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arxiv: 2512.07485 · v3 · submitted 2025-12-08 · 🌀 gr-qc · math-ph· math.MP

The equation of Binet in classical and relativistic orbital mechanics

Jose Luis Alvarez-Perez This is my paper

Pith reviewed 2026-05-17 00:56 UTC · model grok-4.3

classification 🌀 gr-qc math-phmath.MP
keywords Binet equationorbital mechanicsSchwarzschild metricde Sitter spacetimerelativistic orbitscosmological constantphoton trajectories
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The pith

Binet's equation is derived from infinitesimal displacements for both classical and relativistic central force problems in Schwarzschild-(anti-)de Sitter geometry.

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

The paper shows that Binet's equation, which gives the shape of orbits under central forces, emerges from considering small horizontal and vertical movements of a falling body using basic calculus. This same approach is extended to relativity to obtain the Binet equation for the Schwarzschild-(anti-)de Sitter metric by directly connecting the relevant coordinates. The derivation is presented as new because it skips the usual introduction of effective potentials or Killing vector techniques. A reader might care as this provides an elementary geometric view of how orbits form in curved space and comments on how the cosmological constant influences light paths.

Core claim

The central claim is that the relativistic version of Binet's equation for the Schwarzschild-(anti-)de Sitter metric can be derived by directly relating the coordinates involved through infinitesimal horizontal and vertical displacements, without needing potentials or Killing vectors.

What carries the argument

The infinitesimal horizontal and vertical displacement method applied to the metric to relate radial and angular changes in the orbit equation.

If this is right

  • All conic curves arise as solutions for inverse-square force in Newtonian case.
  • The equation applies directly to orbits in Schwarzschild-(anti-)de Sitter spacetime.
  • Controversies about the cosmological constant's role in photon trajectories can be addressed.
  • The derivation uses only elementary infinitesimal calculus concepts.

Where Pith is reading between the lines

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

  • This displacement-based method could be applied to other spacetimes for simplified orbit calculations.
  • It might offer an alternative way to compute perihelion precession without standard geodesic equations.
  • Connections to the role of Lambda in black hole shadows or lensing could be explored further.

Load-bearing premise

The infinitesimal horizontal and vertical displacement method works in the curved Schwarzschild-(anti-)de Sitter geometry while keeping the same geometric meaning without extra curvature corrections.

What would settle it

Computing the orbit using the standard geodesic equation in Schwarzschild-de Sitter and checking if it satisfies the derived Binet equation.

Figures

Figures reproduced from arXiv: 2512.07485 by Jose Luis Alvarez-Perez.

Figure 1
Figure 1. Figure 1: FIG. 1. Representation of the instantaneous position of the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. A massive particle follows a trajectory (solid curve) [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
read the original abstract

Binet's equation provides a direct way to obtain the geometric shape of orbits in a central force field. It is well known that in Newtonian gravitation Binet's equation leads to all the conic curves as solutions for an inverse-square force. In this work, we show how Binet's equation arises from the horizontal and vertical infinitesimal displacements of a body in free fall and in inertial motion. This derivation uses elementary concepts of infinitesimal calculus. Second, we derive the relativistic version of Binet's equation for the Schwarzschild-(anti-)de Sitter metric. This derivation, which is novel, directly relates the coordinates involved in Binet's equation without the need to introduce potentials or the use of Killing vectors. Finally, we tackle some controversies related to the role of the cosmological constant in the trajectory of photons in a Schwarzschild-(anti-)de Sitter or even in Reissner-Nordstr\"om-(anti-)de Sitter spacetimes.

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 / 2 minor

Summary. The manuscript derives Binet's equation from elementary infinitesimal horizontal and vertical displacements of a body in free fall and inertial motion using only basic calculus, first in the Newtonian inverse-square case and then for the relativistic Schwarzschild-(anti-)de Sitter metric. The relativistic derivation is presented as novel because it relates the relevant coordinates directly without introducing potentials or invoking Killing vectors. The paper concludes by addressing controversies concerning the cosmological constant's effect on photon trajectories in SdS and RNdS spacetimes.

