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 freefalling particles (= geodesic deviation) measured in the local reference frame of a noninertial observer in general relativity (GR) is not given by the Riemann tensor. With the gravito-electric field of GR defined as the acceleration of freefalling quasistatic particles relative to the observer, measured in the reference frame of a GR-noninertial observer is different from the curvature . We derive our exact, explicit, and simple gravito-Gauss law for 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 are contributed by all fields including the GR-gravitational fields . In the reference frame of a GR-inertial observer our gravito-Gauss law coincides with with Einstein’s equation, which does not have gravitational fields as sources. We derive the gravito-Ampère law for , the gravito-Faraday law for , and the law for . The densities of energy, momentum, and momentum-flow of GR-gravitational fields are local observables, but they depend on the observer with his local reference frame: if measured by a GR-inertial observer on his worldline in his frame of LONB connections, these quantities are zero. 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. The sources in the gravito-Gauss law contributed by gravitational energy and momentum-flow densities have a repulsive effect on the gravitational acceleration-difference of particles. This contributes to the accelerated expansion of our inhomogeneous Universe today.