Abstract
The model governed by an action measuring the total proper acceleration of trajectories provides a nice framework one to describe the dynamics of massless relativistic particles. In high rigidity cases, metrics with constant curvature, the model is consistent only in spherical three spaces and in three-dimensional anti de Sitter backgrounds, according to a Riemannian or a Lorentzian context, respectively. In contrast to flat gravitational fields, the existence of nontrivial trajectories are shown in a family of three spaces whose metrics admit a certain degree of symmetry. Such trajectories are included in regions with real presence of matter. An algorithm to obtain them is also designed.
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