295 research outputs found
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 or lie outside
it.Comment: 27 pages, 0 figures, typos corrected, version published in APP
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
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