Lanczos tensor

<h2 id="definition">Definition</h2>
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.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> These equations, presented below, were given by Takeno in 1964.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> The way that Lanczos introduced the tensor originally was as a <a href="/facts/Lagrange_multiplier/U5mSI5YP">Lagrange multiplier</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a><a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> on constraint terms studied in the <a href="/facts/Variational_methods_in_general_relativity/XNAvcMdb">variational approach to general relativity</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> Under any definition, the Lanczos tensor H exhibits the following symmetries:

H
          
            a
            b
            c
          
        
        +
        
          H
          
            b
            a
            c
          
        
        =
        0
        ,
        
      
    
    {\displaystyle H_{abc}+H_{bac}=0,\,}

H
          
            a
            b
            c
          
        
        +
        
          H
          
            b
            c
            a
          
        
        +
        
          H
          
            c
            a
            b
          
        
        =
        0.
      
    
    {\displaystyle H_{abc}+H_{bca}+H_{cab}=0.}

The Lanczos tensor always exists in four dimensions<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> but does not generalize to higher dimensions.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> This highlights the <a href="/facts/Spacetime/uRIN1x4b">specialness of four dimensions</a>.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> Note further that the full <a href="/facts/Riemann_tensor/KyeHCWEh">Riemann tensor</a> cannot in general be derived from derivatives of the Lanczos potential alone.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a><a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> The <a href="/facts/Einstein_field_equations/j2rJTzW9">Einstein field equations</a> must provide the <a href="/facts/Ricci_tensor/qujou7QS">Ricci tensor</a> to complete the components of the <a href="/facts/Ricci_decomposition/MLQkLeKj">Ricci decomposition</a>.
The <a href="/facts/Curtright_field/WPqx78sv">Curtright field</a> has a gauge-transformation dynamics similar to that of Lanczos tensor. But Curtright field exists in arbitrary dimensions > 4D.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>

<h3>Weyl–Lanczos equations</h3>
The Weyl–Lanczos equations express the Weyl tensor entirely as derivatives of the Lanczos tensor:<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>

C
                  
                    a
                    b
                    c
                    d
                  
                
              
              
                
                =
                
                  H
                  
                    a
                    b
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                    ;
                    d
                  
                
                +
                
                  H
                  
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                    d
                    a
                    ;
                    b
                  
                
                +
                
                  H
                  
                    b
                    a
                    d
                    ;
                    c
                  
                
                +
                
                  H
                  
                    d
                    c
                    b
                    ;
                    a
                  
                
                +
                (
                
                  H
                  
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(
                    a
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                    ;
                    e
                  
                
                +
                
                  H
                  
                    (
                    a
                    
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;
                    c
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                )
                
                  g
                  
                    b
                    d
                  
                
                +
                (
                
                  H
                  
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(
                    b
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                    )
                    ;
                    e
                  
                
                +
                
                  H
                  
                    (
                    b
                    
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;
                    d
                    )
                  
                
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                  g
                  
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                −
                (
                
                  H
                  
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(
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                    e
                  
                
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                    a
                    
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                  g
                  
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                (
                
                  H
                  
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(
                    b
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                    ;
                    e
                  
                
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                  H
                  
                    (
                    b
                    
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;
                    c
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                )
                
                  g
                  
                    a
                    d
                  
                
                −
                
                  
                    2
                    3
                  
                
                
                  H
                  
                    e
                    f

f
                    ;
                    e
                  
                
                (
                
                  g
                  
                    a
                    c
                  
                
                
                  g
                  
                    b
                    d
                  
                
                −
                
                  g
                  
                    a
                    d
                  
                
                
                  g
                  
                    b
                    c
                  
                
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}C_{abcd}&=H_{abc;d}+H_{cda;b}+H_{bad;c}+H_{dcb;a}+(H^{e}{}_{(ac);e}+H_{(a|e|}{}^{e}{}_{;c)})g_{bd}+(H^{e}{}_{(bd);e}+H_{(b|e|}{}^{e}{}_{;d)})g_{ac}\\&\quad -(H^{e}{}_{(ad);e}+H_{(a|e|}{}^{e}{}_{;d)})g_{bc}-(H^{e}{}_{(bc);e}+H_{(b|e|}{}^{e}{}_{;c)})g_{ad}-{\frac {2}{3}}H^{ef}{}_{f;e}(g_{ac}g_{bd}-g_{ad}g_{bc})\end{aligned}}}

where 
 
 
 
