A Keplerian Limit to Static Spherical Spacetimes in Curvature Coordinates

TL;DR

This paper derives a Keplerian limit solution for test bodies in Schwarzschild and other static spherically symmetric spacetimes, including first-order relativistic corrections, and compares it with exact numerical solutions.

Contribution

It presents a new approximate analytical solution for elliptical orbits in Schwarzschild spacetime with relativistic corrections, extending to Reissner-Nordström and Schwarzschild-de Sitter geometries.

Findings

Analytical solutions closely match numerical results.

Relativistic eccentricity predicts observable orbital effects.

First-order corrections improve Newtonian energy estimates.

Abstract

The problem of a test body in the Schwarzschild geometry is investigated in a Keplerian limit. Beginning with the Schwarzschild metric, a solution to the limited case of approximately elliptical (Keplerian) motion is derived in terms of trigonometric functions. This solution is similar in form to that derived from Newtonian mechanics, and includes first-order corrections describing three effects due to general relativity: precession; reduced radial coordinate; and increased eccentricity. The quantitative prediction of increased eccentricity may provide an additional observational test of general relativity. By analogy with Keplerian orbits, approximate orbital energy parameters are defined in terms of a relativistic eccentricity, providing first-order corrections to Newtonian energies for elliptical orbits. The first-order relativistic equation of orbit is demonstrated to be a limiting case of a very accurate self-consistent solution. This self-consistent solution is supported by exact numerical solutions to the Schwarzschild geometry, displaying remarkable agreement. A more detailed energy parameterization is investigated using the relativistic eccentricity together with the apsides derived from the relativistic effective potential in support of the approximate energy parameters defined using only first-order corrections. The methods and approximations describing this Keplerian limit are applied to more general static spherically-symmetric geometries. Specifically, equations of orbit and energy parameters are also derived in this Keplerian limit for the Reissner-Nordstr\"{o}m and Schwarzschild-de Sitter metrics.