Foundations of general relativistic high-pressure elasticity theory

Abstract

A fully relativistic formulation of the concept of a perfectly elastic solid is developed with a view to application to such problems as (a) the interaction of gravitational radiation with planetary type bodies such as the Earth, and (b) the vibrations and deformations of neutron star crusts. For applications of the former kind a low-pressure Hookean idealization (i.e. a theory postulating a linear stress-strain relation) will often be sufficient. A prototype version of a relativistic Hookean theory suitable for this purpose has already been given by Rayner (1963), but it will be shown that Rayner's theory needs minor corrections in order to be strictly self consistent. For applications to neutron stars a purely Hookean theory will in any case be inadequate and to replace it a high-pressure quasi-Hookean idealization is described. This idealization includes the important special case of the perfect (i.e. isotropic) solid in which the elastic properties may be described in terms of just two (nonlinear) functions of state, namely the pressure and the modulus of rigidity. The theory reduces to the familiar perfect fluid idealization in the limit when the modulus of rigidity is zero. The discussion is conceptually based on the use of two distinct manifolds: a four-dimensional manifold $M$ of space-time events, with a fundamental pseudo-Riemannian metric, and a three-dimensional manifold $\germ{X}$ of idealized particles, with a fundamental measure representing a conserved particle number density. These manifolds are related by a canonical projection which sends world-lines in $M$ onto points in $\germ{X}$.