Relativistic Theory of Elastic Bodies in the Presence of Gravitational Waves

TL;DR

This paper develops a relativistic framework describing how gravitational waves induce elastic oscillations in materials, revealing that gravitational waves act as surface tractions or bulk accelerations depending on the material's geometry.

Contribution

It derives the equations of motion for elastic bodies under gravitational waves using Carter and Quintana's framework, including new descriptions for thin rods and comparisons with existing elasticity theories.

Findings

Gravitational waves act as effective surface tractions in elastic bodies.

For thin rods, gravitational waves induce bulk accelerations related to Riemann curvature.

The derived equations extend existing elasticity theories to relativistic gravitational wave interactions.

Abstract

The equations of motion governing small elastic oscillations of materials, induced by gravitational waves, are derived from the general framework of Carter and Quintana. In transverse-traceless gauge, no bulk forces are present, and the gravitational wave is found to act as an effective surface traction. For thin rods, an equivalent description is given, in which there is no surface traction, but a bulk acceleration, which is related to the Riemann curvature of the gravitational wave. The resulting equations are compared to those of the Synge-Bennoun elasticity theory.