Elasticity in general relativity

Abstract

This paper develops a new theory of elasticity in general relativity distinct from the one proposed by Synge (1959). In the new theory, stress is related to strain by a formula analogous to Hooke’s law, whereas Synge’s theory is formulated in terms of rates of change of stress and strain. Special features of the new theory are: (1) With any elastic body motion there is a uniquely associated rigid motion (in the Born sense), which plays a role analogous to that of the rigid body in the ordinary elasticity theory. (2) The 4-velocity of matter is an eigenvector of the Einstein tensor. (3) At each point of an elastic body there are at most 21 independent elastic coefficients. (4) The differential equations of motion of an elastic body are of the second order. (In Synge’s theory the corresponding equations are of the third order.) (5) As in Synge’s theory, shock waves travel with the same speeds as occur in ordinary elasticity theory. In the formulation of the new theory, consistent use is made of Lie derivatives.