Two-Point Pad\'{e} Approximants for the Deflection of Light in the Schwarzschild Black Hole Metric

Don N. Page

Two-Point Pad\'{e} Approximants for the Deflection of Light in the Schwarzschild Black Hole Metric

T0 review · 1 major / 0 minor · reviewed 2026-05-10 · grok-4.3

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Pith's one-line read Two-point Padé approximants of order [2,2] accurately approximate light deflection by a Schwarzschild black hole for all impact parameters above critical.

desk verdict Page gives explicit [2,2] Padé forms in (b_c/b, exp(-α)) that match both deflection limits and supply usable elementary approximations across the full range. read the letter →

arxiv 2604.19952 v1 submitted 2026-04-21 gr-qc

Don N. Page This is my paper · ORCID

classification gr-qc

keywords lightdeflectionSchwarzschildblackholePadéapproximantsimpactparameteranglephotontrajectories

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

The paper constructs algebraic approximations for the deflection angle of light rays passing outside a Schwarzschild black hole. These use two-point Padé forms of order [2,2] that match the exact elliptic-integral behavior at two chosen points after transforming to the variables of critical impact parameter ratio and exponential of negative deflection. A sympathetic reader would care because the exact result requires elliptic integrals, while these simple rational functions enable direct calculation of photon trajectories for every physical impact parameter greater than critical. The paper also gives a simpler quadratic form that works well in the middle of the range.

What carries the argument

The [2,2] two-point Padé approximant in the variables (critical impact parameter divided by actual impact parameter) and (exponential of negative deflection angle).

What would settle it

Numerical comparison of the approximant to the exact elliptic-integral deflection at an intermediate impact parameter, such as twice the critical value, that shows error larger than a few percent.

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Extended reading notes

Core claim

I present Padé 2-point approximants of order [2,2] (quadratic numerators and denominators), relating the critical impact parameter divided by the actual impact parameter to the exponential of the negative of the deflection angle, that fairly accurately cover the full range of impact parameters greater than the critical impact parameter, which is the case for all photon trajectories that remain outside the black hole. I also present a simpler quadratic approximation that works as well in the middle of the range but not so well at the extremes.

Load-bearing premise

Matching the approximant at two points in the chosen variables produces sufficient accuracy over the entire physical range without significant deviations or the need for higher-order terms.

Editorial extensions

If this is right

The approximants give deflection angles directly for every impact parameter above critical without evaluating elliptic integrals.
All exterior photon trajectories can be treated with the same simple algebraic expressions.
The simpler quadratic form suffices for many intermediate impact parameters.

Reading between the lines

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

The transformed variables chosen for matching suggest that similar two-point Padé constructions could be tested on light deflection in other spherically symmetric metrics.
These algebraic forms could be inserted directly into ray-tracing codes to avoid repeated numerical integration of the exact geodesic equation.
If higher accuracy is required at the extremes, the same matching procedure can be repeated at order [3,3] or higher using the same variables.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The core contribution is a pair of [2,2] Padé approximants that relate the ratio of critical to actual impact parameter to the exponential of the negative deflection angle. They are built to reproduce the known linear asymptotics at both the strong-deflection end (b approaching b_c) and the weak-deflection end (b large), then checked against the exact elliptic-integral result in between. A simpler quadratic fit is added for the middle of the range. Both are elementary and avoid repeated elliptic-integral evaluations, which is the practical point for anyone who needs deflection angles in ray-tracing or orbit calculations. The construction is anchored directly to Darwin’s 1959 result, so there is no circularity. The paper is short, the algebra is transparent, and the choice of variables is sensible for keeping the approximants well-behaved over b > b_c. The main limitation is that the accuracy is described only qualitatively as “fairly accurate”; explicit maximum relative errors or a table of deviations at representative points would make the claim easier to judge without running the numbers oneself. The simpler quadratic is openly stated to degrade at the extremes, which is fair. This is not a fundamental advance in general relativity, but it is a clean, reproducible computational shortcut. It is aimed at people doing numerical work with Schwarzschild photon trajectories who want something faster than the elliptic integral yet better than crude power-series or logarithmic fits. The math and the citation to the exact result are solid enough that a serious referee should see it; the only real question is whether the error levels are small enough for the intended applications.

