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

In this paper we are concerned to reveal that any spacetime structure\slg,D,{\tau}_{[sg]\sslg},\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]\slg(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.


I. INTRODUCTION
In General Relativity, a Lorentzian spacetime structure (LSTS) M, g, D, τ g , ↑ represents a given gravitational field [22], generated by an energy-momentum distribution T ∈ sec T 2 0 M , which dynamics is determined by Einstein equation. In this paper we assume ab initio that the LSTS M, g, D, τ g , ↑ is only an effective description of a gravitational field [6,20], and that in reality all physical fields are to be interpreted in the sense of Faraday and living in a Minkowski spacetime structure M = R 4 ,g,D, τg, ↑ . Under this assumption, our main aim in this paper is to show (Section 2) that when the effective LSTS M, g, D, τ g , ↑ possess at least one nontrivial Killing vector field A ∈ sec T M , if we denote by A = g(A, ) ∈ sec 1 T * M the Killing 1-form field, then given the validity of Einstein equation, the field F = dA satisfies Maxwell like equations with a well determined current 1 . Moreover the Maxwell like equations satisfied by F are compatible with Einstein equation in a sense explained in the proof of Proposition 1. We moreover delve into this approach and prove (Section 3) that the Maxwell like equations (the one that follows from Einstein equation) may be written as a Navier-Stokes equation for an inviscid fluid, once we identify a relation between the Killing 1-forms A and 2Å =g(A, ) and identify the components ofF = dÅ to some variables which appear in the Navier-Stokes equation including as postulate that the electric like components ofF are equal to the components of the Lamb vector field l of the Navier-Stokes fluid plus a well determined vector field d that we impose to be a gradient of a smooth function χ. Of course, not any Killing vector field A of the LSTS M, g, D, τ g , ↑ satisfy that postulate, but as an example we presented in section III shows the postulate has some non trivial realizations. The equation dF = 0 then becomes then the Helmholtz equation for conservation of vorticity and since F =F + G (Remark 7) where G is a closed 2-form field, the homogeneous Maxwell like equation for F is also equivalent to the Helmholtz equation for conservation of vorticity. In addition, taking again into account that the Navier-Stokes fluid flows in Minkowski spacetime structure and that the LSTS structure M, g, D, τ g , ↑ is an effective one according to ideas developed in [6,20] the non homogeneous Maxwell like equation for F (and thus also the one forF ) written in terms of the flat connectionD results in a set of algebraic equations (Eq.(41)) for the components of A (or for A), which constrain the identification of its components to the fields appearing in the Navier-Stokes and Einstein equations since the energy-momentum tensor of the matter field gets related to the field variables associated to the Navier-Stokes equation model .
To summarize, we find Maxwell (like) and Navier-Stokes (like) equations that encodes in precise mathematical sense discussed below the contents of Einstein equation when some well defined conditions are satisfied for a given arbitrary nontrivial Killing vector field of the effective LSTS M, g, D, τ g , ↑ . In Section 4 we present our conclusions, where our achievements are briefly compared with other proposals to identify a correspondence between solutions of Einstein and Navier-Stokes equations in spacetimes of different dimensions.

II. THE MAXWELL LIKE EQUATION EQUIVALENT TO EINSTEIN EQUATION
In this Section we prove two lemmata, a proposition and a corollary whose objective is to obtain given the conditions satisfied by A mentioned in the introduction a Maxwell like equations for the electromagnetic like field F = dA (where A = g(A, )) with a well determined current like term. More precisely we have Killing vector field in a LSTS M, g, D, τ g , ↑ which represents a gravitational field generated by a given energy-momentum distribution T ∈ sec T 2 0 M according to Einstein equation. Then A satisfies the wave equation Before proceeding, note that the field F ∈ sec 2 T * M satisfies Maxwell equations with a current J = RA − 2T (A) that splits in a part J s = RA, of the "superconducting" type.
