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Influence of Observer Inclination and Spacetime Structure on Photon Ring Observables

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Exact photon-ring formulas for non-Kerr black holes: for a pole-on observer, the subring time delay is fixed by spin and shadow size alone, while the azimuthal shift probes frame-dragging.

desk verdict Useful extension of the photon-ring formalism to JP spacetimes, but the unresolved B-dependence of the Lyapunov exponent needs to be fixed before the main claims hold. read the letter →

arxiv 2411.15310 v2 pith:CBTO4RLA submitted 2024-11-22 gr-qc

Kiana Salehi , Rahul Kumar Walia , Dominic Chang , Prashant Kocherlakota This is my paper · ORCID
classification gr-qc PACS 04.70.-s
keywords photonringblackholeshadowJohannsen-PsaltismetricLyapunovexponenttimedelayazimuthalshiftframe-draggingM87*
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper derives closed-form, non-perturbative expressions for the three observables that govern a black hole's photon subrings — the demagnification (Lyapunov) exponent, the time delay between successive subring images, and their azimuthal rotation — for the Johannsen-Psaltis family of stationary, axisymmetric, integrable spacetimes that parameterize deviations from Kerr. The central finding is that, for an observer looking down the spin axis, the time delay is fixed by the spin and the shadow size alone, with all spacetime-deviation dependence cancelling, while the azimuthal shift depends on the ratio of the two metric functions that encode rotation and frame-dragging. That split makes the observables a nearly model-independent probe: combining shadow size, time delay, and azimuthal shift can measure the spin and probe the ergosphere without modeling the accretion flow. The paper shows that the 2017 EHT shadow-size measurement of M87* already predicts a band of allowed time delays, and that a future azimuthal-shift measurement would tighten the constraints on the metric ratio $F/N$. If higher-order photon rings are resolved as expected, these formulas become a direct spectroscopic test of the spacetime near the horizon.

What carries the argument

The engine is the separability of null geodesics in the Johannsen-Psaltis (JP) metric, an axisymmetric spacetime described by four radial functions ($N$, $B$, $F$, $f$) of which only $N$ and $F$ survive in the lensing critical parameters: $f$ is absorbed by the Mino-time parametrization and $B$ is removable by a radial-coordinate redefinition. Because the JP polar potential $\Theta(\theta)$ coincides with Kerr's, the angular half-orbit integrals are identical to the Kerr ones and are evaluated in closed form as complete elliptic integrals of the first, second, and third kinds, $K$, $E$, and $\Pi$. The three critical parameters are assembled from the Mino half-period $\hat{G}_\theta$, the angular integrals $\hat{G}_t$ and $\hat{G}_\varphi$, and the linearized radial Lyapunov rate $\kappa_p = \sqrt{\partial_r^2 \mathcal{R}_p / 2}$, with the demagnification parameter $\gamma_p = \kappa_p \hat{G}_\theta$. For the polar observer the photon shell collapses to the single zero-angular-momentum sphere of radius $r_0 = N_0 / \partial_r N_0$ with shadow radius $I_0 = 1/\partial_r N_0$, reducing the whole apparatus to the two elliptic-integral formulas for $\tau_0$ and $\delta_0$.

What would settle it

Compute null geodesics numerically in a non-separable stationary, axisymmetric spacetime that shares the JP metric functions $N$ and $F$ but has a different polar potential: if the polar-observer subring time delay differs from $2 I_0 E(a^2/I_0^2)$ at the same shadow radius and spin, the closed-form formulas do not generalize. Observational check: resolve the first two photon subrings of a flaring M87* with very-long-baseline interferometry and measure their time delay; a value incompatible with $2 I_0 E(a^2/I_0^2)$ for every spin allowed by the EHT shadow-size measurement would falsify the Kerr-like separability assumption for the central object.

