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| The '''Lanczos tensor''' or '''Lanczos potential''' is a [[tensor rank|rank 3 tensor]] in [[general relativity]] that generates the [[Weyl tensor]].<ref name="Takeno">Hyôitirô Takeno, "On the spintensor of Lanczos", ''Tensor'', '''15''' (1964) pp. 103–119.</ref> It was first introduced by [[Cornelius Lanczos]] in 1949.<ref name="Lanczos1949">Cornelius Lanczos, "Lagrangian Multiplier and Riemannian Spaces", ''Rev. Mod. Phys.'', '''21''' (1949) pp. 497–502. {{doi|10.1103/RevModPhys.21.497}}</ref> The theoretical importance of the Lanczos tensor is that it serves as the [[gauge field]] for the [[gravitational field]] in the same way that, by analogy, the [[electromagnetic four-potential]] generates the [[electromagnetic field]].<ref name="O'DonnellPye">P. O’Donnell and H. Pye, "A Brief Historical Review of the Important Developments in Lanczos Potential Theory", ''EJTP'', '''7''' (2010) pp. 327–350. {{url|http://www.ejtp.com/articles/ejtpv7i24p327.pdf}}</ref><ref name="NovelloVelloso">M. Novello and A. L. Velloso, "The Connection Between General Observers and Lanczos Potential", ''General Relativity and Gravitation'', '''19''' (1987) pp. 1251-1265. {{doi|10.1007/BF00759104}}</ref>
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| ==Definition==
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| The Lanczos tensor can be defined in a few different ways. The most common modern definition is through the Weyl–Lanczos equations, which demonstrate the generation of the Weyl tensor from the Lanczos tensor.<ref name="NovelloVelloso" /> These equations, presented below, were given by Takeno in 1964.<ref name="Takeno" /> The way that Lanczos introduced the tensor originally was as a [[Lagrange multiplier]]<ref name="Lanczos1949" /><ref>Cornelius Lanczos, "The Splitting of the Riemann Tensor", ''Rev. Mod. Phys.'', '''34''' (1962) pp. 379–389. {{doi|10.1103/RevModPhys.34.379}}</ref> on constraint terms studied in the [[variational methods in general relativity|variational approach to general relativity]].<ref>Cornelius Lanczos, "A Remarkable Property of the Riemann–Christoffel Tensor in Four Dimensions", ''Annals of Mathematics'', '''39''' (1938) pp. 842-850. {{url|http://www.jstor.org/stable/1968467}}</ref> Under any definition, the Lanczos tensor exhibits the following symmetries:
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| :<math>
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| H_{abc}+H_{bac}=0,\,</math>
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| :<math>
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| H_{abc}+H_{bca}+H_{cab}=0.
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| </math>
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| The Lanczos tensor always exists in four dimensions<ref name="BC1983">F. Bampi and G. Caviglia, "Third-order tensor potentials for the Riemann and Weyl tensors", ''General Relativity and Gravitation'', '''15''' (1983) pp. 375-386. {{doi|10.1007/BF00759166}}</ref> but does not generalize to higher dimensions.<ref>S. B. Edgar, "Nonexistence of the Lanczos potential for the Riemann tensor in higher dimensions", ''General Relativity and Gravitation'', '''26''' (1994) pp. 329-332. {{doi|10.1007/BF02108015}}</ref> This highlights the [[spacetime#Privileged character of 3+1 spacetime|specialness of four dimensions]].<ref name="O'DonnellPye" /> Note further that the full [[Riemann tensor]] cannot in general be derived from derivatives of the Lanczos potential alone.<ref name="BC1983" /><ref>E. Massa and E. Pagani, "Is the Riemann tensor derivable from a tensor potential?", ''General Relativity and Gravitation'', '''16''' (1984) pp. 805-816. {{doi|10.1007/BF00762934}}</ref> The [[Einstein field equations]] must provide the [[Ricci tensor]] to complete the components of the [[Ricci decomposition]].
