Configurations and Deformations in Relativistic Elasticity

Abstract

Configurations and deformations are the homeomorphisms that relate material manifold and shapes of a solid in conventional non-linear elasticity. They play fundamental role in kinematic description of deformable media and seem to be an essential part of non-linear continuum mechanics. These concepts nevertheless are rooted in Euclidean nature of space and time, adopted in non-relativistic physics, and their direct application in relativistic case is impossible. The paper develops the generalization for them within General Relativity. To this end the concept of relativistic shape is introduced, which is defined as an element of foliation over a congruence of worldlines that constitute the world-tube of the solid. This makes it possible to define displacements as a field of translations along elements of congruence beginning on one element of foliation and ending on another. Geometrically this is similar to Fermi–Walker transport. Each relativistic shape can be endowed with two metrics, one induced by ambient space, and another, given by pushforwarding of the metric induced on another shape (another element of foliation). These metrics are relativistic counterpart of Cauchy–Green measures of deformation. The conditions under which the foliation of congruence of worldlines exists are specified. When these conditions are violated, the natural generalization of the shape leads to a set that cannot be represented by the Riemann submanifold. This situation is similar to that which arises in the continuum theory of defects when trying to find a globally stress-free shape. To solve this problem, it is proposed to construct a material manifold whose elements represent shapes locally. The method of determination of geometry on material manifold, which turns out to be Weyl geometry, is proposed. All constructions are carried out within the framework of a variational approach to the derivation of field equations and Noether symmetries.