Particle motion in weak relativistic gravitational fields

We derive the geodesic equation of motion in the presence of weak gravitational fields produced by relativistic sources such as cosmic strings, decomposed into scalar, vector and tensor parts. To test the result, we perform the first N-body simulations with relativistic weak gravitational external fields. Our test case is a moving straight string, for which we recover the well-known result for the impulse on nonrelativistic particles. We find that the vector (gravito-magnetic) force is an essential contributor. Our results mean that it is now possible to incorporate straightforwardly into N-body simulations all weak relativistic sources, including networks of cosmic defects.