The Lanczos tensor or Lanczos potential is a tensor rank in general relativity that generates the Weyl tensor.[Hyôitirô Takeno, "On the spintensor of Lanczos", Tensor, 15 (1964) pp. 103–119.] It was first introduced by Cornelius Lanczos in 1949.[P. O’Donnell and H. Pye, "A Brief Historical Review of the Important Developments in Lanczos Potential Theory", EJTP, 7 (2010) pp. 327–350. http://www.ejtp.com/articles/ejtpv7i24p327.pdf]
Definition
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.
These equations, presented below, were given by Takeno in 1964.
The way that Lanczos introduced the tensor originally was as a Lagrange multiplier
on constraint terms studied in the variational approach to general relativity.
[Cornelius Lanczos, "A Remarkable Property of the Riemann–Christoffel Tensor in Four Dimensions", Annals of Mathematics, 39 (1938) pp. 842–850.
]
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/a> Under any definition, the Lanczos tensor
H exhibits the following symmetries:
H_{abc}+H_{bac}=0,\,
H_{abc}+H_{bca}+H_{cab}=0.
The Lanczos tensor always exists in four dimensions
The Curtright field has a gauge-transformation dynamics similar to that of Lanczos tensor. But Curtright field exists in arbitrary dimensions > 4D.
Weyl–Lanczos equations
The Weyl–Lanczos equations express the Weyl tensor entirely as derivatives of the Lanczos tensor:
\begin{align}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{align}
where is the Weyl tensor, the semicolon denotes the covariant derivative, and the subscripted parentheses indicate symmetric tensor. 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.
where the subscripted brackets indicate antisymmetrization. An often convenient choice is the Lanczos algebraic gauge, which sets The gauge can be further restricted through the Lanczos differential gauge . These gauge choices reduce the Weyl–Lanczos equations to the simpler form
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}.
Wave equation
The Lanczos potential tensor satisfies a wave equation
& {}- 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{align}
where is the d'Alembert operator and
is known as the
Cotton tensor. Since the Cotton tensor depends only on covariant derivatives of the
Ricci tensor, it can perhaps be interpreted as a kind of matter current.
The additional self-coupling terms have no direct electromagnetic equivalent. These self-coupling terms, however, do not affect the 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 wave equation
in perfect analogy to the vacuum wave equation
of the electromagnetic four-potential. This shows a formal similarity between gravitational waves and electromagnetic waves, with the Lanczos tensor well-suited for studying gravitational waves.
In the weak field approximation where , a convenient form for the Lanczos tensor in the Lanczos gauge is
Example
The most basic nontrivial case for expressing the Lanczos tensor is, of course, for the Schwarzschild metric.
The simplest, explicit component representation in
natural units for the Lanczos tensor in this case is
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
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.
See also
External links
-
Peter O'Donnell, Introduction To 2-Spinors In General Relativity. World Scientific, 2003.
