Epicyclic orbital oscillations in Newton's and Einstein's dynamics

TL;DR

This paper demonstrates that Newton's and Einstein's descriptions of orbital oscillations are equivalent when considering a properly curved space, revealing that Newtonian formulas can accurately predict certain relativistic effects in strong gravitational fields.

Contribution

It establishes a formal equivalence between Newtonian dynamics in curved space and Einstein's general relativity in Schwarzschild spacetime for orbital oscillations.

Findings

Newton's equations in curved space match Einstein's predictions for orbital oscillations.

Newtonian formulas accurately describe the vanishing of radial epicyclic frequency at the marginally stable orbit.

The approach simplifies understanding of relativistic orbital phenomena using Newtonian mechanics.

Abstract

We apply Feynman's principle, ``The same equations have the same solutions'', to Kepler's problem and show that Newton's dynamics in a properly curved 3-D space is identical with that described by Einstein's theory in the 3-D optical geometry of Schwarzschild's spacetime. For this reason, rather unexpectedly, Newton's formulae for Kepler's problem, in the case of nearly circular motion in a static, spherically spherical gravitational potential accurately describe strong field general relativistic effects, in particular vanishing of the radial epicyclic frequency at the marginally stable orbit.