The Elastic Energy-Momentum Tensor in Special Relativity∗

We consider the standard nonrelativistic theory of a continuous, elas-tic medium with finite deformations, according to which the elastic en-ergy 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 covari-ance), 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 elas-tic momentum density turns out to be of order v/c2, and therefore absent in the non-relativistic theory