Space time geometry in the atomic hydrogenoid system. Approach to a dust relativistic model from causal quantum mechanics

We intend to use the description of the electron orbital trajectory in the de Broglie-Bohm (dBB) theory to assimilate to a geodesic corresponding to the General Relativity (GR) and get from it physical conclusions. The dBB approach indicates us the existence of a non-local quantum field (corresponding with the quantum potential), an electromagnetic field and a comparatively very weak gravitatory field, together with a translation kinetic energy of electron. If we admit that those fields and kinetic energy can deform the space time, according to Einstein’s field equations (and to avoid the violation of the equivalence principle as well), we can made the hypothesis that the geodesics of this space-time deformation coincide with the orbits belonging to the dBB approach (hypothesis that is coherent with the stability of matter). From it, we deduce a general equation that relates the components of the metric tensor. Then we find an appropriate metric for it, by modification of an exact solution of Einstein’s field equations, which corresponds to dust in cylindrical symmetry. The found model proofs to be in agreement with the basic physical features of the hydrogen quantum system, particularly with the independence of the electron kinetic momentum in relation with the orbit radius. Moreover, the model can be done Minkowski-like for a macroscopic short distance with a convenient election of a constant. According to this approach, the guiding function of the wave on the particle could be identified with the deformations of the space-time and the stability of matter would be easily justified by the null acceleration corresponding to a geodesic orbit.