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arxiv: 2605.01426 · v1 · submitted 2026-05-02 · 🌀 gr-qc

Recognition: unknown

A note on methods for computing the critical curve of Kerr-like black holes

Siddharth Kumar Sahoo , Indrani Banerjee
Authors on Pith no claims yet

Pith reviewed 2026-05-09 18:23 UTC · model grok-4.3

classification 🌀 gr-qc
keywords black hole shadowcritical curveKerr black holesplasmacelestial coordinatestetradsgravitational lensing
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The pith

Bardeen's definition of the black hole critical curve gives smaller sizes than de Vries's or Grenzebach et al.'s at finite observer distances and in homogeneous plasma.

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

This paper compares three definitions of the critical curve for Kerr-like black holes using celestial coordinates. The definitions produce the same curve for vacuum black holes or those in inhomogeneous plasma when the observer is very far away. They differ when the observer sits at a finite distance, with Bardeen's version always the smallest. In homogeneous plasma Bardeen's curve deviates even at large distances and shrinks further as plasma density rises, opposite to the behavior from the other two definitions and from gravitational lensing calculations. The authors also derive one definition from the assumption of a critical curve on the sky plane and show that switching tetrads only shifts the plotted curve sideways.

Core claim

All three celestial coordinate definitions of the critical curve agree for black holes in vacuum or surrounded by inhomogeneous plasma when viewed from large distances. They diverge for observers at finite distance, where Bardeen's definition yields the smallest curve and de Vries's the largest. When homogeneous plasma is present, Bardeen's definition deviates from the other two even at large distances and the curve contracts compared with the vacuum case as plasma density increases, in contradiction with de Vries's and Grenzebach et al.'s definitions and with prior lensing studies. De Vries's definition is derived by assuming the critical curve lies on the observer's sky plane, which also

What carries the argument

Celestial coordinate definitions of the critical curve (Bardeen's, de Vries's, and Grenzebach et al.'s) that map unstable photon orbits onto the observer's sky plane.

If this is right

  • Bardeen's critical curve is the smallest of the three for any finite-distance observer.
  • In homogeneous plasma Bardeen's curve contracts with rising density even when the observer is far away.
  • De Vries's and Grenzebach et al.'s definitions remain consistent with each other and with gravitational lensing results.
  • Changing from the Bardeen tetrad to the Carter tetrad shifts the plotted critical curve horizontally but does not alter its size for Schwarzschild or Kerr black holes in plasma.

Where Pith is reading between the lines

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

  • Observers using different definitions will infer different black-hole parameters when fitting shadow data from finite-distance or plasma-filled regions.
  • The contradiction with prior lensing studies suggests that one family of definitions may be unsuitable for homogeneous plasma.
  • Explicit choice of coordinate definition will be required in future shadow modeling codes to avoid systematic offsets.

Load-bearing premise

The plasma is perfectly homogeneous and the chosen observer distance plus tetrad introduce no additional effects that would make the three definitions agree.

What would settle it

A ray-tracing calculation of photon orbits around a Kerr black hole in uniform-density plasma, performed for a distant observer, that measures whether the critical curve size grows, shrinks, or stays the same relative to the vacuum case.

Figures

Figures reproduced from arXiv: 2605.01426 by Indrani Banerjee, Siddharth Kumar Sahoo.

Figure 1
Figure 1. Figure 1: A schematic representation of Grenzebach et al. [ view at source ↗
Figure 2
Figure 2. Figure 2: A schematic diagram representing celestial angle view at source ↗
Figure 3
Figure 3. Figure 3: In figures 3a and 3b, O represents the position of the observer and B represents the position of the black hole. B is the origin of the of the Cartesian coordinate system on the observer’s sky plane. The X and Y axes are parallel and anti-parallel to the e µ (ϕ) and e µ (θ) , respectively. In figure 3a, the angular coordinates (x1, y1) corresponding to the point (X1, Y1) are shown. In figure 3b the angular… view at source ↗
Figure 4
Figure 4. Figure 4: Above figure shows the critical curve of a Schwarzschild black hole computed us view at source ↗
Figure 5
Figure 5. Figure 5: Above figure shows the critical curve of a near extremal Kerr black hole com view at source ↗
Figure 6
Figure 6. Figure 6: Above figure shows the critical curve of a Schwarzschild black hole computed us view at source ↗
Figure 7
Figure 7. Figure 7: Above figure shows the critical curve of a near extremal Kerr black hole com view at source ↗
read the original abstract

This study systematically compares Bardeen's, de Vries's, and Grenzebach et al.'s celestial coordinate definitions of the critical curve ("shadow") of Kerr-like black holes. We find that all three definitions agree for black holes in vacuum or surrounded by inhomogeneous plasma observed from large distances. However, they diverge for observers located at a finite distance: Bardeen's definition yields the smallest critical curve, while de Vries's yields the largest. When homogeneous plasma is considered, critical curve computed using Bardeen's definition deviates from the other two even at large distances and contracts compared to the vacuum case with increasing plasma density. This is in clear contradiction with the behaviour predicted by de Vries's, Grenzebach et al.'s definitions, and previous gravitational lensing studies. We derive de Vries's definition assuming a critical curve on the observer's sky plane and explain its discrepancy with Grenzebach et al.'s definition. We further explore the effect of the change of tetrad on the critical curve. Using Bardeen and Carter tetrads, we plot the critical curve for Schwarzschild and Kerr black holes in the presence of plasma, highlighting that tetrad changes introduce only a horizontal shift in the critical curve.

