Relativistic elastic membranes: rotating disks and Dyson spheres

TL;DR

This paper develops a formalism for relativistic elastic membranes, deriving their equations of motion and conserved quantities, and applies it to analyze rotating disks and Dyson spheres, revealing stability nuances.

Contribution

It introduces a variational approach to relativistic elastic membranes and applies it to complex systems like rotating disks and Dyson spheres, highlighting stability insights.

Findings

Rigidly rotating elastic disk analyzed in relativistic context

Dyson sphere in Schwarzschild spacetime in equilibrium

Dipolar perturbations cause instability despite spherical symmetry

Abstract

We derive the equations of motion for relativistic elastic membranes, that is, two-dimensional elastic bodies whose internal energy depends only on their stretching, starting from a variational principle. We show how to obtain conserved quantities for the membrane's motion in the presence of spacetime symmetries, determine the membrane's longitudinal and transverse speeds of sound in isotropic states, and compute the coefficients of linear elasticity with respect to the relaxed configuration. We then use this formalism to discuss two physically interesting systems: a rigidly rotating elastic disk, widely discussed in the context of Ehrenfest's paradox, and a Dyson sphere, that is, a spherical membrane in equilibrium in Schwarzschild's spacetime, with the isotropic tangential pressure balancing the gravitational attraction. Surprisingly, although spherically symmetric perturbations of this system are linearly stable, the axi-symmetric dipolar mode is already unstable. This may be taken as a cautionary tale against misconstruing radial stability as true stability.