arxiv: 2603.09159 · v2 · submitted 2026-03-10 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Shadows of quintessence black holes: spherical accretion, photon trajectories, and geodesic observers

Ji-Wen Li , Zi-Liang Wang , Tao-Tao Sui

Authors on Pith no claims yet

Pith reviewed 2026-05-15 14:13 UTC · model grok-4.3

classification 🌀 gr-qc

keywords quintessence black holesblack hole shadowsspherical accretiongeodesic observersphoton spherecritical impact parameterM87*relativistic aberration

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The pith

In quintessence black holes, freely infalling observers measure smaller shadow angular radii than static ones at the same location, while outgoing observers measure larger ones.

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

The paper shows that quintessence fields modify black-hole spacetimes so they are not asymptotically flat, making the usual assumption of a static observer at infinity invalid. Using perturbative methods, the authors obtain analytic expressions for the horizon, photon sphere, ISCO, and critical impact parameter, then compute intensity profiles for spherical accretion flows viewed by different observers. They find that the photon sphere and critical impact parameter remain fixed properties of the spacetime, yet the apparent angular size of the shadow changes with observer velocity according to relativistic aberration. Applying the results to Event Horizon Telescope data on M87*, they obtain tighter bounds on the quintessence parameter when the equation-of-state parameter is more negative, and these bounds hold across different observer choices.

Core claim

Although the photon-sphere radius and critical impact parameter are invariant features of the spacetime, the observed angular size of the shadow depends sensitively on the observer’s motion and location. Freely infalling observers systematically record smaller angular radii than static observers at the same coordinate radius, while freely outgoing observers record larger radii, consistent with aberration. In contrast to the Schwarzschild case, the impact parameter alone does not fully determine the observed angular structure. When the model is fitted to the Event Horizon Telescope image of M87*, more negative equations of state produce stronger upper limits on the quintessence parameter, and

What carries the argument

Geodesic observers (freely infalling or outgoing) whose four-velocity alters the apparent angular size of the photon-sphere shadow via relativistic aberration in non-asymptotically flat quintessence spacetimes.

If this is right

The angular radius of the shadow is smaller for infalling observers and larger for outgoing observers at fixed coordinate radius.
The impact parameter by itself is insufficient to characterize the observed angular structure of the shadow.
More negative equations of state produce stronger constraints on the quintessence parameter from M87* data, independent of the chosen observer prescription.
Analytical expressions are provided for the event horizon, photon-sphere radius, ISCO, and critical impact parameter as functions of the quintessence parameter.

Where Pith is reading between the lines

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

Future very-long-baseline interferometry arrays could exploit the observer-motion dependence to distinguish quintessence models from other modifications that preserve asymptotic flatness.
The same observer-velocity effect would appear in any non-asymptotically flat spacetime, suggesting that shadow analyses of black holes in de Sitter or other cosmological backgrounds should also specify the observer’s four-velocity.
If accretion flows deviate from perfect spherical symmetry, the reported difference between infalling and static observers may be amplified or reduced, providing a testable signature in polarized intensity maps.

Load-bearing premise

The perturbative expansion used to obtain analytic expressions for the horizon, photon sphere, and critical impact parameter remains valid for the relevant range of the quintessence parameter, and the accretion flow is assumed to be purely spherical and steady-state.

What would settle it

A high-resolution measurement of the M87* shadow angular radius that is inconsistent with the predicted difference between static and infalling observer prescriptions for any value of the quintessence parameter.

Figures

Figures reproduced from arXiv: 2603.09159 by Ji-Wen Li, Tao-Tao Sui, Zi-Liang Wang.

