## The Elastic Energy-Momentum Tensor

in Special Relativity

### David N. Williams

Randall Laboratory of Physics

The University of Michigan

Ann Arbor, MI 48109-1120

June 15, 1989
**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 non-relativistic theory.

*Annals of Physics* **{196}**, no. 2, (1989) 345-360.

`doi:10.1016/0003-4916(89)90181-4`

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