General-relativistic theory of elastic deformable astronomical bodies

Abstract

This paper deals with general perturbations of astronomical bodies in the first post-Newtonian approximation of Einstein's theory of gravity. The formalism starts with a description of some stationary axisymmetric configuration in an asymptotically flat space-time manifold. General perturbations of this configuration are first treated within the Carter-Quintana framework where the bodies are assumed to be composed of some perfectly elastic material. Deviations from the equilibrium configuration are described by means of some displacement field as in the local theory for the global motion of Earth in space. As central results the post-Newtonian perturbation equations for the displacement field, the perturbed field, and local matter variables are presented. Especially for further discussions of global geodynamics (precession, nutation, polar motion, length of day variations) in the framework of general relativity the results presented here might serve as a new starting point. We also derive the perturbed Euler equation for vanishing shear stresses, i.e., for a fluid body, by means of a linearization procedure around the equilibrium state. After taking the Eulerian variations of the local matter variables in terms of the displacement field from the Carter-Quintana framework the two methods are found to produce the same results.