Geodesics of deformed relativity in five dimensions

In a previous paper, we discussed the Killing symmetries of the Kaluza-Klein-like scheme known as Deformed Relativity in five dimensions (DR5), based on a five-dimensional Riemannian space 5 in which the four-dimensional space-time metric is deformed (i.e. it depends on the energy) and energy plays the role of the fifth dimension. In the present paper, we carry on the investigation of the main mathematical aspects of DR5 by studying the geodesic motions in 5. In particular, we consider the case of physical relevance in which the metric coefficients are power functions of the energy (Power Ansatz). The geodesic equations are solved explicitly for all the twelve 5-d. metrics obtained as solutions of the vacuum Einstein equations, and in particular for those describing the four fundamental interactions. It is also shown that it is possible, from the geodesic motion related to one of these Power-Ansatz solutions, to get a time-energy uncertainty relation of the Heisenberg type.