Relativistic elasticity of rigid rods and strings

TL;DR

This paper demonstrates that the motion of a relativistically rigid one-dimensional elastic body is governed by the wave equation, and explores various scenarios including collisions, forces, and black hole horizons.

Contribution

It introduces a simple wave equation model for relativistically rigid rods and strings, providing solutions for several physically interesting cases.

Findings

Wave equation describes relativistic rigidity

Solutions for collisions and forces on rods

Analysis of strings near black hole horizons

Abstract

We show that the equation of motion for a rigid one-dimensional elastic body (i.e. a rod or string whose speed of sound is equal to the speed of light) in a two-dimensional spacetime is simply the wave equation. We then solve this equation in a few simple examples: a rigid rod colliding with an unmovable wall, a rigid rod being pushed by a constant force, a rigid string whose endpoints are simultaneously set in motion (seen as a special case of Bell's spaceships paradox), and a radial rigid string that has partially crossed the event horizon of a Schwarzschild black hole while still being held from the outside.