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arxiv: 2508.00624 · v1 · pith:XJRFTEYRnew · submitted 2025-08-01 · 🌀 gr-qc

Gravitational Lensing in the Schwarzschild Spacetime: Photon Rings in Vacuum and in the Presence of a Plasma

Torben C. Frost This is my paper

Pith reviewed 2026-05-22 00:32 UTC · model grok-4.3

classification 🌀 gr-qc
keywords gravitational lensingphoton ringsSchwarzschild black holeinhomogeneous plasmaaccretion diskmultifrequency observationsnull geodesics
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The pith

Inhomogeneous plasma around a Schwarzschild black hole changes the size and shape of photon rings in a frequency-dependent way that multifrequency observations can use to measure plasma properties.

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

This paper constructs an analytic model of a pressureless, non-magnetised plasma whose electron density rises toward the equatorial plane near a non-rotating black hole. It integrates the null geodesic equations exactly, places a distant observer, and maps the direct image plus the first- and second-order photon rings onto the celestial sphere. The resulting lens equation, redshifts, and travel times differ from the vacuum and uniform-plasma cases. A sympathetic reader cares because real black-hole images contain light that has passed through surrounding gas; frequency-dependent ring distortions therefore supply a direct probe of that gas without assuming uniformity.

Core claim

For the chosen analytic plasma model the geodesic equations integrate in closed form. An orthonormal tetrad at the observer converts the constants of motion into latitude-longitude coordinates on the celestial sphere. The direct image and the first- and second-order photon rings therefore occupy frequency-dependent locations and shapes. These locations are compared with the vacuum and homogeneous-plasma limits, and the redshift and travel time are computed explicitly. The structural differences that appear are shown to be usable for extracting plasma properties from multifrequency data.

What carries the argument

Analytic integration of the null geodesic equations for the specific inhomogeneous plasma density profile, which yields explicit relations between impact parameters and observer angles.

If this is right

  • The first-order photon ring contracts or expands with observing frequency in a manner distinct from the vacuum case.
  • The second-order ring exhibits an even larger relative shift, amplifying the observable signature of the density gradient.
  • Redshift and travel-time differences between frequencies provide independent observables that can be combined with ring geometry.
  • Comparison with homogeneous-plasma results isolates the effect of the equatorial density increase.

Where Pith is reading between the lines

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

  • The same analytic approach could be tested against numerical ray-tracing codes that use more realistic density profiles.
  • If the model is approximately correct, existing black-hole images could be re-analyzed at multiple frequencies to place limits on plasma gradients.
  • Extension to Kerr spacetime would be a natural next step to assess whether spin changes the frequency dependence.

Load-bearing premise

The plasma electron density is given by a functional form that allows the geodesic equations to be integrated completely in closed form.

What would settle it

Multifrequency images of photon rings around a known-mass black hole that show no measurable change in ring radius or shape between frequencies would contradict the predicted effect of the inhomogeneous plasma model.

Figures

Figures reproduced from arXiv: 2508.00624 by Torben C. Frost.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic of the lens equation. In our case we have a [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Lens maps for the direct images in the Schwarzschild spacetime for light rays travelling in vacuum (top left panel), [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Lens maps for the first-order photon rings in the Schwarzschild spacetime for light rays travelling in vacuum (top left [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Lens maps for the second-order photon rings in the Schwarzschild spacetime for light rays travelling in vacuum (top left [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Lens maps for the direct images in the Schwarzschild spacetime for light rays with [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Lens maps for the first-order photon rings in the Schwarzschild spacetime for light rays with [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Lens maps for the second-order photon rings in the Schwarzschild spacetime for light rays with [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Redshift maps for the direct images in the Schwarzschild spacetime for light rays travelling in vacuum (top left panel), [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Redshift maps for the first-order photon rings in the Schwarzschild spacetime for light rays travelling in vacuum [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Redshift maps for the second-order photon rings in the Schwarzschild spacetime for light rays travelling in vacuum [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Travel time maps for the direct images in the Schwarzschild spacetime for light rays travelling in vacuum (top left [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Travel time maps for the first-order photon rings in the Schwarzschild spacetime for light rays travelling in vacuum [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Travel time maps for the second-order photon rings in the Schwarzschild spacetime for light rays travelling in vacuum [PITH_FULL_IMAGE:figures/full_fig_p030_13.png] view at source ↗
read the original abstract

Astrophysical black holes are usually surrounded by an accretion disk. At least parts of these accretion disks consist of a plasma in which light rays with different energies are dispersed. However, we usually do not know the exact configurations of these plasmas. In this paper we will now use the example of a Schwarzschild black hole embedded in an inhomogeneous pressureless and nonmagnetised plasma to investigate how the structural changes of the photon rings can help us to determine the properties of a plasma surrounding a black hole using multifrequency observations. For this purpose we will use a simple analytic model which describes a plasma whose electron density increases towards the equatorial plane when we approach the event horizon. For the chosen model we will first derive and then analytically solve the equations of motion. Then we will place an observer in the domain of outer communication and introduce an orthonormal tetrad to relate the constants of motion to latitude-longitude coordinates on the observer's celestial sphere. In the next step we will use the analytic solutions to investigate the geometric structures of the direct image as well as the photon rings of first and second order on the observer's celestial sphere. We will write down a lens equation and calculate the redshift and the travel time. We will compare the obtained results to results for photon rings in vacuum and in the presence of a homogeneous plasma. Finally, we will discuss which of these quantities can be used to extract information about the properties of the plasma.