Significance. If the central derivation holds, the work supplies an elementary, symmetry-independent route to the relativistic Binet equation that could simplify calculations of orbital shapes and light deflection while clarifying the explicit role of the cosmological constant; such an approach would be useful for pedagogical purposes and for exploring cosmological corrections in strong-field regimes.

major comments (2)
  1. [§3] §3 (relativistic derivation): the infinitesimal horizontal/vertical displacement procedure is transplanted from flat space, yet the text does not display the explicit inclusion of the non-zero Christoffel symbols of the SdS metric when differentiating the trajectory components; without these terms the resulting equation risks omitting curvature corrections that are required for consistency with the geodesic equation.
  2. [relativistic Binet equation] Eq. (relativistic Binet form): the final expression is stated to reduce to the Newtonian limit, but the manuscript does not show the intermediate steps that recover the standard 1/r^2 force term when the metric coefficients are expanded to first post-Newtonian order; this verification is load-bearing for the claim that the method preserves the correct Newtonian correspondence.
minor comments (2)
  1. [Abstract] The abstract would be strengthened by quoting the key relativistic Binet equation so that readers can immediately compare it with the standard form obtained via Killing vectors.
  2. [Notation] Notation for the angular coordinate and its derivative is introduced without a dedicated table or list; a short glossary would improve readability for readers unfamiliar with the displacement-based approach.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which help strengthen the presentation of our derivation. We address each major comment below and indicate the changes we will make in the revised version.

read point-by-point responses

  1. Referee: [§3] §3 (relativistic derivation): the infinitesimal horizontal/vertical displacement procedure is transplanted from flat space, yet the text does not display the explicit inclusion of the non-zero Christoffel symbols of the SdS metric when differentiating the trajectory components; without these terms the resulting equation risks omitting curvature corrections that are required for consistency with the geodesic equation.

    Authors: We agree that an explicit display of the Christoffel symbols strengthens the connection to the geodesic equation. Although the derivation in §3 uses the metric to define the infinitesimal displacements in Schwarzschild-(anti-)de Sitter coordinates, the intermediate differentiation steps were not written out in full. In the revised manuscript we will add these steps, showing how the non-zero Christoffel symbols enter when differentiating the trajectory components and confirming that the curvature corrections are correctly incorporated. revision: yes

  2. Referee: [relativistic Binet equation] Eq. (relativistic Binet form): the final expression is stated to reduce to the Newtonian limit, but the manuscript does not show the intermediate steps that recover the standard 1/r^2 force term when the metric coefficients are expanded to first post-Newtonian order; this verification is load-bearing for the claim that the method preserves the correct Newtonian correspondence.

    Authors: We concur that the explicit recovery of the Newtonian limit is important for validating the correspondence. The manuscript states that the relativistic form reduces to the classical Binet equation but omits the intermediate algebra. In the revision we will insert the expansion of the SdS metric coefficients to first post-Newtonian order and demonstrate step by step how the standard 1/r² force term is recovered, thereby confirming the Newtonian correspondence. revision: yes

Circularity Check

0 steps flagged

Derivation from elementary displacements is self-contained with no reduction to inputs

full rationale

The paper starts from infinitesimal horizontal/vertical displacements in free fall and inertial motion, applies elementary calculus to obtain Binet's equation in the Newtonian case, then transplants the same displacement relations to the Schwarzschild-(anti-)de Sitter metric to obtain the relativistic form without potentials or Killing vectors. No step equates the final coordinate relation to a fitted parameter, a self-cited uniqueness theorem, or a renamed empirical pattern; the central result is generated from the metric coefficients and the displacement definitions themselves. The transplant assumption may raise correctness questions about omitted Christoffel terms, but that is external to circularity. No load-bearing self-citation chain or self-definitional loop is present.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard Schwarzschild-(anti-)de Sitter metric and the applicability of infinitesimal calculus to both Newtonian and relativistic settings. No free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption The Schwarzschild-(anti-)de Sitter metric correctly describes spacetime around a spherical mass with a cosmological constant.
    Invoked for the relativistic derivation of Binet's equation.

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Reference graph

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