De Sitter geodesics

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The geodesics on the (1 + 3)-dimensional de Sitter (dS) spacetime are considered studying how their parameters are determined by the conserved quantities in the conformal Euclidean, Friedmann–Lemaître–Robertson–Walker, de Sitter–Painlevé and static local charts with Cartesian space coordinates. Moreover, it is shown that there exists a special static chart in which the geodesics are genuine hyperbolas whose asymptotes are given by the conserved momentum and the associated dual momentum.