A General Relativistic Pendulum: Isochronous vs. Geodesic Motion

TL;DR

This paper demonstrates that General Relativity ensures the isochrony of a pendulum and explores novel effects in oscillation frequencies near black hole horizons, revealing differences from Newtonian predictions.

Contribution

It provides a comprehensive analysis of pendulum oscillations within GR, including novel effects and divergences near the event horizon, extending classical pendulum theory into relativistic regimes.

Findings

Proper and coordinate frequencies depend on relativistic parameters.

GR effects cause divergences and instabilities in frequencies near the event horizon.

Isochrony is maintained within the framework of GR.

Abstract

General Relativity (GR) is shown to be a complete theory with respect to the isochrony of the pendulum. This guarantees that time can be measured with a mechanical clock within the theory itself as a matter of principle. The proper and coordinate oscillation frequencies of a simple pendulum are computed as a function of 3 coordinate parameters: the distance of the fulcrum to the gravity source, its length and the Schwarzschild radius. Novel GR effects appear in the Schwarzschild coordinates as divergences, zeroes and other instabilities of the oscillation frequencies that are absent in Newtonian gravity. The anomalies in the proper and coordinate frequencies occur under extreme conditions: either the fulcrum or the pendulum mass or both remain inside the Event Horizon.