In order to present proofs for the propositions above, the bundle of differential forms is embedded in the Clifford bundle 5 -T * M = 4 r=0 T r M ֒→ Cℓ(M, g), where Cℓ(M, g) is the Clifford bundle of differential forms [18] where g is the metric of the cotangent bundle. In this way, given any basis [13]. To start the proof of the propositions we need two lemmata.
Lemma 3 If a vector field A ∈ sec T M , with M part of the structure M, g, D, τ g , ↑ is a Killing vector field then δ [19]. 3 In this paper we use the nomenclature and (whenever possible) the notations in [18]. The covariant D'Alembertian is = ∂ · ∂ where ∂ = ϑ µ De µ is the Dirac operator acting on sections of the Clifford bundle Cℓ(M, g). 4 Keep in mind that the validity of Einstein equation is an hypothesis in the proposition. 5 A ֒→ B means that A is embedded in B and A ⊆ B.
Proof. To prove the Lemma 3 it is only necessary to recall that (2) and that in the Clifford bundle formalism [18] it follows that δ g A = −∂ A. Then it reads and the lemma is proved.
is a Killing vector field then we have where ∂ = ϑ µ D eµ is the Dirac operator acting on the sections of the Clifford bundle Cℓ(M, g) and ∂ ∧ ∂ is the Ricci operator acting on sec where R µ ν are the components of the Ricci tensor.
Proof. To prove that ∂ ∧ ∂A = R µ A µ , it is well known that the Ricci operator is an extensorial entity [18], namely it satisfies ∂∧∂A = A µ ∂∧∂ϑ µ , and since 6 ∂∧∂ϑ µ = R µ it reads In order to prove that A = R µ A µ the definition of the covariant D'Alembertian is used [18], and it follows that Taking into account that and in addition that multiplying Eq.(4) by g σν it follows that and thus which proves the Lemma. 6 See Chapter 4 of [18]. Now all pre-requisites necessary to prove Proposition 1 are demonstrated and then its proof is provided in what follows:

Proof. (of Proposition1) Under the hypothesis of Proposition 1, Einstein equation (in geometrical units) is written as
and can be rewritten in the equivalent form Now, we use the fact that the Ricci operator satisfies ∂∧∂ϑ µ = R µ and moreover that it is an extensorial operator 7 , i. e., A µ (∂ ∧ ∂ϑ µ ) = ∂ ∧ ∂A. After multiplying Eq.(6) by A µ it follows that where which proves the first part of the proposition. To prove that Eq. (1) is compatible with Einstein equation we start using (1) can be written after some trivial algebra as (R µ ν − 1 2 Rδ µ ν + T µ ν )A µ = 0. Next we observe that even if some of the A µ are zero in a given coordinate basis it is always possible to find a new coordinate basis where all the A ′ µ = ∂x′ ν ∂x µ A µ = 0. In this new basis we have Now, Eq. (8) is a homogeneous system of linear equations for the variables A ′ µ and since all the Proof. (of Corollary 2) To prove the corollary we sum A = ∂ · ∂A to both members of Eq. (7) and take into account that for any C ∈ sec Cℓ(M, g) the following expression [18] ∂ 2 C = ∂ ∧ ∂C+∂ · ∂C holds. Then Of course, (1.7) in Papapetrou paper [16] obtained in component form. See also Eq.(5) of [5]. When Einstein equation is valid we can immediately write taking into account Eq.(6) and the corollary is proved.
