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 term s 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 fourdimensional manifold of space-time events, with a fundamental pseudo-Riemannian metric, and a three-dimensional manifold X of idealized particles, with a fundam ental measure representing a conserved particle number density. These manifolds are related by a canonical projection which sends world-lines in M onto points in X.

Foundations of general relativistic high-pressure elasticity theory

Relativistic elastomechanics as a lagrangian field theory

The paper is concerned with the Einstein equations for a spherically symmetric static distribution of anisotropic matter. The equations are cast into a system of Fuchsian type ODE for certain scalar invariants of the strain. And then the existence and regularity of this ODE is studied under general constitutive relation. In the case the constitutive relation is given by a quadratic form of strain, it is also shown that the solutions stay regular up to the boundary of the material ball.

Static Solutions of the Einstein Equations for Spherically Symmetric Elastic Bodies

The relativistic interior dynamics of a spherically symmetric, non rotating star composed of an elastic material is analyzed. A suitable formulation of relativistic elasticity is used, and the corresponding second order Einstein equations are found. The equations arise naturally in a non-comoving frame and reduce, in the static case, to the “generalized” Tohnan-Oppenheimer-Volkhoff equations considered by J. Kijowski and the author in a previous paper. The charged case is also discussed.

The dynamical structure of the Einstein equations for a non-rotating starss

In this article it is shown that the equation derived by weber from the equation of geodesic deviation — the equation that constitutes the basis for the theoretical studies concerning the detection of gravitational waves — can be obtained as an approximation to an equation deduced from Cauchy's equation which governs an elastic continuum. This is achieved within the frame of the formalism of the theory of relativistic continua developed by A. C. Eringen and his collaborators and the present author. The use of piezoelectric crystals in order to measure the stresses that result from incident gravitational waves is also examined within the framework provided by this theory.

Harmonic oscillations of elastic continua and detection of gravitational waves

Elasticity theory in general relativity formulated from classical nonlinear three dimensional theory, discussing thermodynamics and weak field limit

Walter C. Hernandez

Papers

Elasticity Theory in General Relativity