On geodesic deviation in Schwarzschild spacetime

TL;DR

This paper analyzes geodesic deviation in Schwarzschild spacetime, deriving solutions for satellite orbit deviations relevant to space-based gravimetry, and compares these with Newtonian analogs to enhance understanding of spacetime curvature effects.

Contribution

It provides a detailed derivation of the Jacobi equation for geodesic deviation in Schwarzschild spacetime and constructs explicit solutions for satellite orbit deviations, including Newtonian comparisons.

Findings

Solutions for geodesic deviation under various initial conditions

Quantitative impact of spacetime curvature on satellite orbits

Comparison between relativistic and Newtonian deviations

Abstract

For metrology, geodesy and gravimetry in space, satellite based instruments and measurement techniques are used and the orbits of the satellites as well as possible deviations between nearby ones are of central interest. The measurement of this deviation itself gives insight into the underlying structure of the spacetime geometry, which is curved and therefore described by the theory of general relativity (GR). In the context of GR, the deviation of nearby geodesics can be described by the Jacobi equation that is a result of linearizing the geodesic equation around a known reference geodesic with respect to the deviation vector and the relative velocity. We review the derivation of this Jacobi equation and restrict ourselves to the simple case of the spacetime outside a spherically symmetric mass distribution and circular reference geodesics to find solutions by projecting the Jacobi equation on a parallel propagated tetrad as done by Fuchs. Using his results, we construct solutions of the Jacobi equation for different physical initial scenarios inspired by satellite gravimetry missions and give a set of parameter together with their precise impact on satellite orbit deviation. We further consider the Newtonian analog and construct the full solution, that exhibits a similar structure, within this theory.