Remark 5 . We remark that since ∂ = d − δ g we can write a single Maxwell like equation 8 for the field F associated to the Killing form A, i.e., 7 Note that the covariant D'Alembertian ∂ · ∂ is not an extensorial operator, i.e., in general ∂ · ∂A = Aµ∂ · ∂ϑ µ . 8 No misprint here! In [6] it was shown that if the manifold M is parallelizable 9 , i. e., there exists four global vector fields e a ∈ sec T M , a = 0, 1, 2, 3 with {e a } a basis for T M Take {g a } as the dual basis of the {e a }. If a LSTS M, g, D, τ g , ↑ is introduced by postulating that g = η ab g a ⊗ g b , then the gravitational field is described by field equations -equivalent in a precise mathematical sense to Einstein equation -satisfied by the potentials g a . In addition, they are derived through a variational principle from a Lagrangian density The field equations for the fields F a = dg a ∈ sec 2 T * M are: where the T a , as above, are the energy-momentum 1-form fields of the matter fields and the t a are energy-momentum 1-form fields of the gravitational field. They are indeed legitimate tensor objects since in [20] it has been proved that they have the very nice and straightforward expression Under the conditions above, if Eq. (6) is rewritten in the orthonormal cobasis {g a } it reads Then, as above, taking into account the definitions of the Ricci, the covariant D'Alembertian, and the Hodge Laplacian operators, together with Eq. (14) and denoting t(A) := t a A a − A a (∂ · ∂)g a , the equations of motion of our theory under those conditions are expressed: that can be summarized in a single equation with the use of the Dirac operator ∂ acting on sections of the Clifford bundle:

III. FROM MAXWELL EQUATION TO A NAVIER-STOKES EQUATION
In this Section we obtain a Navier-Stokes equation that follows from the Maxwell like equation obtained above (that as just showed above encodes Einstein equation) once we impose that the electric like components of F = dA satisfy Eq. (22). In order to do that we start with the observation that the original Navier-Stokes equation describes the non relativistic motion of a general fluid in Newtonian spacetime. It is not thus adequate to use -at least in principle -a general Lorentzian spacetime structure M, g, D, τ g , ↑ to describe a fluid motion. In fact we want to describe a fluid motion in a background spacetime such that the fluid medium, together with its dynamics, is equivalent to a Lorentzian spacetime governed by Einstein equation in the sense described below.
In order to proceed, it was proposed in [6] a theory of the gravitational field, where gravitation is interpreted as a plastic distortion of the Lorentz vacuum. In that theory the gravitational field is represented by a (1, 1)-extensor field h : sec 1 T * M → sec 1 T * M living in Minkowski spacetime 10 . The field h -generated by a given energymomentum distribution in some region U of Minkowski spacetime -distorts the Lorentz vacuum described by the global cobasis 11 {γ µ = dx µ }, dual with respect to the basis {e µ = ∂/∂x µ } of T M , thus generating the gravitational potentials g a = h(δ a µ γ µ ). 9 The motivation being Geroch theorem [9] which says that a necessary and sufficient condition for a 4-dimensional Lorentzian manifold M, g to admit spinor fields is that the orthonormal frame bundle be trivial, which implies that the manifold is parallelizable. 10 Minkowski spacetime is the structure M = R 4 , • g,D, τ• g , ↑ ), where • g is Minkowski metric,D is its Levi-Civita connection, and the remaining symbols define the spacetime orientation and the time orientation. 11 The x µ are global coordinate functions in Einstein-Lorentz Poincaré gauge for the Minkowskispacetime structure that are naturally adapted to an inertial reference frame e 0 = ∂/∂x 0 ,De 0 = 0 . More details on the concept of reference frames can be found, e.g., in [8,18] Now, in the inertial reference frame e 0 = ∂/∂x 0 (according to the Minkowski spacetime structure M = R 4 ,g,D, τg, ↑ ), we write using the global coordinate functions x µ for M ≃ R 4 , 12 where the vector function v = (v 1, v 2 , v 3 ) is to be identified with the 3-velocity of a Navier-Stokes fluid -in the inertial frame e 0 according to the conditions disclosed below. Also, V 0 denotes a scalar function representing an external potential acting on the fluid, and where the functions p and ρ are identified respectively with the pressure and density of the fluid and supposed functionally related, i.e., dp ∧ dq = 0. Furthermore v 2 : Before proceeding note that φ looks like the relativistic energy per unit mass of the fluid. Then we will write A as where the new potential function V is the sum of V 0 with the the sum of the Taylor expansion terms of We. proceed by identifying the magnetic like and the electric like components of the fieldF with components of the vorticity field and the Lamb vector field of the fluid. We remark that whereas the identification of the magnetic like components ofF with the vorticity field is natural, the identification of the electric like components of F is here postulated, i.e. we suppose thatF besides being derived from a Killing vector field associated to the structure also satisfies the constraint given by Eq.(22) below. We thus writeF µν = (dÅ) µν as were w is vorticity of the velocity field and is the so called Lamb vector and moreover where χ is a smooth function. 12 The basis {e µ } is the reciprocal basis of the basis {eµ}, i.e.,g(e µ , eν ) = δ µ ν .

