Relativistic Linear Restoring Force

TL;DR

This paper explores different relativistic formulations of a linear restoring force, including retarded interactions, revealing oscillatory behavior with amplitude growth in both relativistic and non-relativistic regimes.

Contribution

It introduces a relativistic version of a linear restoring force incorporating retardation effects, extending classical Hooke's law into relativistic contexts.

Findings

Retarded force leads to oscillations with amplitude growth.

Both relativistic and non-relativistic formulations exhibit amplitude increase.

New relativistic force models recover classical behavior in the non-relativistic limit.

Abstract

We consider two different forms for a relativistic version of a linear restoring force. The pair comes from taking Hooke's law to be the force appearing on the right of the relativistic expressions: dp/dt or dp/dtau . Either formulation recovers Hooke's law in the non-relativistic limit. In addition to these two forces, we introduce a form of retardation appropriate for the description of a linear (in displacement) force arising from the interaction of a pair of particles with a relativistic field. The procedure is akin to replacing Coulomb's law in E&M with a retarded form (the first correction in the full relativistic case). This retardation leads to the expected oscillation, but with amplitude growth in both its relativistic and non-relativistic incarnations.