The Gravito-Maxwell Equations of General Relativity in the local reference frame of a GR-noninertial observer

Abstract

We show that the acceleration-difference of neighboring free-falling particles (= geodesic deviation) measured in the local reference frame of a GR-noninertial observer is not given by the Riemann tensor. With the gravito-electric field of GR defined as the acceleration of free-falling quasistatic particles relative to the observer, the divergence of the gravito-electric field measured in the reference frame of a GR-noninertial observer is different from the Ricci curvature $R^0_{\,\,0}$. We derive our exact, explicit, and simple gravito-Gauss law for the divergence of the gravito-electric field in our new reference frame of a GR-noninertial observer with his LONB (Local Ortho-Normal Basis) and his LONB-connections in his time and 3-directions: the sources of the divergence of the gravito-electric field are contributed by all fields including the GR-gravitational fields, gravito-electric and gravito-magnetic. In the reference frame of a GR-inertial observer our gravito-Gauss law coincides with Einstein's $R^0_{\,\,0}$ equation, which does not have gravitational fields as sources. We derive the gravito-Ampere law, the gravito-Faraday law and the law for the divergence of the gravito-magnetic field. The densities of energy, momentum, and momentum-flow of GR-gravitational fields are local observables, but depend on observer with his local reference frame: these quantities are zero if measured by a GR-inertial observer. For a GR-noninertial observer the sources of gravitational energy, momentum, and momentum-flow densities have the opposite sign from the electromagnetic and matter sources. In the gravito-Gauss law the sources contributed by gravitational energy and momentum-flow densities have a repulsive effect on the gravitational acceleration-difference of particles.