Alternative derivation of the relativistic contribution to perihelic precession

TL;DR

This paper presents an alternative, undergraduate-accessible derivation of the first-order relativistic contribution to perihelic precession using coordinate transformations and the correspondence principle, aligning with established results.

Contribution

It introduces a novel derivation method for relativistic precession that avoids standard perturbation techniques, making the concept more accessible to students.

Findings

Derived orbit equation similar to Newtonian form with relativistic corrections

Relativistic precession result matches established first-order calculations

Reduced radius for circular orbit agrees with Schwarzschild potential calculations

Abstract

An alternative derivation of the first-order relativistic contribution to perihelic precession is presented. Orbital motion in the Schwarzschild geometry is considered in the Keplerian limit, and the orbit equation is derived for approximately elliptical motion. The method of solution makes use of coordinate transformations and the correspondence principle, rather than the standard perturbative approach. The form of the resulting orbit equation is similar to that derived from Newtonian mechanics and includes first-order corrections to Kepler's orbits due to general relativity. The associated relativistic contribution to perihelic precession agrees with established first-order results. The reduced radius for the circular orbit is in agreement to first-order with that calculated from the Schwarzschild effective potential. The method of solution is understandable by undergraduate students.