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Several physical problems such as the `twin paradox' in curved spacetimeshave purely geometrical nature and may be reduced to studying properties ofbundles of timelike geodesics. The paper is a general introduction tosystematic investigations of the geodesic structure of physically relevantspacetimes. The investigations are focussed on the search of locally andglobally maximal timelike geodesics. The method of dealing with the localproblem is in a sense algorithmic and is based on the geodesic deviationequation. Yet the search for globally maximal geodesics is non-algorithmic andcannot be treated analytically by solving a differential equation. Here onemust apply a mixture of methods: spacetime symmetries (we have effectivelyemployed the spherical symmetry), the use of the comoving coordinates adaptedto the given congruence of timelike geodesics and the conjugate points on thesegeodesics. All these methods have been effectively applied in both the localand global problems in a number of simple and important spacetimes and theiroutcomes have already been published in three papers. Our approach shows thateven in Schwarzschild spacetime (as well as in other static sphericallysymetric ones) one can find a new unexpected geometrical feature: instead ofone there are three different infinite sets of conjugate points on each stablecircular timelike geodesic curve. Due to problems with solving differentialequations we are dealing solely with radial and circular geodesics.

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