  The general relativistic theory of elasticity is reviewed from a Lagrangian,
as opposed to Eulerian, perspective. The equations of motion and
stress-energy-momentum tensor for a hyperelastic body are derived from the
gauge-invariant action principle first considered by DeWitt. This action is a
natural extension of the action for a single relativistic particle. The central
object in the Lagrangian treatment is the Landau-Lifshitz radar metric, which
is the relativistic version of the right Cauchy-Green deformation tensor. We
also introduce relativistic definitions of the deformation gradient, Green
strain, and first and second Piola-Kirchhoff stress tensors. A gauge-fixed
description of relativistic hyperelasticity is also presented, and the
nonrelativistic theory is derived in the limit as the speed of light becomes
infinite.
