A new approximation of photon geodesics in Schwarzschild spacetime

1); 2); (2) Research Centre for Computational Physics; 3; (3) M. R. \v{S}tefanik Observatory; (4) International Space Science Institute - ISSI; 5) ((1) INAF - Osservatorio Astronomico di Roma; (5) International Space Science Institute Beijing; Bern; Czech Republic

arxiv: 1907.11786 · v1 · pith:OXF7SYRLnew · submitted 2019-07-25 · 🌀 gr-qc

A new approximation of photon geodesics in Schwarzschild spacetime

Riccardo La Placa (1 , 2) , Pavel Bakala (2 , 3 , 1) , Luigi Stella (1) , Maurizio Falanga (4 , 5) ((1) INAF - Osservatorio Astronomico di Roma

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Monte Porzio Catone Italy (2) Research Centre for Computational Physics Data Processing Silesian University in Opava Czech Republic (3) M. R. \v{S}tefanik Observatory Planetarium Hlohovec Slovak Republic (4) International Space Science Institute - ISSI Bern Switzerland (5) International Space Science Institute Beijing P.R. China)

This is my paper

Pith reviewed 2026-05-24 16:09 UTC · model grok-4.3

classification 🌀 gr-qc

keywords photon geodesicsSchwarzschild spacetimeapproximationgravitational deflectionISCOblack holeray tracing

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The pith

A new approximation tracks photon geodesics in Schwarzschild spacetime for emission angles near 180 degrees with under 1 percent error to the ISCO.

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

The paper introduces a functional approximation for photon paths in Schwarzschild spacetime that targets the most strongly deflected trajectories. These are the paths emitted from locations behind the central mass, where the angle between emission direction and observer line of sight approaches 180 degrees. The approximation reproduces exact numerical geodesics to within 1 percent all the way inward to the innermost stable circular orbit at six gravitational radii. This removes the need to integrate the full geodesic equation for each ray in that regime.

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.

What carries the argument

A functional approximation to the photon geodesic equation in Schwarzschild spacetime constructed to match behavior at large deflection angles.

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.

Where Pith is reading between the lines

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

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.

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.

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.

read the original 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 $6~GM/c^2$.

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

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 paper introduces an approximation aimed at photon paths where the emission angle approaches π, the strongly deflected trajectories behind the central mass. That regime shows up in some ray-tracing work for compact objects, so a fast analytic handle could be handy for modeling. The abstract states the approximation stays within 1% of numerical results out to the ISCO at 6M, which would make it usable for quick estimates in that range if it holds. No free parameters are mentioned, and the result is framed as validated against independent integration rather than circular fitting. That is the positive side: it targets a concrete gap in existing approximations for bent null geodesics. The main limitation is that the abstract supplies neither the explicit functional form nor any steps showing how it was built or matched to the geodesic equation. Without that, the accuracy bound cannot be assessed for robustness or for how it behaves near the claimed limit. The stress-test concern stands on the information given. The work is a short research note rather than a full derivation paper, so the citation pattern is light and not a problem. Readers who do numerical photon tracing in Schwarzschild for astrophysical applications might find the final formula useful once it is written down, but the current text does not let anyone judge whether the central claim is reliable. I would not send this to peer review in its present form; the authors would need to add the missing construction and comparison details first.

Referee Report

1 major / 0 minor

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

1 responses · 0 unresolved

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

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

0 steps flagged

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.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5735 in / 1053 out tokens · 27709 ms · 2026-05-24T16:09:28.150065+00:00 · methodology

discussion (0)

Forward citations

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