An exact solution of Einstein's equations for two particles falling freely in an external gravitational field

Abstract

I begin with a Weyl axially symmetric, static metric representing a spherical particle in equilibrium under the attraction of a semi-infinite rod (s.i.r.) of line density 1/2 and another, pseudo-uniform, gravitational field. A coordinate transformation is then used to remove the s.i.r., enlarge the spacetime, and make the solution time-dependent. The result represents two spherical particles (which do not interact because each is outside the null cone of the other) moving in a certain gravitational wave field. It is shown that the particles move on geodesies of the background wave field. The sources of the wave field are briefly investigated.