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| ===Weyl–Lanczos equations===
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| The Weyl–Lanczos equations express the Weyl tensor entirely as derivatives of the Lanczos tensor:<ref name="O'Donnell">P. O’Donnell, "A Solution of the Weyl–Lanczos Equations for the Schwarzschild Space-Time", ''General Relativity and Gravitation'', '''36''' (2004) pp. 1415-1422. {{doi|10.1023/B:GERG.0000022577.11259.e0}}</ref>
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| :<math>\begin{align}C_{abcd}&=H_{abc;d}+H_{cda;b}+H_{bad;c}+H_{dcb;a} \\
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| &\, \, \, \, \, + (H^e{}_{(ac);e} + H_{(a|e|}{}^e{}_{;c)})g_{bd} + (H^e{}_{(bd);e} + H_{(b|e|}{}^e{}_{;d)})g_{ac} \\
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| &\, \, \, \, \, - (H^e{}_{(ad);e} + H_{(a|e|}{}^e{}_{;d)})g_{bc} - (H^e{}_{(bc);e} + H_{(b|e|}{}^e{}_{;c)})g_{ad} \\
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| &\, \, \, \, \, -\frac{2}{3} H^{ef}{}_{f;e}(g_{ac}g_{bd}-g_{ad}g_{bc})\end{align}</math>
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| where <math>C_{abcd}</math> is the Weyl tensor, the semicolon denotes the [[covariant derivative]], and the subscripted parentheses indicate [[symmetric tensor|symmetrization]]. Although the above equations can be used to define the Lanczos tensor, they also show that it is not unique but rather has [[gauge freedom]] under an [[affine group]].<ref>K. S. Hammon and L. K. Norris "The Affine Geometry of the Lanczos H-tensor Formalism", ''General Relativity and Gravitation'','''25''' (1993) pp. 55-80. {{doi|10.1007/BF00756929}}</ref> If <math>\Phi^a</math> is an arbitrary [[vector field]], then the Weyl–Lanczos equations are invariant under the gauge transformation | |
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| :<math>H'_{abc} = H_{abc}+\Phi_{[a} g_{b]c}</math>
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| where the subscripted brackets indicate [[antisymmetric tensor|antisymmetrization]]. An often convenient choice is the Lanczos algebraic gauge, <math>\Phi_a=-\frac{2}{3}H_{ab}{}^b,</math> which sets <math>H'_{ab}{}^b=0.</math> The gauge can be further restricted through the Lanczos differential gauge <math>H_{ab}{}^c{}_{;c}=0</math>. These gauge choices reduce the Weyl–Lanczos equations to the simpler form
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| :<math>\begin{align}C_{abcd}&=H_{abc;d}+H_{cda;b}+H_{bad;c}+H_{dcb;a} \\
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| &\, \, \, \, \, +H^e{}_{ac;e}g_{bd}+H^e{}_{bd;e}g_{ac}-H^e{}_{ad;e}g_{bc}-H^e{}_{bc;e}g_{ad}.\end{align}</math>
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| ==Wave equation==
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| The Lanczos potential tensor satisfies a wave equation<ref>P. Dolan and C. W. Kim "The wave equation for the Lanczos potential", ''Proc. R. Soc. Lond. A'', '''447''' (1994) pp. 557-575. {{doi|10.1098/rspa.1994.0155}}</ref>
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| :<math>\begin{align}\Box H_{abc} = & J_{abc}\\
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| & {}- 2{R_c}^d H_{abd}+{R_a}^d H_{bcd}+{R_b}^d H_{acd}\\
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| & {}+ \left( H_{dbe}g_{ac}-H_{dae}g_{bc} \right)R^{de}+\frac{1}{2}RH_{abc},\end{align}</math>
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| where <math>\Box</math> is the [[d'Alembert operator]] and
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| :<math>J_{abc} = R_{ca;b}-R_{cb;a}-\frac{1}{6}\left( g_{ca}R_{;b}-g_{cb}R_{;a} \right)</math>
