Maxwell, Einstein, Dirac and Navier-Stokes Equations

Abstract

In the previous chapters we exhibit several different faces of Maxwell, Einstein and Dirac equations. In this chapter we show that given certain conditions we can encode the contents of Einstein equation in Maxwell like equations for a field \(F = dA \in \sec \bigwedge \nolimits ^{2}T^{{\ast}}M\) (see below), whose contents can be also encoded in a Navier-Stokes equation. For the particular cases when it happens that F2 ≠ 0 we can also using the Maxwell-Dirac equivalence of the first kind discussed in Chap. 13 to encode the contents of the previous quoted equations in a Dirac-Hestenes equation for \(\psi \in \sec (\bigwedge \nolimits ^{0}T^{{\ast}}M + \bigwedge \nolimits ^{2}T^{{\ast}}M + \bigwedge \nolimits ^{4}T^{{\ast}}M)\) such that \(F =\psi \gamma ^{21}\tilde{\psi }.\) Specifically, we first show in Sect. 15.1 how each LSTS \((M,\boldsymbol{g},D,\tau _{\boldsymbol{g}},\uparrow )\) which, as we already know, is a model of a gravitational field generated by \(\mathbf{T} \in \sec T_{2}^{0}M\) (the matter plus non gravitational fields energy-momentum tensor) in Einstein GRT is such that for any \(\mathbf{K} \in \sec TM\) which is a vector field generating a one parameter group of diffeomorphisms of M we can encode Einstein equation in Maxwell like equations satisfied by F = dK where \(K =\boldsymbol{ g}(\boldsymbol{K},\) ) with a well determined current term named the Komar current\(J_{\boldsymbol{K}}\), whose explicit form is given. Next we show in Sect. 15.2 that when K=A is a Killing vector field, due to some noticeable results [Eqs. (15.28) and (15.29)] the Komar current acquires a very simple form and is then denoted \(J_{\boldsymbol{A}}\). Next, interpreting, as in Chap. 11 the Lorentzian spacetime structure \((M,\boldsymbol{g},D,\tau _{\boldsymbol{g}},\uparrow )\) as no more than an useful representation for the gravitational field represented by the gravitational potentials \(\{\mathfrak{g}^{a}\}\) which live in Minkowski spacetime (here denoted by \((M = \mathbb{R}^{4},\boldsymbol{ \mathring{g} }, \mathring{D} ,\tau _{\boldsymbol{ \mathring{g} }},\uparrow )\)) we show in Sect. 15.3 that we can find a Navier-Stokes equation which encodes the contents of the Maxwell like equations (already encoding Einstein equations) once a proper identification is made between the variables entering the Navier-Stokes equations and the ones defining \( \mathring{A} =\boldsymbol{ \mathring{g} }(\boldsymbol{A},)\) and \( \mathring{F} = d \mathring{A} \), objects clearly related [see Eq. (15.49)] to A and F = dA. We also explicitly determine also the constraints imposed by the nonhomogeneous Maxwell like equation \(\mathop{\delta }\limits_{\boldsymbol{g}}F = -J_{\boldsymbol{A}}\) on the variables entering the Navier-Stokes equations and the ones defining A (or \( \mathring{A} \)).