# A relativistic gravity train

## TL;DR

This paper explores the relativistic motion of a particle in a uniform sphere under gravity and electrostatics, deriving exact relations and analyzing relativistic effects on oscillation periods and time spent near turning points.

## Contribution

It provides exact closed-form solutions for relativistic oscillations in gravitational and electrostatic contexts, extending classical harmonic motion to relativistic regimes.

## Key findings

- Oscillation periods depend on amplitude in relativistic cases
- Particles spend most proper time near turning points at high velocities
- Relativistic effects significantly alter oscillation dynamics and timing

## Abstract

A nonrelativistic particle released from rest at the edge of a ball of uniform charge density or mass density oscillates with simple harmonic motion. We consider the relativistic generalizations of these situations where the particle can attain speeds arbitrarily close to the speed of light; generalizing the electrostatic and gravitational cases requires special and general relativity, respectively. We find exact closed-form relations between the position, proper time, and coordinate time in both cases, and find that they are no longer harmonic, with oscillation periods that depend on the amplitude. In the highly relativistic limit of both cases, the particle spends almost all of its proper time near the turning points, but almost all of the coordinate time moving through the bulk of the ball. Buchdahl's theorem imposes nontrivial constraints on the general-relativistic case, as a ball of given density can only attain a finite maximum radius before collapsing into a black hole. This article is intended to be pedagogical, and should be accessible to those who have taken an undergraduate course in general relativity.