REVIEW 1 major objections
A new approximation of photon geodesics in Schwarzschild spacetime
T0 review · 1 major / 0 minor · reviewed 2026-05-24 · grok-4.3
Pith's one-line read A new approximation tracks photon geodesics in Schwarzschild spacetime for emission angles near 180 degrees with under 1 percent error to the ISCO.
desk verdict This note claims a new approximation for highly bent photon geodesics in Schwarzschild with sub-1% error, but the abstract gives no functional form, derivation, or validation steps to check the claim. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
A functional approximation to the photon geodesic equation in Schwarzschild spacetime constructed to match behavior at large deflection angles.
What would settle it
Direct numerical integration of the geodesic equation for emission angles between 150 and 180 degrees at radii from 6 to 10 GM/c² compared against the approximation output.
Extended reading notes
Core claim
The authors present an approximation to photon geodesics in Schwarzschild spacetime that remains accurate to better than 1 percent for highly bent trajectories with emission angles approaching π, extending inward to the ISCO at 6 GM/c².
Load-bearing premise
A functional form exists that reproduces the exact geodesic behavior near emission angle π without the paper specifying the derivation or fitting steps used to reach the reported accuracy.
Editorial extensions
If this is right
- Light rays emitted from the far side of a non-rotating compact object can be traced without repeated numerical integration.
- Ray-tracing calculations for emission near the ISCO gain speed while preserving sub-percent accuracy.
- Models of strong gravitational deflection around Schwarzschild sources become feasible for larger numbers of rays.
Reading between the lines
- The same style of approximation could be tested for adaptation to the Kerr metric around spinning black holes.
- Implementation in existing ray-tracing codes would allow faster generation of synthetic images or light curves from accretion flows.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript introduces a new approximation for photon geodesics in Schwarzschild spacetime, with emphasis on highly bent trajectories where the angle between the initial emission position and the observer's line of sight approaches π (i.e., points behind the central mass). It asserts that the approximation deviates by less than 1% from exact numerical integration results for source radii down to the ISCO at 6 GM/c².
Significance. A simple, accurate approximation for strongly deflected null geodesics would be useful for semi-analytic ray-tracing in strong-field lensing and black-hole imaging contexts. The significance cannot be assessed because the manuscript supplies neither the explicit functional form nor the derivation or validation protocol that would allow independent verification of the stated accuracy bound.
major comments (1)
- [Abstract] Abstract: the central accuracy claim (deviations <1% up to the ISCO) is load-bearing for the paper's contribution, yet the abstract provides neither the explicit functional form of the approximation nor any derivation steps, matching procedure, or error-analysis details against numerical integration. Without these elements the reported bound cannot be evaluated for robustness or generality.
Simulated Author's Rebuttal
We thank the referee for their comments on our research note. We address the major comment below and agree that revisions are needed to make the central claim more verifiable.
read point-by-point responses
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Referee: [Abstract] Abstract: the central accuracy claim (deviations <1% up to the ISCO) is load-bearing for the paper's contribution, yet the abstract provides neither the explicit functional form of the approximation nor any derivation steps, matching procedure, or error-analysis details against numerical integration. Without these elements the reported bound cannot be evaluated for robustness or generality.
Authors: We agree that the abstract is too terse to allow independent evaluation of the accuracy bound. In the revised version we will expand the abstract to include the explicit functional form of the approximation, a concise outline of the derivation approach, the matching procedure to numerical geodesics, and the error-analysis protocol. This will make the <1% deviation claim directly assessable from the abstract while preserving its length constraints. revision: yes
Circularity Check
No circularity: approximation introduced and validated against independent numerical integration
full rationale
The paper introduces a new functional approximation for photon geodesics in Schwarzschild spacetime focused on near-π emission angles and reports <1% deviation from exact numerical results up to the ISCO. The abstract and available text frame this as an empirical validation against external numerical integration of the geodesic equation, with no quoted equations showing the approximation defined in terms of its own outputs, no fitted parameters renamed as predictions, and no load-bearing self-citations or uniqueness theorems. The central claim therefore remains independent of its inputs.
Assumptions & free parameters
Cite this review
Pith. "Pith review of A new approximation of photon geodesics in Schwarzschild spacetime." pith.science (2026). https://pith.science/paper/OXF7SYRL
@misc{pith2026190711786,
author = {Pith},
title = {Pith review of: A new approximation of photon geodesics in Schwarzschild spacetime},
year = {2026},
howpublished = {\url{https://pith.science/paper/OXF7SYRL}},
note = {Machine review of arXiv:1907.11786}
}
abstract
In this research note we introduce a new approximation of photon geodesics in Schwarzschild spacetime which is especially useful to describe highly bent trajectories, for which the angle between the initial emission position and the line of sight to the observer approaches $\pi$: this corresponds to the points behind the central mass of the Schwarzschild metric with respect to the observer. The approximation maintains very good accuracy overall, with deviations from the exact numerical results below $1\%$ up to the innermost stable circular orbit (ISCO) located at $6~GM/c^2$.
Reviewed May 24, 2026 · model on record in the stance chip above.
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