The local and global geometrical aspects of the twin paradox in static spacetimes: I. Three spherically symmetric spacetimes

We investigate local and global properties of timelike geodesics in three static spherically symmetric spacetimes. These properties are of its own mathematical relevance and provide a solution of the physical `twin paradox' problem. The latter means that we focus our studies on the search of the longest timelike geodesics between two given points. Due to problems with solving the geodesic deviation equation we restrict our investigations to radial and circular (if exist) geodesics. On these curves we find general Jacobi vector fields, determine by means of them sequences of conjugate points and with the aid of the comoving coordinate system and the spherical symmetry we determine the cut points. These notions identify segments of radial and circular gepdesics which are locally or globally of maximal length. In de Sitter spacetime all geodesics are globally maximal. In CAdS and Bertotti--Robinson spacetimes the radial geodesics which infinitely many times oscillate between antipodal points in the space contain infinite number of equally separated conjugate points and there are no other cut points. Yet in these two spacetimes each outgoing or ingoing radial geodesic which does not cross the centre is globally of maximal length. Circular geodesics exist only in CAdS spacetime and contain an infinite sequence of equally separated conjugate points. The geodesic curves which intersect the circular ones at these points may either belong to the two-surface $\theta=\pi/2$ or lie outside it.Comment: 27 pages, 0 figures, typos corrected, version published in APP

Every timelike geodesic in anti--de Sitter spacetime is a circle of the same radius

We refine and analytically prove an old proposition due to Calabi and Markus on the shape of timelike geodesics of anti--de Sitter space in the ambient flat space. We prove that each timelike geodesic forms in the ambient space a circle of the radius determined by $\Lambda$, lying on a Euclidean two--plane. Then we outline an alternative proof for $AdS_4$. We also make a comment on the shape of timelike geodesics in de Sitter space.Comment: An expanded version of the work published in International Journal of Modern Physics D. 8 pages, 0 figure

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

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

Jacobi fields, conjugate points and cut points on timelike geodesics in special spacetimes

Several physical problems such as the `twin paradox' in curved spacetimes have purely geometrical nature and may be reduced to studying properties of bundles of timelike geodesics. The paper is a general introduction to systematic investigations of the geodesic structure of physically relevant spacetimes. The investigations are focussed on the search of locally and globally maximal timelike geodesics. The method of dealing with the local problem is in a sense algorithmic and is based on the geodesic deviation equation. Yet the search for globally maximal geodesics is non-algorithmic and cannot be treated analytically by solving a differential equation. Here one must apply a mixture of methods: spacetime symmetries (we have effectively employed the spherical symmetry), the use of the comoving coordinates adapted to the given congruence of timelike geodesics and the conjugate points on these geodesics. All these methods have been effectively applied in both the local and global problems in a number of simple and important spacetimes and their outcomes have already been published in three papers. Our approach shows that even in Schwarzschild spacetime (as well as in other static spherically symetric ones) one can find a new unexpected geometrical feature: instead of one there are three different infinite sets of conjugate points on each stable circular timelike geodesic curve. Due to problems with solving differential equations we are dealing solely with radial and circular geodesics.Comment: A revised and expanded version, self-contained and written in an expository style. 36 pages, 0 figures. A substantially abridged version appeared in Acta Physica Polonica

Canonical gauge-invariant variables for scalar perturbations in synchronous coordinates

Under an appropriate change of the perturbation variable Lifshitz-Khalatnikov propagation equations for the scalar perturbation reduce to d'Alembert equation. The change of variables is based on the Darboux transform.Comment: LaTeX, 9 page