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The local and global geometrical aspects of the twin paradox in static spacetimes. II: Reissner-Nordström and ultrastatic metrics. (English) Zbl 1371.83046

Summary: This is a consecutive paper (for Part I, see [ibid. 45, No. 5, 1051–1075 (2014; Zbl 1371.83045)]) on the timelike geodesic structure of static spherically symmetric spacetimes. First, we show that for a stable circular orbit (if it exists) in any of these spacetimes all the infinitesimally close to it timelike geodesics constructed with the aid of the general geodesic deviation vector have the same length between a pair of conjugate points. In Reissner–Nordström black hole metric, we explicitly find the Jacobi fields on the radial geodesics and show that they are locally (and globally) maximal curves between any pair of their points outside the outer horizon. If a radial and circular geodesics in R–N metric have common endpoints, the radial one is longer. If a static spherically symmetric spacetime is ultrastatic, its gravitational 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 motion is uniform) and its total energy always exceeds the rest energy, i.e. it has no gravitational energy. Previously, the absence of the gravitational force has been known only for the global Barriola–Vilenkin monopole. In the spacetime of the monopole, we explicitly find all timelike geodesics, the Jacobi fields on them and the condition under which a generic geodesic may have conjugate points.

MSC:

83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory
53C22 Geodesics in global differential geometry
83C50 Electromagnetic fields in general relativity and gravitational theory

Citations:

Zbl 1371.83045
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