Behavior of spacelike geodesics in the extended Schwarzschild manifold

Abstract

We demonstrate that there exist unstable, spacelike, circular orbits in the Schwarzschild field for all radii in the range $0<r<3m$. The conditions under which spacelike trajectories bend toward or away from the source of the field are derived for the entire $r\ensuremath{-}\ensuremath{\varphi}$ plane. We show that any nonradial spacelike geodesic with turning point less than $2m$ will appear spacelike over the entire $u\ensuremath{-}v$ plane. The scattering and capture cross sections for a particle on a spacelike trajectory are evaluated. Finally, we suggest that there are compelling reasons for rejecting the usual assumption of a global past-future relation in the extended Schwarzschild manifold.