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 F 2 ≠ 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 \( \mathring{A} \)).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Komar called a related quantity the generalized flux.
- 2.
V denotes a spacelike hypersurface and S = ∂ V its boundary. Usually the integral \(\mathfrak{E}\) is calculated at a constant x 0 time hypersurface and the limit is taken for S being the boundary at infinity.
- 3.
Note that since \(\mathop{\delta }\limits_{\boldsymbol{g}}(\mathcal{G}^{\mu }K_{\mu }) = 0\) it follows from Eq. (15.16) that indeed \(\mathop{\delta }\limits_{\boldsymbol{g}}L = 0\).
- 4.
Something that is not given in [10].
- 5.
Observe that when A is a Killing vector field the quantities \(\int \nolimits _{V } \star \boldsymbol{ T}(A)\) and \(\int \nolimits _{V }\frac{1} {2} \star TA\) are separately conserved as it is easily verified.
- 6.
An equivalent formula appears, e.g., as Eq. (11.2.10) in [23]. However, it is to be emphasized here the simplicity and transparency of our approach concerning traditional ones based on classical tensor calculus.
- 7.
The total system is the system consisting of the gravitational plus matter and non gravitational fields.
- 8.
- 9.
More details may be found in [7].
- 10.
The {x μ} are global coordinate functions in Einstein-Lorentz Poincaré gauge for the Minkowski spacetime that are naturally adapted to an inertial reference frame \(e_{0} = \partial /\partial \mathtt{x}^{0}, \mathring{D} e_{0} = 0\).
- 11.
The basis {e μ} is the reciprocal basis of the basis {e μ }, i.e., \(\boldsymbol{ \mathring{g} }(e^{\mu },e_{\nu }) =\delta _{ \nu }^{\mu }.\)
- 12.
- 13.
We have (details in [7]) \(\mathit{g} =\boldsymbol{ h}^{\dag }\boldsymbol{h}\) and \(\boldsymbol{ \mathring{g} }(\mathit{g}(\boldsymbol{\gamma }_{\mu }),\boldsymbol{\gamma }_{\nu }) = g_{\mu \nu } =\boldsymbol{ \mathring{g} }(\boldsymbol{h}(\boldsymbol{\gamma }_{\mu }),\boldsymbol{h}(\boldsymbol{\gamma }_{\nu })) =\boldsymbol{ \mathring{g} (}\mathfrak{g}_{\mu }, \mathfrak{g}_{\nu })\).
- 14.
The spherical coordinate functions are (r,\(\theta,\varphi )\).
- 15.
We use that \(\boldsymbol{ \mathring{g} } = \mathring{g} _{\mu \nu }\vartheta ^{\mu } \otimes \vartheta ^{\nu } = \mathring{g} ^{\mu \nu }\vartheta _{\mu } \otimes \vartheta _{\nu }\), where \(\{\vartheta _{\mu }\}\) is the reciprocal basis of \(\{\vartheta ^{\mu }\}\), namely \(\vartheta _{\mu } = \mathring{g} _{\alpha \mu }\vartheta ^{\alpha }\) and \( \mathring{g} ^{\mu \nu } \mathring{g} _{\mu \kappa } =\delta _{ \kappa }^{\nu }\). In the bases associated to {x μ} it is \(\boldsymbol{ \mathring{g} } =\eta _{\mu \nu }\boldsymbol{\gamma }^{\mu } \otimes \boldsymbol{\gamma }^{\nu } =\eta ^{\mu \nu }\boldsymbol{\gamma }_{\mu } \otimes \boldsymbol{\gamma }_{\nu }\).
- 16.
Of course, it is a partial differential equation that needs to be satisfied by the components of the stress tensor of the connection.
- 17.
- 18.
For other papers relating Einstein equations and Navier-Stokes equations we quote also here that the authors in [2] show that cosmic censorship might be associated to global existence for Navier-Stokes or the scale separation characterizing turbulent flows, and in the context of black branes in AdS5, Einstein equations are shown to be led to the nonlinear equations of boundary fluid dynamics [3]. In addition, gravity variables can provide a geometrical framework for investigating fluid dynamics, in a sense of a geometrization of turbulence [6].
- 19.
Although, as asserted in Weinberg [24], all Lorentzian spacetimes that represent gravitational fields of physical interest possess some Killing vector fields.