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Extended reading notes

Core claim

Working within the Johannsen-Psaltis metric, whose polar null-geodesic potential is identical to Kerr's, the paper obtains the critical photon-ring parameters as explicit functions of the photon-sphere radius, the spin, and the two deviation functions $N$ and $F$. For a polar observer these reduce to $\gamma_0 = \left[\sqrt{-N^3 \partial_r^2 N / ((\partial_r N)^2 B^2)}\right]_0 \frac{2}{I_0} K(a^2/I_0^2)$, $\tau_0 = 2 I_0 E(a^2/I_0^2)$, and $\delta_0 = a \left(F/(N I_0) - 1\right) (2/I_0) K(a^2/I_0^2) + \pi$, where $I_0$ is the shadow radius and $K$ and $E$ are complete elliptic integrals of the first and second kind. The time delay carries no deviation functions at all, so a shadow-size-plus-time-delay measurement is a direct spin measurement; the azimuthal shift carries the ratio $F/N$ at the polar photon sphere, linking it to frame-dragging and the ergosphere. For small inclinations the time delay remains unchanged to first order, whereas the shadow size and the other parameters acquire corrections that depend on $F$ and its derivative, a regime relevant to M87*'s roughly $17^\circ$ inclination. The expressions reduce properly to the known Kerr and Schwarzschild limits.

Load-bearing premise

The load-bearing premise is that the Johannsen-Psaltis metric admits a Carter constant that makes null geodesics separable, with the polar potential identical to Kerr's, so every elliptic-integral expression for the photon-ring parameters rests on this separability and the paper offers no route to the observables when it fails.

Editorial extensions

If this is right

  • A shadow-size plus polar time-delay measurement fixes the spin directly and without accretion-flow modeling, because $\tau_0 = 2 I_0 E(a^2/I_0^2)$ contains no deviation-function dependence.
  • Adding the azimuthal shift converts the observables into a probe of the metric ratio $F/N$ at the photon sphere — in effect a light-based test of frame-dragging and of whether an ergosphere exists.
  • In the non-spinning limit the parameters collapse to $\gamma_p = \pi \kappa_p / I_p$, $\tau_p = \pi I_p$, and $\delta_p = \pi$, so the time delay measures the shadow radius and successive subring images appear at antipodal points.
  • For M87*'s near-polar inclination (about $17^\circ$), the time delay equals its polar value to first order, so current EHT shadow-size constraints already yield a prediction band for the delay across all spins.
  • A measured azimuthal shift different from $\pi$ necessarily implies a spinning black hole, making $\delta_p$ a spin indicator that does not rely on the shadow-size calibration.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The blindness of all three parameters to the metric functions $f$ and $B$ delimits what photon-ring tests of gravity can certify: two spacetimes differing only in $f$ or $B$ would be indistinguishable by these observables, so other probes such as polarization or extended-source lensing would be needed to close that gap.
  • Because $\tau_0$ is degenerate in spin and shadow size, a time-delay measurement alone cannot distinguish non-Kerr geometries with equal shadow radii; the azimuthal shift is the observable that breaks the degeneracy.
  • The formulas double as a design calculation for future space-VLBI missions: for a target spin and deviation $F/N$, the precision required on $\tau_0$ and $\delta_0$ follows directly from the polar expressions, giving a resolvability criterion before launch.
  • If the first-order inclination independence of $\tau$ holds in practice, near-polar observers see essentially the same ring clock regardless of small pointing uncertainties, which would simplify flare-monitoring campaigns for M87*.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This paper extends the Gralla–Lupsasca photon-ring lensing framework to the Johannsen–Psaltis (JP) family of stationary, axisymmetric, integrable spacetimes. It derives closed-form expressions for the three critical photon-ring parameters—the demagnification Lyapunov exponent \gamma_p, the time delay \tau_p, and the azimuthal shift \delta_p—in Eqs. (31), (32), and (39), specializes them to polar observers in Eq. (45), and analyzes small-inclination corrections in Appendix C. The paper then uses the EHT M87* shadow-size measurement to argue that combining future measurements of time delay and azimuthal shift could jointly constrain the spin and the metric-function ratio F/N, thereby probing frame-dragging and the ergosphere.