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 / 3 minor

Summary. The paper systematically compares Bardeen's, de Vries's, and Grenzebach et al.'s celestial coordinate definitions of the critical curve (shadow) for Kerr-like black holes. It reports agreement among the definitions in vacuum and for inhomogeneous plasma when observed from large distances, but discrepancies at finite observer distances (Bardeen smallest, de Vries largest). For homogeneous plasma, Bardeen's definition deviates even at large distances, with the curve contracting as plasma density increases; this contradicts the other two definitions and prior gravitational lensing results. The paper derives de Vries's definition from a sky-plane assumption, explains its difference from Grenzebach et al., and shows that tetrad changes (Bardeen vs. Carter) produce only a horizontal shift in plots for Schwarzschild and Kerr cases with plasma.

Significance. If the reported discrepancies and the homogeneous-plasma contraction hold after addressing the asymptotic limit, the work is significant for clarifying definitional choices in black-hole shadow calculations in plasma, directly relevant to EHT data interpretation. The systematic side-by-side comparison, explicit derivations, and concrete plots for Schwarzschild/Kerr cases constitute a useful methodological contribution that can prevent inconsistent results in future studies.

major comments (1)
  1. [homogeneous plasma discussion] The central claim that Bardeen's definition deviates from de Vries and Grenzebach even at large distances for homogeneous plasma (and contracts with increasing density) is load-bearing for the paper's main result. However, the computations appear to be performed at finite (though large) r_obs; because homogeneous plasma extends to infinity, an explicit analytic or numerical demonstration of the r_obs → ∞ limit is required to rule out a residual finite-distance effect. Without this, the reported contradiction with prior lensing studies may not be intrinsic to the definitions.
minor comments (3)
  1. [Abstract] The abstract states that the definitions 'agree for black holes in vacuum or surrounded by inhomogeneous plasma observed from large distances' but does not specify the numerical value or criterion used for 'large distances'; adding this would improve reproducibility.
  2. [derivation section] The derivation of de Vries's definition (assuming a critical curve on the observer's sky plane) is presented clearly, but a short side-by-side comparison table of the three coordinate expressions would help readers see the exact algebraic origin of the discrepancy with Grenzebach et al.
  3. [tetrad exploration] The statement that tetrad changes introduce 'only a horizontal shift' is useful; confirming this holds for all plotted cases (including the homogeneous-plasma Kerr example) would strengthen the claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed and constructive review. The concern regarding the need for an explicit r_obs → ∞ demonstration in the homogeneous plasma case is well taken, and we address it below while committing to a revision that strengthens this aspect of the manuscript.

read point-by-point responses

  1. Referee: The central claim that Bardeen's definition deviates from de Vries and Grenzebach even at large distances for homogeneous plasma (and contracts with increasing density) is load-bearing for the paper's main result. However, the computations appear to be performed at finite (though large) r_obs; because homogeneous plasma extends to infinity, an explicit analytic or numerical demonstration of the r_obs → ∞ limit is required to rule out a residual finite-distance effect. Without this, the reported contradiction with prior lensing studies may not be intrinsic to the definitions.

    Authors: We agree that an explicit demonstration of the asymptotic limit is required to confirm the discrepancy is intrinsic to the definitions rather than a finite-distance artifact. In the revised manuscript we will add numerical results computed at substantially larger observer radii (r_obs up to 10^5 M and beyond) to show convergence of the reported trends. We will also include a short analytic argument for the r_obs → ∞ limit, derived from the asymptotic form of the null geodesic equations in uniform plasma, demonstrating that Bardeen's celestial coordinates continue to produce a contracted curve while de Vries and Grenzebach definitions remain consistent with the vacuum case. This addition will directly address the potential inconsistency with prior lensing studies and reinforce the central claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity; comparisons rest on independent literature definitions and explicit derivations

full rationale

The paper systematically compares three pre-existing celestial coordinate definitions (Bardeen, de Vries, Grenzebach et al.) drawn from the literature, derives de Vries's version from the assumption of a critical curve on the observer sky plane, and numerically explores tetrad dependence via explicit plots for Schwarzschild and Kerr cases in plasma. No equations reduce a claimed result to a fitted parameter or self-referential input by construction; no load-bearing self-citations are invoked to justify uniqueness or ansatzes; all reported discrepancies follow directly from the stated definitions and coordinate choices without circular reduction. The homogeneous-plasma deviation at large but finite distance is a computed outcome, not a definitional tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard general relativity for light propagation in curved spacetime with plasma but introduces no new free parameters or postulated entities.

axioms (1)
  • domain assumption Standard general relativity framework for null geodesics in Kerr-like spacetimes with refractive plasma
    Invoked throughout the comparisons of critical curve definitions.

pith-pipeline@v0.9.0 · 5515 in / 1165 out tokens · 40861 ms · 2026-05-09T18:23:16.998817+00:00 · methodology

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Reference graph

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