**Figure 2.** Figure 2: FIG. 2: Left panel: Angular radius of the photon sphere observed by static observers at different radii for several [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: Difference in the total emission coefficient, [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Metric time component [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5: Difference in the redshift factor [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6: Same as Fig. 5, but now with the location of [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

**Figure 9.** Figure 9: Overall, the qualitative behavior closely resem [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7: Observed intensity [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8: Same as Fig. 7, but with the static observer located at a fixed radius [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9: Observed intensity [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10: Angular radius of the photon sphere observed by different observers located at various radii. Left panel: [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗

**Figure 11.** Figure 11: FIG. 11: Relative deviation of the angular radius of the [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗

read the original abstract

The presence of a quintessence-like field can influence the black hole shadow through three primary mechanisms: the dynamics of accretion flows, the trajectories of photons, and the motion of observers. Unlike standard shadow analyses that assume a static observer at spatial infinity, the non-asymptotically flat nature of quintessence-corrected spacetimes motivates the consideration of freely falling (geodesic) observers. Using a perturbative approach, we derive analytical expressions for the event-horizon location, photon-sphere radius, innermost stable circular orbit, and critical impact parameter. We compute the observed intensity profiles for both static and infalling spherical accretion flows. We find that, although the photon-sphere radius and the critical impact parameter are invariant properties of the spacetime, the apparent angular size of the shadow depends sensitively on the observer's motion and location. Freely infalling observers systematically measure smaller angular radii than static observers at the same radius, whereas freely outgoing observers measure larger ones, in agreement with relativistic aberration. In contrast to the Schwarzschild case, the impact parameter alone is insufficient to characterize the observed angular structure in non-asymptotically flat spacetimes. Applying our results to the Event Horizon Telescope observation of M87$^\ast$, we show that more negative equations of state lead to stronger constraints on the quintessence parameter, largely independent of the observer prescription. Our analysis highlights the importance of carefully specifying the observer in shadow studies of non-asymptotically flat black-hole spacetimes.

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 paper shows observer motion via aberration changes apparent shadow size in quintessence spacetimes, with EHT bounds on the parameter staying roughly independent of observer choice, but the perturbative expressions need validation for the relevant parameter range.

read the letter

The core result is that infalling geodesic observers measure smaller shadow angular radii than static ones at fixed coordinate radius, while outgoing ones measure larger, as expected from aberration. In these non-asymptotically flat metrics the usual critical impact parameter no longer directly sets the observed angle, so the observer four-velocity has to be tracked explicitly. They derive perturbative analytic forms for the horizon, photon sphere, ISCO, and critical impact parameter, then compute intensity profiles for both static and infalling spherical accretion flows before applying the setup to M87* data. The claim that more negative equations of state tighten the bounds on the quintessence parameter, largely independent of observer prescription, follows directly from those expressions. That observer dependence and the resulting EHT application are the concrete additions beyond standard static-at-infinity analyses. The derivations stay internally consistent within the perturbative framework and reproduce the expected aberration effect. The spherical-accretion idealization is handled cleanly enough to produce usable profiles. The main limitation is the perturbative expansion itself. If the quintessence strength values that shift the shadow by the EHT uncertainty already push the expansion parameter outside its validity range, both the reported angular-size differences and the claimed independence of constraints become unreliable. No numerical cross-checks or error estimates on the analytic forms are supplied, so the quantitative bounds rest on an untested assumption about the size of the correction term. The spherical steady-state flow is another standard but restrictive choice. This work is for people modeling black-hole shadows in modified gravity who need to account for observer motion in non-flat asymptotics. A reader focused on EHT constraints or geodesic observers will find usable expressions and a clear application. It deserves peer review because the central claim is specific, the math is explicit, and the EHT comparison is direct, even though referees will need to verify the expansion range and accretion assumptions.

Referee Report

1 major / 2 minor

Summary. The paper analyzes shadows of quintessence black holes using a perturbative approach to derive analytical expressions for the event horizon, photon-sphere radius, ISCO, and critical impact parameter. It computes observed intensity profiles for static and infalling spherical accretion flows, shows that geodesic observer motion alters the apparent shadow angular size via relativistic aberration, and applies the results to EHT M87* data to constrain the quintessence parameter, finding stronger bounds for more negative equations of state that are largely independent of observer choice.