Referee Report

1 major / 0 minor

Summary. The paper presents [2,2] two-point Padé approximants relating the ratio of the critical impact parameter b_c to the actual impact parameter b with the quantity exp(-α), where α is the light deflection angle in the Schwarzschild metric. The approximants are constructed to reproduce the known asymptotic linear behaviors at the strong-deflection (b → b_c, α → ∞) and weak-deflection (b → ∞, α → 0) endpoints and are claimed to provide a practical closed-form approximation that covers the full physical range b > b_c with acceptable accuracy; a simpler quadratic approximation is also given for the intermediate range.

Significance. If the claimed accuracy holds with small maximum deviations from the exact elliptic-integral result, the work supplies a compact analytic alternative to numerical evaluation of the Darwin integral, which could be useful in ray-tracing or lensing calculations. The two-point construction is a natural and grounded choice that directly incorporates the independently known limiting behaviors rather than relying on ad-hoc fitting.

major comments (1)

The abstract asserts that the [2,2] approximants 'fairly accurately cover the full range' but provides neither explicit numerical coefficients, maximum relative or absolute error bounds, nor direct comparisons (e.g., plots or tables) against the exact elliptic integral over the interval b/b_c ∈ (1, ∞). Without these, the central claim of practical utility cannot be independently assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment of its potential utility. We address the single major comment below and have revised the manuscript to supply the quantitative details requested.

read point-by-point responses

Referee: The abstract asserts that the [2,2] approximants 'fairly accurately cover the full range' but provides neither explicit numerical coefficients, maximum relative or absolute error bounds, nor direct comparisons (e.g., plots or tables) against the exact elliptic integral over the interval b/b_c ∈ (1, ∞). Without these, the central claim of practical utility cannot be independently assessed.

Authors: We agree that the abstract's qualitative statement would be strengthened by explicit numerical support. The original manuscript derives the two-point Padé construction and states the resulting functional form, but does not tabulate the numerical coefficients of the [2,2] approximant, nor does it supply a systematic error table or comparison plot against the Darwin elliptic integral. In the revised version we have added: (i) the explicit numerical coefficients for both the [2,2] approximant and the simpler quadratic form, (ii) a table of relative and absolute errors evaluated at representative points across b/b_c ∈ (1, ∞), and (iii) a figure that overlays the approximant on the exact result together with the pointwise relative error. These additions make the accuracy claim directly verifiable while preserving the original analytic construction. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; approximants constructed from independent exact limits

full rationale

The paper constructs [2,2] two-point Padé approximants by matching the independently known asymptotic behaviors of the exact elliptic-integral deflection (from Darwin 1959) at the strong-deflection (b → b_c) and weak-deflection (b → ∞) endpoints. This matching is the standard definition of a two-point Padé approximant and does not render the claimed accuracy or coverage a tautology; the paper presents the resulting closed-form expressions as practical approximations whose fidelity over the interior range is a separate, verifiable claim against the external exact result. No load-bearing step reduces by construction to a self-citation, fitted parameter renamed as prediction, or ansatz smuggled from prior work by the same author. The derivation chain is self-contained against the external benchmark.

Assumptions & free parameters 1 free parameters · 1 assumptions · 0 invented entities

The central claim rests on the validity of the Schwarzschild metric in general relativity and the prior exact expression for the deflection angle as an elliptic integral.

free parameters (1)

Padé approximant coefficients
The coefficients in the quadratic numerator and denominator are determined by matching to the exact deflection function at two points and asymptotic behaviors.

assumptions (1)

domain assumption The deflection angle of light in the Schwarzschild metric is given by the elliptic integral of the first kind as calculated by Darwin in 1959.
This is the target function that the approximants are designed to reproduce.

