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In this paper we are concerned to reveal that any spacetime structure<M,[g]<LaTeX>\slg</LaTeX>,D,{\tau}_{[sg]<LaTeX>\sslg</LaTeX>},\uparrow>, whichis a model of a gravitational field in General Relativity generated by anenergy-momentum tensor T --- and which contains at least one nontrivial Killingvector field A --- is such that the 2-form field F=dA (whereA=[g]<LaTeX>\slg</LaTeX>(A,)) satisfies a Maxwell like equation --- with a welldetermined current that contains a term of the superconducting type--- whichfollows directly from Einstein equation. Moreover, we show that the resultingMaxwell like equations, under an additional condition imposed to the Killingvector field, may be written as a Navier-Stokes like equation as well. As aresult, we have a set consisting of Einstein, Maxwell and Navier-Stokesequations that follows sequentially from the first one under precisemathematical conditions and once some identifications about field variables areevinced, as detailed explained throughout the text. We compare and emulate ourresults with others on the same subject appearing in the literature.

The Maxwell and Navier-Stokes that Follow from Einstein Equation in a Spacetime Containing a Killing Vector Field