 
 C
 
 a
 b
 c
 d
 
 
 
 
 {\displaystyle C_{abcd}}
 
 is the Weyl tensor, the semicolon denotes the <a href="/facts/Covariant_derivative/sRGN0Xh6">covariant derivative</a>, and the subscripted parentheses indicate <a href="/facts/Symmetric_tensor/CWKBfgnv">symmetrization</a>. Although the above equations can be used to define the Lanczos tensor, they also show that it is not unique but rather has <a href="/facts/Gauge_freedom/eJW4uffl">gauge freedom</a> under an <a href="/facts/Affine_group/pe8CxKXV">affine group</a>.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a> If 
 
 
 
 
 Φ
 
 a
 
 
 
 
 {\displaystyle \Phi ^{a}}
 
 is an arbitrary <a href="/facts/Vector_field/ccFNuPPp">vector field</a>, then the Weyl–Lanczos equations are invariant under the gauge transformation

H
          
            a
            b
            c
          
          ′
        
        =
        
          H
          
            a
            b
            c
          
        
        +
        
          Φ
          
            [
            a
          
        
        
          g
          
            b
            ]
            c
          
        
      
    
    {\displaystyle H'_{abc}=H_{abc}+\Phi _{[a}g_{b]c}}

where the subscripted brackets indicate <a href="/facts/Antisymmetric_tensor/atcSaPST">antisymmetrization</a>. An often convenient choice is the Lanczos algebraic gauge, 
 
 
 
 
 Φ
 
 a
 
 
 =
 −
 
 
 2
 3
 
 
 
 H
 
 a
 b

b
          
        
        ,
      
    
    {\displaystyle \Phi _{a}=-{\frac {2}{3}}H_{ab}{}^{b},}
  
 which sets 
  
    
      
        
          H
          
            a
            b
          
          ′

b
          
        
        =
        0.
      
    
    {\displaystyle H'_{ab}{}^{b}=0.}
  
  The gauge can be further restricted through the Lanczos differential gauge 
  
    
      
        
          H
          
            a
            b

;
 c
 
 
 =
 0
 
 
 {\displaystyle H_{ab}{}^{c}{}_{;c}=0}
 
. These gauge choices reduce the Weyl–Lanczos equations to the simpler form

C
          
            a
            b
            c
            d
          
        
        =
        
          H
          
            a
            b
            c
            ;
            d
          
        
        +
        
          H
          
            c
            d
            a
            ;
            b
          
        
        +
        
          H
          
            b
            a
            d
            ;
            c
          
        
        +
        
          H
          
            d
            c
            b
            ;
            a
          
        
        +
        
          H
          
            e

a
            c
            ;
            e
          
        
        
          g
          
            b
            d
          
        
        +
        
          H
          
            e

b
            d
            ;
            e
          
        
        
          g
          
            a
            c
          
        
        −
        
          H
          
            e

a
            d
            ;
            e
          
        
        
          g
          
            b
            c
          
        
        −
        
          H
          
            e

b
            c
            ;
            e
          
        
        
          g
          
            a
            d
          
        
        .
      
    
    {\displaystyle C_{abcd}=H_{abc;d}+H_{cda;b}+H_{bad;c}+H_{dcb;a}+H^{e}{}_{ac;e}g_{bd}+H^{e}{}_{bd;e}g_{ac}-H^{e}{}_{ad;e}g_{bc}-H^{e}{}_{bc;e}g_{ad}.}

<h2 id="wave-equation">Wave equation</h2>
The Lanczos potential tensor satisfies a wave equation<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a>

◻
                
                  H
                  
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                    c
                  
                
                =
              
              
                
                
                  J
                  
                    a
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                    c

−
                2
                
                  
                    
                      R
                      
                        c
                      
                    
                  
