The elastic energy-momentum tensor in special relativity

Abstract

We consider the standard nonrelativistic theory of a continuous, elastic medium with finite deformations, according to which the elastic energy is a function only of the state of strain, and the elastic stress tensor is proportional to the strain gradient of the elastic energy in appropriate coordinates. We derive a special relativistic, energy-momentum tensor, which yields the standard class of theories in the nonrelativistic limit, from the requirement that it depend only on the state of deformation (including the minimal dependence on velocity consistent with covariance), plus conservation laws. The result agrees with an earlier theory proposed by B. DeWitt ( in “Gravitation: An Introduction to Current Research” (L. Witten, Ed.), pp. 305–318, Wiley, New York, 1962), who generalized the nonrelativistic Lagrangian to general relativity. The elastic momentum density turns out to be of order v c 2 , and therefore absent in the nonrelativistic theory.