Significance. If the derivations are accepted, the paper fills a genuine gap: previous non-Kerr photon-ring studies were mostly restricted to spherical symmetry or to polar observers, whereas here the critical parameters are written for arbitrary observer inclination in a non-perturbative way. The explicit reduction to the known Kerr and spherical limits and the use of the EHT shadow size as an external input rather than a fitted parameter are strengths. The paper also clearly identifies the physical interpretation of F/N as an ergosphere indicator. The main obstacle is the unresolved coordinate-dependence of \gamma_0, which affects the paper's claim that lensing observables constrain only the metric functions N and F. The separability assumption is honestly stated as a scope condition rather than hidden, which I regard as appropriate.

major comments (3)
  1. [§II.A, Eq. (45), footnote 6] The text in §II.A states that 'neither f nor B appear in the expressions for any of the critical parameters' and that photon orbits are determined solely by N and F. This is directly contradicted by Eq. (45), where \gamma_0 contains the factor B^{-2}, and by footnote 6, which concedes that B appears in \gamma_p. Because B can be removed by the radial redefinition d\rho=B\,dr, the physical demagnification exponent should be invariant under this redefinition; however, the combination in Eq. (45) is not manifestly invariant—under d\rho=B\,dr the derivative \partial_r^2 N transforms with an additional B' term, so the expression changes unless a compensating term is included. The paper must either prove the invariance of Eq. (45) or replace it with a manifestly coordinate-invariant definition of \gamma_p. As written, the claim that lensing measurements constrain only N and F is not established for the Lyapunov exponent.
  2. [Eq. (45), second line] The second equality for \delta_0 in Eq. (45) appears algebraically inconsistent with the first. From Eq. (42), N_0 \mathcal{I}_0 = r_0, so F/(N\mathcal{I})_0 = F_0/r_0. The second line instead has F/(r\mathcal{I}_0^2), which is not equal to F_0/r_0 and is also dimensionally suspect, since F has dimension of inverse length and F/(r\mathcal{I}_0^2) would not be dimensionless. Please correct this expression and check whether any appendix or figure uses the incorrect form; Eq. (B1), which solves for F/N, appears to rely on the correct first form.
  3. [§III and Appendix C] The paper's illustrative EHT constraints in Figs. 1 and 2 use \tau_0 and \delta_0, which do not contain B, so the B-dependence of \gamma does not invalidate those specific figures. However, the general framework section promises that all three critical parameters are determined only by N and F. The small-inclination expansion reinforces the problem: Eq. (C7) puts B into \kappa, and Eq. (C9) states that \Delta\kappa depends on B, its derivative, and the third derivative of N. The statement that 'neither f nor B appear' must therefore be corrected globally, and the coordinate-invariance question must be resolved before the general framework can be used as advertised.
minor comments (5)
  1. [Introduction and §IV] The Introduction says 'Section V provides conclusions' but the conclusion is numbered IV; please renumber or fix the cross-reference.
  2. [§III, Fig. 2, Appendix A] There are numerous typographical errors, including 'obsever', 'Schwarzchild', 'purturbative', 'crtical', and 'dependeds'; please proofread the manuscript carefully.
  3. [Eq. (B1)] The notation G_\theta(m) in Eq. (B1) should be \hat{G}_\theta to match the definition in Eq. (44).
  4. [Ref. [2]] The listed arXiv identifier 2311.08680 for the first Sgr A* EHT paper should be verified; it appears to be incorrect or mismatched with the citation.
  5. [§II.A after Eq. (8)] The statement that \Theta(\theta) is identical to the Kerr potential 'due to their asymptotic flatness' is not the correct explanation; the polar potential is identical because the angular metric factor \Sigma cancels in the null geodesic equations for this metric family.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the critical-parameter expressions are derived from the JP null-geodesic equations and checked against external Kerr and spherical benchmarks; the main self-citation is a formalism tool, not a forced premise.