Remark 6
Before proceeding it is important to emphasize that the identification of the components ofF with the components of the Lamb and vorticity fields of a fluid selects a preferred reference frame, namely the inertial reference frame e 0 = ∂/∂x 0 introduced above.
At this point we recall that the non relativistic Navier-Stokes equation for an inviscid fluid is given by [2,7] ∂v ∂t or using a well known vector identity, By these identifications 13 , we get a Navier-Stokes like equation from the straightforward identification of l − d = (F 01 ,F 02 ,F 03 ) and w = (F 32 ,F 13 ,F 21 ). Indeed, we have where ǫ ijk is the 3-dimensional Kronecker symbol. Eq. (26) becomes and since d = −∇χ for some smooth function χ then Eq.(28) can be written as which is now a Navier-Stokes like equation for a fluid moving in an external potetntial V ′ = V + χ. Moreover, the homogeneous Maxwell equation dF = 0 is equivalent to which express Helmholtz equation for conservation of vorticity.
Remark 7 Note that since A = g(A, ) =A µ γ µ =Å µ g µν γ ν , we can write 14 where the extensor field g is defined by g(γ µ ) = g µν γ ν . Thus, in general we can write since d(F −F ) = 0 , F =F + G where G is a closed 2-form field 15 . So, dF = dF = 0 express the same content, namely Eq.(30), the Helmholtz equation for conservation of vorticity.
Remark 8 Of course, the main idea of this paper that we can get from the Maxwell like equation that follows from the Einstein equation a Navier-Stokes euation for an invisicid fluid depends on the existence of nontrivial solutions of Eq.(20), i.e.,F 0i = (w × v) i + d i . So, it is important to show that this equation has nontrivial realizations for at least some Killing vector fields living in M , such that M, g, D, τ g , ↑ models a gravitational field. That this is the case, is easily seem if we take, e.g., the Schwarzschild spacetime structure. As well known [12] the vector field A =∂ ϕ = −x 2 ∂ x 1 + x 1 ∂ x 2 is a Killing vector field for the Schwarzschild metric 16 . The 1-form field corresponding to it and living in Minkowski spacetime isÅ = x 2 dx 1 − x 1 dx 2 . Thus φ = 0 and v = (x 2 , −x 1 , 0). This gives and Eq.(29) holds.

