Infinitesimal deformations of finite conjugacies in non-linear classical or general relativistic theory of elasticity

Abstract

We develop here an exact theory of infinitesimal perturbation of relativistic elastic continua. We use our exact, non-linear, covariant theory of relativistic elasticity based on the notion of finite conjugacy from a space-time ( M ,g) to another space-time ( M ′,g′): we intrinssically identify the perturbations with vector fields on M ′ and get many different infinitesimal relativistic strain-tensors. We can then write down the well-determined system of partial differential equations for such vector fields, which for example governs neutron stars as well as detectors and emitters of gravitational waves. This relativistic system contains in the particular case of Newtonian elasticity, the infinitesimal displacement equations of a continuum around a finitely-deformed configuration which here may be no equilibrium. It also gives as other particular cases both classical and relativistic hypoelasticity theories. Many terms, including curvature terms and new elastic coefficients arise in this exact theory, even if the perturbed finite conjugacy is simply the identity, because non-linear effects, prestresses of any kind, the curvature of space-time and the elastic behaviour of the specific internal scalar energy have been taken into account in the finite equations.