References
Bergmann, P.G.: Conservation laws in general relativity as the generators of coordinate transformations. Phys. Rev. 112, 287–289 (1958)
Bhattacharyya, S., Hubeny, V.E., Minwalla, S., Rangamani, M.: Nonlinear fluid dynamics from gravity. J. High Energy Phys. 2, 045 (2008). arXiv:0712.2456 [hep-th]
Bhattacharyya, S., Minwalla, S., Wadia, S.R.: The incompressible non-relativistic Navier-Stokes equation from gravity. J. High Energy Phys. 8, 059 (2009). arXiv:0810.1545 [hep-th]
Breedberg, I., Keller, C., Lysov, V., Strominger, A.: From Navier-Stokes to Einstein. (2011) arXiv: 1101.2451 [hep-ph]
Chorin, A.J., Marsden, J.E.: A Mathematical Introduction to Fluid Mechanics. Springer, New-York (1993)
Eling, C., Fouxon, I., Oz, Y.: Gravity and a Geometrization of Turbulence: An Intriguing Correspondence. (2010) arXiv:1004.2632 [hep-th]
Fernández, V.V., Rodrigues, W.A. Jr.: Gravitation as Plastic Distortion of the Lorentz Vacuum. Fundamental Theories of Physics, vol. 168. Springer, Heidelberg (2010). Errata for the book at: http://www.ime.unicamp.br/~walrod/errataplastic
Flanders, H.: Differential Forms with Applications to the Physical Sciences. Academic, New York (1963)
Hubney, V.E.: The fluid/gravity correspondence: a new perspective on the membrane paradigm. Classical Quantum Gravity 28, 114007 (2011). arXiv:1011.4948 [gr-qc]
Komar, A.: Asymptotic covariant conservation laws for gravitational radiation. Phys. Rev. 113, 934–936 (1958)
Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields, 4th revised English edn. Pergamon, New York (1975)
Misner, C.M., Thorne, K.S., Wheeler, J.A.: Gravitation. W. H. Freeman, San Francisco (1973)
Padmanabhan, T.: The hydrodynamics of atoms of spacetime: gravitational field equations is Navier-Stokes equation. Int. J. Mod. Phys. D 20, 2817–2822 (2011)
Rangamani, M.: Gravity & hydrodynamics: lecture on the fluid-gravity correspondence. Classical Quantum Gravity 26, 224003 (2009). arXiv:0905.4352 [hep-th]
Rodrigues, W.A. Jr.: Killing vector fields, Maxwell equations and Lorentzian spacetimes. Adv. Appl. Clifford Algebras 20, 871–884 (2010)
Rodrigues, W.A. Jr.: The nature of the gravitational field and its legitimate energy-momentum tensor. Rep. Math. Phys. 69, 265–279 (2012)
Rodrigues, F.G., da Rocha, R., Rodrigues, W.A. Jr.: The Maxwell and Navier-Stokes equations equivalent to Einstein equation in a spacetime containing a Killing vector field. AIP Conf. Proc. 1483, 277–295 (2012). arXiv:1109.5274v1 [math-ph]
Sachs, R.K., Wu, H.: General Relativity for Mathematicians. Springer, New York (1977)
Sánchez, M.: Lorentzian manifolds admitting a Killing vector field. Nonlinear Anal. Methods Appl. 30, 634–654 (1977)
Schmelzer, I.: A generalization of the Lorentz ether to gravity with general relativity limit. Adv. Appl. Clifford Algebras 22, 203–242 (2012). arXiv:gr-qc/0205035
Sulaiman, A., Handoko, L.T.: Relativistic Navier-Stokes equation from a gauge invariant lagrangian. Int. J. Mod. Phys. A 24, 3630–3637 (2009). arXiv:physics/0508219 [physics.flu-dyn]
Sulaiman, A., Djun, T.P., Handoko, L.T.: Invariant fluid lagrangian and its application to cosmology. J. Theor. Comput. Stud. 0401 (2006). arXiv:physics/0508086 [physics.flu-dyn]
Wald, R.: General Relativity. University of Chicago Press, Chicago (1984)
Weinberg, S.: Photons and gravitons in perturbation theory: derivation of Maxwell’s and Einstein’s equations. Phys. Rev. B 138, 988–1002 (1965)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Rodrigues, W.A., Capelas de Oliveira, E. (2016). Maxwell, Einstein, Dirac and Navier-Stokes Equations. In: The Many Faces of Maxwell, Dirac and Einstein Equations. Lecture Notes in Physics, vol 922. Springer, Cham. https://doi.org/10.1007/978-3-319-27637-3_15
Download citation
DOI: https://doi.org/10.1007/978-3-319-27637-3_15
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-27636-6
Online ISBN: 978-3-319-27637-3
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)