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| is known as the [[Cotton tensor]]. Since the Cotton tensor depends only on [[covariant derivative]]s of the [[Ricci tensor]], it can perhaps be interpreted as a kind of matter current.<ref name="Roberts1996">Mark D. Roberts, "The Physical Interpretation of the Lanczos Tensor." ''Nuovo Cim.B'' '''110''' (1996) 1165-1176. {{doi|10.1007/BF02724607}} {{arXiv|gr-qc/9904006}}</ref> The additional self-coupling terms have no direct electromagnetic equivalent. These self-coupling terms, however, do not affect the [[vacuum solution (general relativity)|vacuum solutions]], where the Ricci tensor vanishes and the curvature is described entirely by the Weyl tensor. Thus in vacuum, the [[Einstein field equations]] are equivalent to the [[homogeneous differential equation|homogeneous]] wave equation <math>\Box H_{abc} = 0,</math> in perfect analogy to the vacuum wave equation <math>\Box A_{a} = 0</math> of the electromagnetic four-potential. This shows a formal similarity between [[gravitational wave]]s and [[electromagnetic wave]]s, with the Lanczos tensor well-suited for studying gravitational waves.<ref>J. L. López-Bonilla, G. Ovando and J. J. Peña, "A Lanczos Potential for Plane Gravitational Waves." ''Foundations of Physics Letters'' '''12''' (1999) 401-405. {{doi|10.1023/A:1021656622094}}</ref>
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| In the weak field approximation where <math>g_{ab}=\eta_{ab}+h_{ab}</math>, a convenient form for the Lanczos tensor in the Lanczos gauge is<ref name="Roberts1996" />
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| :<math>4H_{abc} \approx h_{ac,b}-h_{bc,a}-\frac{1}{6}(\eta_{ac}{h^d}_{d,b}-\eta_{bc}{h^d}_{d,a}) .</math>
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| ==Example==
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| The most basic nontrivial case for expressing the Lanczos tensor is, of course, for the [[Schwarzschild metric]].<ref name="NovelloVelloso" /> The simplest, explicit component representation in [[natural units]] for the Lanczos tensor in this case is
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| :<math>H_{trt}=\frac{GM}{r^2}</math>
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| with all other components vanishing up to symmetries. This form, however, is not in the Lanczos gauge. The nonvanishing terms of the Lanczos tensor in the Lanczos gauge are
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| :<math>H_{trt}=\frac{2GM}{3r^2}</math>
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| :<math>H_{r\theta\theta}=\frac{-GM}{3(1-2GM/r)}</math>
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| :<math>H_{r\phi\phi}=\frac{-GM\sin^2 \theta}{3(1-2GM/r)}</math>
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| It is further possible to show, even in this simple case, that the Lanczos tensor cannot in general be reduced to a linear combination of the spin coefficients of the [[Newman–Penrose formalism]], which attests to the Lanczos tensor's fundamental nature.<ref name="O'Donnell" /> Similar calculations have been used to construct arbitrary [[Petrov type D]] solutions.<ref>Zafar Ahsan and Mohd Bilal, "A Solution of Weyl-Lanczos Equations for Arbitrary Petrov Type D Vacuum Spacetimes." ''Int J Theor Phys'' '''49''' (2010) 2713-2722. {{doi|10.1007/s10773-010-0464-5}}</ref>
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| ==See also==
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| *[[Bach tensor]]
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| *[[Schouten tensor]]
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| *[[Ricci calculus]]
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| ==References==
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| <references />
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| ==External links==
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| *Peter O'Donnell, ''Introduction To 2-Spinors In General Relativity''. [http://www.worldscibooks.com/physics/5222.html World Scientific], 2003.
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| [[Category:Gauge theories]]
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| [[Category:Differential geometry]]
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| [[Category:Tensors]]
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| [[Category:Tensors in general relativity]]
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| [[Category:1949 introductions]]
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