A four-dimensional Hooke's law can encompass linear elasticity and inertia

TL;DR

This paper extends Hooke's law into four-dimensional spacetime, integrating linear elasticity with inertia, and demonstrates its consistency with classical elasticity equations in the slow motion limit.

Contribution

It introduces a four-dimensional formulation of Hooke's law that unifies elasticity and inertia within relativistic frameworks.

Findings

Four-dimensional Hooke's law can model inertia of matter.

In flat spacetime, the equations reduce to classical elasticity.

The approach unifies elasticity and inertia in relativistic context.

Abstract

The question is examined, whether the formally straightforward extension of Hooke's time-honoured stress-strain relation to the four dimensions of special and of general relativity can make physical sense. The four-dimensional Hooke's law is found able to account for the inertia of matter; in the flat space, slow motion approximation the field equations for the ``displacement'' four-vector field can encompass both linear elasticity and inertia. In this limit one just recovers the equations of motion of the classical theory of elasticity.