Several physical problems such as the "twin paradox" in curved spacetimes have a purely geometrical nature and are 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. These are focussed on the search of locally maximal timelike geodesics. The method is based on determining conjugate points on chosen geodesic curves. The method presented here is effective at least in the case of radial and circular geodesics in static spherically symmetric spacetimes. Our approach shows that even in Schwarzschild spacetime (as well as in other static spherically symmetric ones), one can find a new unexpected geometrical feature: each stable circular orbit contains besides the obvious set of conjugate points two other sequences of conjugate points. The obvious limitations of the approach arise from one’s inability to solve involved ordinary differential equations and the recent progress in the field allows one to increase the range of metrics and types of geodesic curves tractable by this method.
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