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

TL;DR

This paper investigates the geometrical properties of timelike geodesics in static spherically symmetric spacetimes, focusing on Reissner--Nordström and ultrastatic metrics, revealing maximality of radial geodesics and properties of ultrastatic spacetimes.

Contribution

It provides explicit analysis of Jacobi fields and geodesic maximality in Reissner--Nordström and ultrastatic spacetimes, extending understanding of their geodesic structure.

Findings

Radial geodesics in Reissner--Nordström are locally and globally maximal.

In ultrastatic spacetimes, particles experience no gravitational force and have constant energy.

Explicit conditions for conjugate points in monopole spacetime.

Abstract

This is a consecutive paper 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--Nordstr\"{o}m 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.