Space and Space-Time geodesics in Schwarzschild geometry

TL;DR

This paper explores the differences between geodesic equations in Schwarzschild geometry for space and space-time, highlighting the importance of relativistic time and correcting common misconceptions about planetary orbits.

Contribution

It clarifies the role of relativistic time in geodesic equations and challenges the simplified rubber sheet analogy for planetary orbits in general relativity.

Findings

Geodesic equations for space alone imply unphysical repulsion.

Proper space-time geodesics reduce to Newtonian orbits in the non-relativistic limit.

Misleading nature of the rubber sheet analogy for planetary orbits.

Abstract

Geodesic orbit equations in the Schwarzschild geometry of general relativity reduce to ordinary conic sections of Newtonian mechanics and gravity for material particles in the non-relativistic limit. On the contrary, geodesic orbit equations for a proper spatial submanifold of Schwarzschild metric at any given coordinate-time correspond to an unphysical gravitational repulsion in the non-relativistic limit. This demonstrates at a basic level the centrality and critical role of relativistic time and its intimate pseudo-Riemannian connection with space. Correspondingly, a commonly popularized depiction of geodesic orbits of planets as resulting from the curvature of space produced by the sun, represented as a rubber sheet dipped in the middle by the weighing of that massive body, is mistaken and misleading for the essence of relativity, even in the non-relativistic limit.