                  
                    d
                  
                
                
                  H
                  
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                +
                
                  
                    
                      R
                      
                        a
                      
                    
                  
                  
                    d
                  
                
                
                  H
                  
                    b
                    c
                    d
                  
                
                +
                
                  
                    
                      R
                      
                        b
                      
                    
                  
                  
                    d
                  
                
                
                  H
                  
                    a
                    c
                    d

+
                
                  (
                  
                    
                      H
                      
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                        b
                        e
                      
                    
                    
                      g
                      
                        a
                        c
                      
                    
                    −
                    
                      H
                      
                        d
                        a
                        e
                      
                    
                    
                      g
                      
                        b
                        c
                      
                    
                  
                  )
                
                
                  R
                  
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                +
                
                  
                    1
                    2
                  
                
                R
                
                  H
                  
                    a
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                ,
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\Box H_{abc}=&\;J_{abc}\\&{}-2{R_{c}}^{d}H_{abd}+{R_{a}}^{d}H_{bcd}+{R_{b}}^{d}H_{acd}\\&{}+\left(H_{dbe}g_{ac}-H_{dae}g_{bc}\right)R^{de}+{\frac {1}{2}}RH_{abc},\end{aligned}}}

where 
 
 
 
 ◻
 
 
 {\displaystyle \Box }
 
 is the <a href="/facts/D%2527Alembert_operator/hCzCvJyD">d'Alembert operator</a> and

J
          
            a
            b
            c
          
        
        =
        
          R
          
            c
            a
            ;
            b
          
        
        −
        
          R
          
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            ;
            a
          
        
        −
        
          
            1
            6
          
        
        
          (
          
            
              g
              
                c
                a
              
            
            
              R
              
                ;
                b
              
            
            −
            
              g
              
                c
                b
              
            
            
              R
              
                ;
                a
              
            
          
          )
        
      
    
    {\displaystyle J_{abc}=R_{ca;b}-R_{cb;a}-{\frac {1}{6}}\left(g_{ca}R_{;b}-g_{cb}R_{;a}\right)}

is known as the <a href="/facts/Cotton_tensor/NpsP7YzX">Cotton tensor</a>. Since the Cotton tensor depends only on <a href="/facts/Covariant_derivative/sRGN0Xh6">covariant derivatives</a> of the <a href="/facts/Ricci_tensor/qujou7QS">Ricci tensor</a>, it can perhaps be interpreted as a kind of matter current.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a> The additional self-coupling terms have no direct electromagnetic equivalent. These self-coupling terms, however, do not affect the <a href="/facts/Vacuum_solution_(general_relativity)/M8BSxN4x">vacuum solutions</a>, where the Ricci tensor vanishes and the curvature is described entirely by the Weyl tensor. Thus in vacuum, the <a href="/facts/Einstein_field_equations/j2rJTzW9">Einstein field equations</a> are equivalent to the <a href="/facts/Homogeneous_differential_equation/UmTJCNEh">homogeneous</a> wave equation 
 
 
 
 ◻
 
 H
 
 a
 b
 c
 
 
 =
 0
 ,
 
 
 {\displaystyle \Box H_{abc}=0,}
 
 in perfect analogy to the vacuum wave equation 
 
 
 
 ◻
 
 A
 
 a
 
 
 =
 0
 
 
 {\displaystyle \Box A_{a}=0}
 
 of the electromagnetic four-potential. This shows a formal similarity between <a href="/facts/Gravitational_wave/4XWN3NtG">gravitational waves</a> and <a href="/facts/Electromagnetic_wave/Ynx7ujCW">electromagnetic waves</a>, with the Lanczos tensor well-suited for studying gravitational waves.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a>
In the weak field approximation where 
 
 
 
 
 g
 
 a
 b
 
 
 =
 
 η
 
 a
 b
 
 
 +
 
 h
 
 a
 b
 
 
 
 
 {\displaystyle g_{ab}=\eta _{ab}+h_{ab}}
 
, a convenient form for the Lanczos tensor in the Lanczos gauge is<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a>

4
        
          H
          
            a
            b
            c
          
        
        ≈
        
          h
          
            a
            c
            ,
            b
          
        
        −
        
          h
          
            b
            c
            ,
            a
          
        
        −
        
          
            1
            6
          
        
        (
        
          η
          
            a
            c
          
        
        
          
            
              h
              
                d
              
            
          
          
            d
            ,
            b
          
        
        −
        
          η
          
            b
            c
          
        
        
          
            
              h
              
                d
              
            
          
          
            d
            ,
            a
          
        
        )
        .
      
