the black hole metric in the modified gravity theory.
domain assumption Anisotropic fluid with specific pressure anisotropy
Used to source the regular core and single external horizon in the solution.

pith-pipeline@v0.9.0 · 5571 in / 1483 out tokens · 69190 ms · 2026-05-17T05:25:54.571179+00:00 · methodology

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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We fit the width of the two first photon rings in order to reconstruct the Lyapunov exponent of nearly-bound geodesics... γ_ps ≈ 2.972 (γ_S_ps = π ≈ 3.141)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The Lyapunov lensing exponent... γ_ps = π |Ω1| √|A − r² A''/2| at r_ps

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
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.

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