Relativistic theory of elastic deformable astronomical bodies: perturbation equations in rotating spherical coordinates and junction conditions

In this paper, the dynamical equations and junction conditions at the interface between adjacent layers of different elastic properties for an elastic deformable astronomical body in the first post-Newtonian approximation of Einstein theory of gravity are discussed in both rotating Cartesian coordinates and rotating spherical coordinates. The unperturbed rotating body (the ground state) is described as uniformly rotating, stationary and axisymmetric configuration in an asymptotically flat space-time manifold. Deviations from the equilibrium configuration are described by means of a displacement field. In terms of the formalism of relativistic celestial mechanics developed by Damour, Soffel and Xu, and the framework established by Carter and Quintana the post Newtonian equations of the displacement field and the symmetric trace-free shear tensor are obtained. Corresponding post-Newtonian junction conditions at interfaces also the outer surface boundary conditions are presented. The PN junction condition is an extension of Wahr's one which is a Newtonian junction conditions without rotating.