    
    {\displaystyle 4H_{abc}\approx h_{ac,b}-h_{bc,a}-{\frac {1}{6}}(\eta _{ac}{h^{d}}_{d,b}-\eta _{bc}{h^{d}}_{d,a}).}

<h2 id="example">Example</h2>
The most basic nontrivial case for expressing the Lanczos tensor is, of course, for the <a href="/facts/Schwarzschild_metric/gkm25ScM">Schwarzschild metric</a>.<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a> The simplest, explicit component representation in <a href="/facts/Natural_units/nf8megYV">natural units</a> for the Lanczos tensor in this case is

H
          
            t
            r
            t
          
        
        =
        
          
            
              G
              M
            
            
              r
              
                2
              
            
          
        
      
    
    {\displaystyle H_{trt}={\frac {GM}{r^{2}}}}

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

H
          
            t
            r
            t
          
        
        =
        
          
            
              2
              G
              M
            
            
              3
              
                r
                
                  2
                
              
            
          
        
      
    
    {\displaystyle H_{trt}={\frac {2GM}{3r^{2}}}}

H
          
            r
            θ
            θ
          
        
        =
        
          
            
              −
              G
              M
            
            
              3
              (
              1
              −
              2
              G
              M
              
                /
              
              r
              )
            
          
        
      
    
    {\displaystyle H_{r\theta \theta }={\frac {-GM}{3(1-2GM/r)}}}

H
          
            r
            ϕ
            ϕ
          
        
        =
        
          
            
              −
              G
              M
              
                sin
                
                  2
                
              
              ⁡
              θ
            
            
              3
              (
              1
              −
              2
              G
              M
              
                /
              
              r
              )
            
          
        
      
    