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

2 major / 3 minor

Summary. The manuscript examines gravitational lensing by a Schwarzschild black hole surrounded by an inhomogeneous, pressureless, non-magnetized plasma whose electron density increases toward the equatorial plane near the horizon. It derives the plasma-modified equations of motion, claims to solve them analytically, places an observer in the outer domain, introduces an orthonormal tetrad to map constants of motion to celestial-sphere coordinates, computes the direct image and first- and second-order photon rings, writes a lens equation, and evaluates redshift and travel time. These quantities are compared with the vacuum and homogeneous-plasma cases to argue that multifrequency observations of ring-structure changes can extract plasma properties.

Significance. If the analytic solutions are valid, the work supplies an explicit, parameter-free example in which plasma inhomogeneity produces traceable modifications to photon-ring geometry on the observer’s sky. This is directly relevant to interpreting multifrequency EHT or ngEHT data on black-hole shadows and rings. The absence of fitted parameters and the explicit comparison to vacuum and homogeneous cases are strengths that enhance the falsifiability of the proposed extraction method.

major comments (2)
  1. [§3] §3 (derivation of equations of motion): the central claim that the chosen n_e(r,θ) permits fully analytic integration rests on the separability of the Hamilton-Jacobi equation once the ω_p²(r,θ)/ω² term is included. The manuscript must demonstrate explicitly that no residual r–θ coupling survives after the two Killing constants are imposed; otherwise the reported closed-form r(λ) and θ(λ) are at best approximate and the subsequent lens equation and ring-coordinate expressions in §4 lose their analytic traceability.
  2. [§4] §4 (photon rings and lens equation): the argument that structural changes in the first- and second-order rings can be used to extract plasma parameters via multifrequency observations presupposes that the analytic solutions of §3 remain valid across the relevant impact-parameter range. Without explicit checks against limiting cases (equatorial plane, large-r asymptotics, or homogeneous-plasma reduction) or error estimates on the integration, the extraction claim is not yet load-bearing.
minor comments (3)
  1. [Abstract] The abstract states that the equations are “derived and solved analytically” but does not indicate the domain of validity or any approximations; this should be stated explicitly.
  2. [§2] Clarify the precise definition of the orthonormal tetrad and the mapping from conserved quantities to observer latitude-longitude coordinates; a short appendix or figure would help.
  3. [§5] Travel-time and redshift expressions should be compared numerically to the vacuum case for at least one concrete set of plasma parameters to illustrate the magnitude of the effect.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for highlighting its potential relevance to multifrequency EHT observations. We address each major comment below and will incorporate the requested clarifications and validations into a revised version.

read point-by-point responses

  1. Referee: [§3] §3 (derivation of equations of motion): the central claim that the chosen n_e(r,θ) permits fully analytic integration rests on the separability of the Hamilton-Jacobi equation once the ω_p²(r,θ)/ω² term is included. The manuscript must demonstrate explicitly that no residual r–θ coupling survives after the two Killing constants are imposed; otherwise the reported closed-form r(λ) and θ(λ) are at best approximate and the subsequent lens equation and ring-coordinate expressions in §4 lose their analytic traceability.

    Authors: We agree that an explicit verification of separability strengthens the presentation. The plasma density profile n_e(r,θ) was specifically constructed so that the term ω_p²(r,θ)/ω², when combined with the two Killing constants, produces an additive separation of the Hamilton-Jacobi equation into purely r-dependent and θ-dependent parts with no cross terms remaining. In the revised manuscript we will insert a short subsection in §3 that substitutes the conserved quantities, cancels all mixed r–θ contributions term by term, and arrives at the decoupled ordinary differential equations whose solutions are the reported closed-form expressions for r(λ) and θ(λ). revision: yes

  2. Referee: [§4] §4 (photon rings and lens equation): the argument that structural changes in the first- and second-order rings can be used to extract plasma parameters via multifrequency observations presupposes that the analytic solutions of §3 remain valid across the relevant impact-parameter range. Without explicit checks against limiting cases (equatorial plane, large-r asymptotics, or homogeneous-plasma reduction) or error estimates on the integration, the extraction claim is not yet load-bearing.

    Authors: We accept that additional validation improves the robustness of the extraction argument. The revised §4 will contain direct comparisons of the analytic ring coordinates and lens equation against three limiting cases: (i) the equatorial-plane restriction, (ii) the large-r asymptotic expansion, and (iii) the exact reduction to the homogeneous-plasma solution. We will also report the maximum relative deviation obtained when the analytic expressions are cross-checked against numerical integration of the geodesic equations over the impact-parameter interval relevant to the first- and second-order rings, thereby quantifying the accuracy of the closed-form results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained from metric and plasma ansatz

full rationale

The paper begins with the Schwarzschild metric and a postulated analytic electron-density profile for the pressureless plasma, derives the modified geodesic equations, and states that this specific profile permits explicit integration in quadratures. It then constructs the observer tetrad, lens equation, and redshift expressions directly from those solutions without any fitted parameters, self-citation chains, or uniqueness theorems that reduce the outputs to the inputs by construction. The reported ring coordinates and structural changes are computed quantities, not redefinitions or statistical fits of the same data. This matches the default expectation of an independent derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Schwarzschild vacuum solution, the geometric-optics limit for light in a cold plasma, and the specific functional form chosen for the electron density; no new particles or forces are introduced.

axioms (2)
  • domain assumption Light propagation obeys the geometric-optics limit in a cold, pressureless, non-magnetised plasma whose refractive index depends only on electron density.
    Invoked to reduce the problem to null geodesics modified by a position-dependent refractive index.
  • standard math The background spacetime is exactly the Schwarzschild metric.
    Standard vacuum solution of Einstein's equations used throughout.

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