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

TL;DR

This paper demonstrates that in certain spacetimes with a Killing vector field, Einstein's equations imply Maxwell-like equations with superconducting currents, which can further be reformulated as Navier-Stokes equations under specific conditions.

Contribution

It reveals a direct derivation of Maxwell and Navier-Stokes equations from Einstein's equations in spacetimes with Killing vectors, establishing a novel link between gravitational and fluid dynamics.

Findings

Maxwell-like equations with superconducting currents derived from Einstein equations.

Under additional conditions, Maxwell-like equations can be expressed as Navier-Stokes equations.

Sequential derivation of Einstein, Maxwell, and Navier-Stokes equations from the spacetime structure.

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.