Deformed Spacetime pp 379-388 | Cite as

# Complete Solutions of Geodesic Equations

We discussed in Chap. 24 the solutions of the geodesic equations for the four phenomenological metrics of the fundamental interactions, obtained as special cases of the classes of solutions of the vacuum Einstein equations in the Power Ansatz. However, it is easily seen that they hold only for the energy ranges where the metrics are not Minkowskian (namely below threshold for the electromagnetic and weak metrics, and above threshold for the strong and gravitational ones). Moreover, in most cases the value of the parameter *r* was fixed (as functions of the other coefficients *qμ*, *μ* = 0, 1, 2, 3) by the structure of the Einstein equations. We want now to give the general solutions of the geodesic equations for the four interactions, starting from the general form of the metrics (20.21)–(20.23), obtained by the 5D embedding of the 4D DSR phenomenological metrics in the DR5 framework. As already stressed, such a procedure leaves undetermined the fifth metric coefficient *f*(*x*5), and therefore yields *r*-parametrized metrics.

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