In this paper we analyze trajectories of spacelike curves which are critical points of a Lagrangian depending on its total torsion. We focus on two important families of spacetimes, Generalized Robertson-Walker and standard static spacetimes. For the former, we show that such trajectories are those with constant curvature. For the latter we also obtain a characterization in terms of the curvature of the trajectory, but in this case measured with an appropriate conformal metric.
In this work, we study spacelike trajectories of a relativistic particle model whose Lagrangian is given by the total torsion on its worldline. Once the field equations are established, we investigate the trajectories in two physically relevant families of four-dimensional spacetimes: Generalized Robertson-Walker and standard static spacetimes. For the former we obtain both, characterization as well as non-existence results for Frenet curves of different orders, which are the curves capable of modeling trajectories. For standard static spacetimes, we obtain a full characterization of the critical curves and present an example in which we can explicitly compute non-trivial trajectories.
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