full rationale

The central derivation is self-contained: the critical parameters (31), (32), and (39) follow from the separated null-geodesic equations (5)-(12), the SNG conditions (17), and the linearized radial equation (27), with the polar limits (45) obtained by algebraic substitution into the polar turning-point solution (42)-(44). No target observable is used as an input to derive itself. The EHT shadow size is introduced only as an external observational datum in Fig. 1 and Fig. 2, so the time-delay and azimuthal-shift statements are conditional relations rather than fitted-parameter predictions. The Kerr limit reproduces the independent framework of Ref. [19], and the spherically symmetric limit reproduces Ref. [24], providing external benchmarks. The self-citation to the authors' prior work [33] for the JP reformulation and for polar-orbit formulas is not load-bearing, because those formulas are re-derivable directly from the paper's own Eq. (17) and Eq. (42). The paper's stated limitations—restriction to Carter-separable, integrable spacetimes and lack of a method for non-separable metrics—are scope conditions, not circular premises. One in-text anomaly should be noted: Sec. II.A claims that neither f nor B appears in the critical parameters, while the paper's own Eq. (45) contains an explicit B factor in gamma0, and footnote 6 concedes that B appears there. This is an internal consistency and gauge-invariance concern, not a circularity: it does not reduce any derived result to its own input. Overall, the derivation chain is not circular; the only blemish is a minor self-citation and the unresolved B-dependence of gamma0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numerical parameters are fitted to data in the derivation; the EHT shadow-size measurement is used as an external input. The paper's model freedom sits in the arbitrary JP metric functions N, F, B, which are not determined by a field theory, and the Kerr photon-ring scaling assumptions are inherited from [19]. No new particles, forces, or dimensions are introduced.

assumptions (5)
  • domain assumption The spacetime belongs to the Johannsen-Psaltis metric family with arbitrary radial functions N, F, B and a Carter constant enabling separable null geodesics.
    Section II, metric Eq. (1) and geodesic equations Eq. (5). The paper relies on the JP metric's integrability, citing [46] and [49].
  • domain assumption The polar angular potential Theta(theta) is identical to Kerr's in all JP spacetimes, so polar turning points and Mino half-periods are shared with Kerr.
    Section II A, statement after Eq. (5): Theta is identical in both metrics due to asymptotic flatness; this drives the use of Kerr elliptic integrals.
  • domain assumption The spacetime has a single photon shell containing only unstable spherical null geodesics in the exterior; no stable photon spheres or multiple shells are considered.
    Section II C, footnote 4 restricts to non-Kerr BHs with a single photon shell of unstable SNGs.
  • domain assumption The universal scaling relations for photon subrings, Eq. (26), hold for JP spacetimes: successive rings demagnify exponentially, delay by tau_p, and rotate by delta_p.
    Section II E, Eq. (26) is imported from Gralla-Lupsasca [19] and assumed for the non-Kerr family; the paper defines gamma_p, tau_p, delta_p as these coefficients.
  • domain assumption The 2017 EHT shadow-size measurement of M87* is a valid input for the polar shadow radius, and the M87* inclination of about 17 degrees can be treated as a small perturbation.
    Section III and Appendix C Eq. (C1); the forecasts use rho0 from [10,12,13] and the first-order expansion in inclination.

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Pith. "Pith review of Influence of Observer Inclination and Spacetime Structure on Photon Ring Observables." pith.science (2026). https://pith.science/paper/CBTO4RLA

@misc{pith2026241115310,
  author       = {Pith},
  title        = {Pith review of: Influence of Observer Inclination and Spacetime Structure on Photon Ring Observables},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CBTO4RLA}},
  note         = {Machine review of arXiv:2411.15310}
}
read the original abstract

Recent observations of the near-horizon regions of BHs, particularly the images captured by the Event Horizon Telescope (EHT) collaboration, have greatly advanced our understanding of gravity in extreme conditions. These images reveal a bright, ring-like structure surrounding the central dark area of supermassive BHs, created by the images of unstable photon orbits. As observational capabilities improve, future studies are expected to resolve higher-order rings, providing new opportunities to test gravity through observables such as the Lyapunov exponent, time delay, and azimuthal shift. These observables offer valuable insights into the structure of spacetime, BH properties, and the inclination of the observer. In this study, we employ a non-perturbative and non-parametric framework to examine how these observables change with deviations from the no-hair theorem and varying inclinations. We focus particularly on polar observers, which are highly relevant for the supermassive compact object at the centre of the galaxy M87. Our analysis explores how each of these observables can reveal information about the structure of spacetime and the morphology and existence of the ergosphere and event horizon. Furthermore, we illustrate this characterization for several specific alternative spacetimes, investigating how these current and potential future measurements, including those of the shadow size, can provide direct insights into the spin parameter values for each of these spacetimes.

Figures

Figures reproduced from arXiv: 2411.15310 by the authors.

Figure 1
Figure 1. FIG. 1. Shown in green is the region of combined values of time [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Shown here is a parameter space that captures frame-dragging in BH spacetimes, with the ratio of the metric functions [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

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