Motion of a small body through an external field in general relativity calculated by matched asymptotic expansions

Abstract

This paper shows that a small body with possibly strong internal gravity moves through an empty region of a curved, and not necessarily asymptotically flat, external space-time on an geodesic. By approximate geodesic, one means the following: Suppose the ratio $\ensuremath{\epsilon}\ensuremath{\equiv}\frac{m}{L}$, where $m$ is the body's mass and $L$ is a curvature reference length of the unperturbed external field, is a small parameter. Then $O(L)$ deviations from geodesic motion in the unperturbed external field vanish over times of $O(L)$, with possible $O(L)$ corrections occurring only over times of order $\frac{L}{\ensuremath{\epsilon}}$ or longer. The world line is here calculated directly from the Einstein field equation using a generalized method of matched asymptotic expansions based on a previous paper concerning singular perturbations on manifolds and related to a technique used by D'Eath. Aside from D'Eath's work, previous results on the motion of realistic bodies have assumed weak internal gravity, in some cases incorporating additional assumptions such as perfect fluids or high symmetry. This calculation makes no assumptions about the details of the body, such as weak fields, symmetry, the equations of state for matter, or even the presence of matter. Most previous treatments assumed asymptotic flatness of the external field. Here, it is only assumed that, in the region of interest, the external spacetime is empty and free of singularities. The results extend the work of D'Eath to a more general class of objects that includes nonstationary black holes, naked singularities, and neutron stars, as well as ordinary astrophysical objects. This method can be applied to related problems, such as the motion of a charged black hole through an external gravitational and electromagnetic field. A future paper will combine this method with Burke's method of obtaining radiation reaction to calculate the orbital-period shortening of gravitationally bound, slow-motion systems, such as the binary pulsar PSR 1913 + 16, containing objects with strong internal gravity.