Epicyclic orbital oscillations in Newton's and Einstein's gravity from the geodesic deviation equation

TL;DR

This paper demonstrates that the differences in epicyclic frequencies between Newtonian and Einsteinian gravity are purely geometric, using the geodesic deviation equation to unify the analysis of both regimes.

Contribution

It provides a new perspective by representing Newtonian and Einsteinian test body motions as geodesic deviations, clarifying the geometric origin of frequency differences.

Findings

Epicyclic frequency differences are purely geometric in origin.

Geodesic deviation equation reproduces known frequency results.

Unified geometric framework for Newtonian and Einsteinian gravity.

Abstract

In a recent paper Abramowicz and Klu{\'z}niak have discussed the problem of epicyclic oscillations in Newton's and Einstein's dynamics and have shown that Newton's dynamics in a properly curved three-dimensional space is identical to test-body dynamics in the three-dimensional optical geometry of Schwarzschild space-time. One of the main results of this paper was the proof that different behaviour of radial epicyclic frequency and Keplerian frequency in Newtonian and General Relativistic regimes had purely geometric origin contrary to claims that nonlinearity of Einstein's theory was responsible for this effect. In this paper we obtain the same result from another perspective: by representing these two distinct problems (Newtonian and Einstein's test body motion in central gravitational field) in a uniform way -- as a geodesic motion. The solution of geodesic deviation equation reproduces the well known results concerning epicyclic frequencies and clearly demonstrates geometric origin of the difference between Newtonian and Einstein's problems.