Significance. If the perturbative regime holds for quintessence strengths that shift the shadow by amounts comparable to EHT uncertainties, the work usefully demonstrates the necessity of specifying observer motion in non-asymptotically flat spacetimes and supplies concrete, observer-robust constraints on the quintessence equation of state from existing shadow observations.

major comments (1)

[perturbative derivations of r_h, r_ph, b_crit and § on M87* application] The perturbative expansions for r_ph, b_crit and related quantities (used throughout the intensity calculations and M87* constraints) are not accompanied by an explicit validity check or error estimate for the parameter values that produce ~10% shadow-size deviations. Because the central claim of observer-independent constraints rests on these analytic forms, their accuracy in the relevant regime must be demonstrated, for example by direct numerical root-finding of the effective potential or by reporting the size of higher-order terms.

minor comments (2)

[Introduction and §3] Notation for the quintessence parameter and equation-of-state w should be introduced once with a clear definition and then used consistently; occasional redefinitions in later sections obscure the comparison between static and infalling cases.
[figures showing intensity vs. impact parameter] Figure captions for the intensity profiles should explicitly state the radial location and four-velocity of each observer so that the aberration effect can be reproduced without returning to the text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The single major comment concerns the lack of explicit validation for our perturbative expansions in the regime relevant to the M87* constraints. We address this point directly below and will incorporate the requested checks in the revised manuscript.

read point-by-point responses

Referee: [perturbative derivations of r_h, r_ph, b_crit and § on M87* application] The perturbative expansions for r_ph, b_crit and related quantities (used throughout the intensity calculations and M87* constraints) are not accompanied by an explicit validity check or error estimate for the parameter values that produce ~10% shadow-size deviations. Because the central claim of observer-independent constraints rests on these analytic forms, their accuracy in the relevant regime must be demonstrated, for example by direct numerical root-finding of the effective potential or by reporting the size of higher-order terms.

Authors: We agree that an explicit accuracy assessment strengthens the paper. In the revised manuscript we will add a dedicated subsection (or appendix) that directly compares the perturbative expressions for r_ph, b_crit, and the critical impact parameter against numerical root-finding of the effective-potential equation for the precise range of the quintessence parameter that produces ~10% shadow-size shifts. We will report the relative truncation errors (including the magnitude of the next-order terms) and show that they remain well below the EHT uncertainty level for the values used in the M87* analysis. This numerical validation will be performed for both the static and infalling observer cases to confirm that the observer-independent nature of the resulting constraints is not an artifact of the perturbative truncation. revision: yes

Circularity Check

0 steps flagged

Derivations from modified metric and geodesic equations remain independent of final M87* constraints

full rationale

The paper starts from the quintessence-corrected metric and applies standard geodesic equations to obtain perturbative analytic forms for r_h, r_ph, and b_crit. Observer-dependent angular radii are then computed via relativistic aberration applied to the critical impact parameter. M87* constraints follow from direct numerical comparison of these shadow sizes to EHT data, without any refitting of the quintessence parameter inside the derivation itself. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the chain. The perturbative assumption is stated as an explicit limitation rather than smuggled in via citation.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The analysis rests on the standard quintessence-modified Schwarzschild metric, perturbative expansions around small quintessence strength, and the idealization of spherical accretion; the quintessence parameter and equation-of-state parameter are constrained rather than freely fitted inside the derivation.

free parameters (2)

quintessence parameter
Strength of the quintessence field, constrained by comparison with M87* data rather than fitted inside the derivation.
equation of state w
Parameter controlling the quintessence field behavior, varied to obtain different constraint strengths.

axioms (2)

domain assumption General relativity coupled to a quintessence scalar field yielding a specific non-asymptotically flat metric
Invoked throughout the derivation of horizons, photon spheres, and geodesics.
domain assumption Spherical, steady-state accretion flow with given emissivity profile
Used to compute observed intensity profiles for static and infalling cases.

invented entities (1)

quintessence field no independent evidence
purpose: To source the modified spacetime metric around the black hole
Standard extension of GR to model dark-energy effects; no new independent evidence supplied in the paper.

pith-pipeline@v0.9.0 · 5570 in / 1622 out tokens · 75278 ms · 2026-05-15T14:13:02.471061+00:00 · methodology

discussion (0)

Forward citations

Cited by 4 Pith papers

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Reference graph

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