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

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 Eg of GR defined as the acceleration of freefalling quasi-static particles relative to the observer, divEg measured in the reference frame of a GR-noninertial observer is different from the curvature R00. We derive our exact, explicit, and simple gravito-Gauss law for divEg in our new reference frame of a GR-noninertial observer with his LONB (Local Ortho-Normal Basis ¯eˆa) and his LONB-connections (ωˆbˆa)ˆc in his time- and 3-directions: the sources of divEg are contributed by all fields including the GR-gravitational fields (Eg,Bg). In the reference frame of a GR-inertial observer our gravito-Gauss law coincides with with Einstein’s R00 equation, which does not have gravitational fields as sources. We derive the gravito-Ampère law for curlBg, the gravito-Faraday law for curlEg, and the law for divBg. The densities of energy, momentum, and momentum-flow of GR-gravitational fields (Eg,Bg) 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 con nections, 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. ; ISSN:0264-9381 ; ISSN:1361-6382