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

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