The equation of Binet in classical and relativistic orbital mechanics

TL;DR

This paper derives Binet's equation in classical and relativistic contexts, linking geometric orbit shapes to infinitesimal displacements and addressing the role of the cosmological constant in photon trajectories.

Contribution

It presents a novel derivation of the relativistic Binet's equation directly from coordinate relations without potentials or Killing vectors.

Findings

Classical Binet's equation yields conic orbits for inverse-square law.

Relativistic Binet's equation derived directly from coordinate displacements.

Discusses the influence of the cosmological constant on photon trajectories.

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.