    {\displaystyle H_{r\phi \phi }={\frac {-GM\sin ^{2}\theta }{3(1-2GM/r)}}}

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 <a href="/facts/Newman%25E2%2580%2593Penrose_formalism/U18aSKry">Newman–Penrose formalism</a>, which attests to the Lanczos tensor's fundamental nature.<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a> Similar calculations have been used to construct arbitrary <a href="/facts/Petrov_type_D/0Y4sNsgS">Petrov type D</a> solutions.<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Bach_tensor/g5mNz0Q9">Bach tensor</a></li>
<li><a href="/facts/Ricci_calculus/We4yudeq">Ricci calculus</a></li>
<li><a href="/facts/Schouten_tensor/uEXCjlMF">Schouten tensor</a></li>
<li><a href="/facts/Tetradic_Palatini_action/gFwFGjEL">tetradic Palatini action</a></li>
<li><a href="/facts/Self-dual_Palatini_action/mq48BxtX">Self-dual Palatini action</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li>Peter O'Donnell, Introduction To 2-Spinors In General Relativity. <a href="http://www.worldscibooks.com/physics/5222.html">World Scientific</a>, 2003.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Hyôitirô Takeno, "On the spintensor of Lanczos", Tensor, 15 (1964) pp. 103–119. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Lanczos, Cornelius (1949-07-01). "Lagrangian Multiplier and Riemannian Spaces". Reviews of Modern Physics. 21 (3). American Physical Society (APS): 497–502. Bibcode:1949RvMP...21..497L. doi:10.1103/revmodphys.21.497. ISSN 0034-6861. <a href="https://doi.org/10.1103%2Frevmodphys.21.497" target="_blank">https://doi.org/10.1103%2Frevmodphys.21.497</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">P. O’Donnell and H. Pye, "A Brief Historical Review of the Important Developments in Lanczos Potential Theory", EJTP, 7 (2010) pp. 327–350. www.ejtp.com/articles/ejtpv7i24p327.pdf <a href="http://www.ejtp.com/articles/ejtpv7i24p327.pdf" target="_blank">http://www.ejtp.com/articles/ejtpv7i24p327.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Novello, M.; Velloso, A. L. (1987). "The connection between general observers and Lanczos potential". General Relativity and Gravitation. 19 (12). Springer Science and Business Media LLC: 1251–1265. Bibcode:1987GReGr..19.1251N. doi:10.1007/bf00759104. ISSN 0001-7701. S2CID 122998917. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Novello, M.; Velloso, A. L. (1987). "The connection between general observers and Lanczos potential". General Relativity and Gravitation. 19 (12). Springer Science and Business Media LLC: 1251–1265. Bibcode:1987GReGr..19.1251N. doi:10.1007/bf00759104. ISSN 0001-7701. S2CID 122998917. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Hyôitirô Takeno, "On the spintensor of Lanczos", Tensor, 15 (1964) pp. 103–119. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Lanczos, Cornelius (1949-07-01). "Lagrangian Multiplier and Riemannian Spaces". Reviews of Modern Physics. 21 (3). American Physical Society (APS): 497–502. Bibcode:1949RvMP...21..497L. doi:10.1103/revmodphys.21.497. ISSN 0034-6861. <a href="https://doi.org/10.1103%2Frevmodphys.21.497" target="_blank">https://doi.org/10.1103%2Frevmodphys.21.497</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Lanczos, C. (1962-07-01). "The Splitting of the Riemann Tensor". Reviews of Modern Physics. 34 (3). American Physical Society (APS): 379–389. Bibcode:1962RvMP...34..379L. doi:10.1103/revmodphys.34.379. ISSN 0034-6861. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Cornelius Lanczos, "A Remarkable Property of the Riemann–Christoffel Tensor in Four Dimensions", Annals of Mathematics, 39 (1938) pp. 842–850. www.jstor.org/stable/1968467 <a href="https://www.jstor.org/stable/1968467" target="_blank">https://www.jstor.org/stable/1968467</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Bampi, Franco; Caviglia, Giacomo (1983). "Third-order tensor potentials for the Riemann and Weyl tensors". General Relativity and Gravitation. 15 (4). Springer Science and Business Media LLC: 375–386. Bibcode:1983GReGr..15..375B. doi:10.1007/bf00759166. ISSN 0001-7701. S2CID 122782358. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Edgar, S. Brian (1994). "Nonexistence of the Lanczos potential for the Riemann tensor in higher dimensions". General Relativity and Gravitation. 26 (3). Springer Science and Business Media LLC: 329–332. Bibcode:1994GReGr..26..329E. doi:10.1007/bf02108015. ISSN 0001-7701. S2CID 120343522. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">P. O’Donnell and H. Pye, "A Brief Historical Review of the Important Developments in Lanczos Potential Theory", EJTP, 7 (2010) pp. 327–350. www.ejtp.com/articles/ejtpv7i24p327.pdf <a href="http://www.ejtp.com/articles/ejtpv7i24p327.pdf" target="_blank">http://www.ejtp.com/articles/ejtpv7i24p327.pdf</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Bampi, Franco; Caviglia, Giacomo (1983). "Third-order tensor potentials for the Riemann and Weyl tensors". General Relativity and Gravitation. 15 (4). Springer Science and Business Media LLC: 375–386. Bibcode:1983GReGr..15..375B. doi:10.1007/bf00759166. ISSN 0001-7701. S2CID 122782358. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Massa, Enrico; Pagani, Enrico (1984). "Is the Riemann tensor derivable from a tensor potential?". General Relativity and Gravitation. 16 (9). Springer Science and Business Media LLC: 805–816. Bibcode:1984GReGr..16..805M. doi:10.1007/bf00762934. ISSN 0001-7701. S2CID 120457526. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Curtright, Thomas (December 1985). "Generalized gauge fields". Physics Letters B. 165 (4–6): 304–308. Bibcode:1985PhLB..165..304C. doi:10.1016/0370-2693(85)91235-3. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
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</ol>

Lanczos tensor open-in-new

Lanczos tensor