The local and global geometrical aspects of the twin paradox in static spacetimes: II. Reissner--Nordstr'{o}m and ultrastatic metrics

摘要

This is a consecutive paper on the timelike geodesic structure of staticspherically symmetric spacetimes. First we show that for a stable circularorbit (if it exists) in any of these spacetimes all the infinitesimally closeto it timelike geodesics constructed with the aid of the general geodesicdeviation vector have the same length between a pair of conjugate points. InReissner--Nordstr\"{o}m black hole metric we explicitly find the Jacobi fieldson the radial geodesics and show that they are locally (and globally) maximalcurves between any pair of their points outside the outer horizon. If a radialand circular geodesics in R--N metric have common endpoints, the radial one islonger. If a static spherically symmetric spacetime is ultrastatic, itsgravitational field exerts no force on a free particle which may stay at rest;the free particle in motion has a constant velocity (in this sense the motionis uniform) and its total energy always exceeds the rest energy, i.~e.~it hasno gravitational energy. Previously the absence of the gravitational force hasbeen known only for the global Barriola--Vilenkin monopole. In the spacetime ofthe monopole we explicitly find all timelike geodesics, the Jacobi fields onthem and the condition under which a generic geodesic may have conjugatepoints.