Alternative derivation of the relativistic contribution to perihelic precession

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.

💡 Research Summary

The paper presents an alternative derivation of the first‑order relativistic contribution to the perihelion precession of a test particle moving in the Schwarzschild spacetime. Rather than employing the standard perturbative expansion of the geodesic equation, the authors adopt a “Keplerian limit” approach: they assume that the orbit is nearly elliptical, as in Newtonian mechanics, and treat general‑relativistic effects as small corrections.

Starting from the Schwarzschild metric in static, spherically symmetric coordinates, the authors write down the Lagrangian for a test particle confined to the equatorial plane (θ = π/2). Conserved quantities—energy E and angular momentum L—lead to an effective radial potential
V_eff(r) = −GM/r + L²/(2r²) − GM L²/(c² r³).
The first two terms reproduce the Newtonian Kepler problem, while the third term is the leading relativistic correction of order 1/c².

To obtain an orbit equation that resembles the familiar Newtonian form, the authors introduce the reciprocal radius u = 1/r and the standard Keplerian differential equation d²u/dφ² + u = GM/L². The relativistic correction appears as an additional term 3GM u²/c² on the right‑hand side of the exact geodesic equation. Instead of expanding this nonlinear term directly, the authors perform a small coordinate transformation of the angular variable: they define a new angle ψ = φ + δ(φ), where δ(φ) is a tiny phase shift induced by the relativistic term. Assuming δ ≪ 1, they substitute ψ into the differential equation and retain only first‑order contributions in both δ and 1/c². The resulting equation retains the same structure as the Newtonian one, but with an extra 1/c² term that can be evaluated using the unperturbed Keplerian solution u₀ = (GM/L²)