Remark 9
For the example given in Remark 8 we have simply A = fÅ where f = r 2 cos 2 θ. Thus in this case, we have the simple expression There are many examples of Killing vector fields for which A = fÅ and for such fields that developments given below in terms of A are easily translated in terms ofÅ. In particular, when A = fÅ we have the following identification of the components of the Lamb and vorticity vector fields with the components of F , To continue, we recall that in order for the Navier-Stokes equation just obtained to be compatible with Einstein equation it is necessary yet to take into account Eq.(9), the non homogeneous Maxwell equation written in the structure M, g, D, τ g , ↑ , and the fact that A is in the Lorenz gauge, namely δ g A = 0, since these equations produce as we are going to see constraints among the several fields involved, The constraints involving the components of A = g(A,) with A defined in Eq. (17) are also encoded in Eq.(1), which must now be expressed in terms of the objects defining the Minkowski spacetime structure M = R 4 ,g,D, τg, ↑ . Now, taking into account thatDg = 0, we have where A ∈ sec T 2 0 M ⊗ 1 T * M is the non metricity tensor of D with respect tog. In the coordinates x µ introduced above it follows that Then, as it is well known 17 the relation between the coefficients Γ ν µα andΓ ν µα associated to the connections D andD (D e µ ϑ ν = −Γ ν µα ϑ α ,D e µ ϑ ν = −Γ ν µα ϑ α ) in an arbitrary coordinate vector { ∂ ∂x µ } and covector {ϑ ν = dx ν } basesassociated to arbitrary coordinate functions {x µ } covering U ⊂ M -are given by 18 where are the components of the so called strain tensor of the connection. In the coordinate bases { ∂ ∂x µ } and {γ µ = dx µ }, associated to the coordinate functions x µ , it follows thatΓ ν µα = 0 and in addition the following relation for the Ricci tensor of D holds: R µν = J (µν) . 16 The spherical coordinate functions are (r,θ, ϕ). 17 See, e.g., [18]. 18 We use thatg =gµν ϑ µ ⊗ ϑ ν =g µν ϑµ ⊗ ϑν , where {ϑµ} is the reciprocal basis of {ϑ µ }, namely ϑµ =gαµϑ α andg µνg µκ = δ ν κ . In the bases associated to x µ it isg = ηµν γ µ ⊗ γ ν = η µν γ µ ⊗ γ ν [3,4].
Denoting K ρ αβ = − 1 2 S ρ αβ , the J (µν) is the symmetric part of Now, if the Dirac operator associated to the Levi-Civita connectionD ofg is introduced by it can be shown that 19 where A = A µ γ µ ,Ǎ κ := η βκ g βσ A σ , and L α = η αβ J βσ γ σ . The symbol · g denotes the scalar product accomplished According to Eq.(3) ∂ ∧ ∂A = A and thus Eq.(1) can also be written as Taking into account Eq.(40), the following algebraic equation 20 , relating the components A σ to the components of the energy-momentum tensor of matter and the components of the g field that is part of the original LSTS, is obtained: Eq.(41) are the constraints need to be satisfied by the variables of our theory in order for the Navier-Stokes equation to be compatible with the contents of Einstein equation. As a last remark we observe that Eq.(41) may be also interpreted as an equation providing the energy-momentum tensor of the matter field as a function of the variables entering the Navier-Stokes identification.

IV. CONCLUSIONS
We demonstrated that for each Lorentzian spacetime representing a gravitation field in General Relativity which contains an arbitrary Killing vector field A, the field F = dA (where A = g(A, )) satisfies Maxwell like equations with well determined current 1-form field. Moreover we showed that for all Killing 1-form fieldsÅ =g(A, ), when some identifications of the components ofÅ and the variables entering the Navier-Stokes equation are accomplished and in particular when the postulated nontrivial condition (Eq.(20)-F 0i = (dÅ) 0i = l i − d i -is satisfied, the Maxwell like equations forF and thus the ones for F can be written as a Navier-Stokes equation representing an inviscid fluid. Thus, the Maxwell and Navier-Stokes like equations found in this paper 21 are almost directly obtained from Einstein equation through thoughtful identification of fields. All fields in our approach live in a 4-dimensional background spacetime, namely Minkowski spacetime and the Lorentzian spacetime structures M, g, D, τ g , ↑ is considered only a (sometimes useful) description of gravitational fields. Thus our approach is in contrast with the very interesting and important studies in, e. g., [1,10,17] where it is shown through some identifications that every solution for an incompressible Navier-Stokes equation in a (p + 1)-dimensional spacetime gives rise to a solution of Einstein equation in (p+2)-dimensional spacetime. It is worth also to quote [15] where it is also suggested an interesting relation between Einstein equations and the Navier-Stokes equation. Finally we remark that it is clear that we can find examples [21] of Lorentzian spacetimes that do not have any nontrivial Killing vector field. However, as asserted in Weinberg [26], all Lorentzian spacetimes that represent gravitational fields of physical interest possess some Killing vector fields, and we exhibit an example (Remark 8) the approach of the present paper applies.