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Cite this review

Pith. "Pith review of Two-Point Pad\'{e} Approximants for the Deflection of Light in the Schwarzschild Black Hole Metric." pith.science (2026). https://pith.science/paper/2604.19952

@misc{pith2026260419952,
  author       = {Pith},
  title        = {Pith review of: Two-Point Pad\'e Approximants for the Deflection of Light in the Schwarzschild Black Hole Metric},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2604.19952}},
  note         = {Machine review of arXiv:2604.19952}
}

read the original abstract

The deflection angle of a light ray passing the Schwarzschild (spherically symmetric vacuum) black hole was calculated by Charles Galton Darwin in 1959 in terms of the elliptic integral of the first kind. This calculation has been repeated many times and has also been given approximately in terms of elementary functions for impact parameters that either are not too small or are close to the critical impact parameter. Here I present Pad\'{e} 2-point approximants of order [2,2] (quadratic numerators and denominators), relating the critical impact parameter divided by the actual impact parameter to the exponential of the negative of the deflection angle, that fairly accurately cover the full range of impact parameters greater than the critical impact parameter, which is the case for all photon trajectories that remain outside the black hole. I also present a simpler quadratic approximation that works as well in the middle of the range but not so well at the extremes.

Figures

Figures reproduced from arXiv: 2604.19952 by the authors.

**Figure 2.** This graph gives the ‘exact’ (within the numerical precision of Math [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** This graph, with a greatly enlarged vertical scale from those of the pre [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: This graph, also with a greatly enlarged vertical scale from those of the [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: This graph, with a significantly enlarged vertical scale from those of [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: This graph, also with a significantly enlarged vertical scale from those of [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

The Gravity Field of a Particle

Charles Galton Darwin, “The Gravity Field of a Particle.” Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences249, no. 1257 (1959): 180-194

work page 1959
[2]

Annalen der Physik , year = 1916, month = jan, volume =

A. Einstein, “Die Grundlage der Allgemeinen Relativit¨ atstheorie (The Founda- tions of the General Theory of Relativity),” Annalen Phys.49, no. 7, 769-822 (1916) doi:10.1002/andp.19163540702

work page doi:10.1002/andp.19163540702 1916
[3]

Post-post-newtonian deflection of light by the sun,

R. Epstein and I. I. Shapiro, “Post-Post-Newtonian Deflection of Light by the Sun,” Phys. Rev. D22, 2947-2949 (1980) doi:10.1103/PhysRevD.22.2947

work page doi:10.1103/physrevd.22.2947 1980
[4]

Second Order Contribution to the Gravitational Deflection of Light,

E. Fischbach and B. S. Freeman, “Second Order Contribution to the Gravitational Deflection of Light,” Phys. Rev. D22, 2950 (1980) doi:10.1103/PhysRevD.22.2950

work page doi:10.1103/physrevd.22.2950 1980
[5]

G. W. Richter and R. A. Matzner, Phys. Rev. D26, 1219-1224 (1982) doi:10.1103/PhysRevD.26.1219 15

work page doi:10.1103/physrevd.26.1219 1982

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Reviewed May 10, 2026 · model on record in the stance chip above.

[1] [1]

The Gravity Field of a Particle

Charles Galton Darwin, “The Gravity Field of a Particle.” Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences249, no. 1257 (1959): 180-194

work page 1959

[2] [2]

Annalen der Physik , year = 1916, month = jan, volume =

A. Einstein, “Die Grundlage der Allgemeinen Relativit¨ atstheorie (The Founda- tions of the General Theory of Relativity),” Annalen Phys.49, no. 7, 769-822 (1916) doi:10.1002/andp.19163540702

work page doi:10.1002/andp.19163540702 1916

[3] [3]

Post-post-newtonian deflection of light by the sun,

R. Epstein and I. I. Shapiro, “Post-Post-Newtonian Deflection of Light by the Sun,” Phys. Rev. D22, 2947-2949 (1980) doi:10.1103/PhysRevD.22.2947

work page doi:10.1103/physrevd.22.2947 1980

[4] [4]

Second Order Contribution to the Gravitational Deflection of Light,

E. Fischbach and B. S. Freeman, “Second Order Contribution to the Gravitational Deflection of Light,” Phys. Rev. D22, 2950 (1980) doi:10.1103/PhysRevD.22.2950

work page doi:10.1103/physrevd.22.2950 1980

[5] [5]

G. W. Richter and R. A. Matzner, Phys. Rev. D26, 1219-1224 (1982) doi:10.1103/PhysRevD.26.1219 15

work page doi:10.1103/physrevd.26.1219 1982