Lanczos tensor

The Lanczos tensor or Lanczos potential is a rank 3 tensor in general relativity that generates the Weyl tensor. It was first introduced by Cornelius Lanczos in 1949. 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.

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. Under any definition, the Lanczos tensor exhibits the following symmetries:

$H_{abc}+H_{bac}=0,\,$

$H_{abc}+H_{bca}+H_{cab}=0.$

The Lanczos tensor always exists in four dimensions but does not generalize to higher dimensions. This highlights the specialness of four dimensions. Note further that the full Riemann tensor cannot in general be derived from derivatives of the Lanczos potential alone. The Einstein field equations must provide the Ricci tensor to complete the components of the Ricci decomposition.

Weyl–Lanczos equations

The Weyl–Lanczos equations express the Weyl tensor entirely as derivatives of the Lanczos tensor:

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

where $C_{abcd}$ is the Weyl tensor, the semicolon denotes the covariant derivative, and the subscripted parentheses indicate 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. If $\Phi ^{a}$ is an arbitrary vector field, then the Weyl–Lanczos equations are invariant under the gauge transformation

$H'_{abc}=H_{abc}+\Phi _{[a}g_{b]c}$

where the subscripted brackets indicate antisymmetrization. An often convenient choice is the Lanczos algebraic gauge, $\Phi _{a}=-{\frac {2}{3}}H_{ab}{}^{b},$ which sets $H_{ab}{}^{b}=0.$ The gauge can be further restricted through the Lanczos differential gauge $H_{ab}{}^{c}{}_{;c}=0$. These gauge choices reduce the Weyl–Lanczos equations to the simpler form

${\begin{aligned}C_{abcd}&=H_{abc;d}+H_{cda;b}+H_{bad;c}+H_{dcb;a}\\&\,\,\,\,\,+H^{e}{}_{ac;e}g_{bd}+H^{e}{}_{bd;e}g_{ac}-H^{e}{}_{ad;e}g_{bc}-H^{e}{}_{bc;e}g_{ad}.\end{aligned}}$

Wave equation

The Lanczos potential tensor satisfies a wave equation:

${\begin{aligned}\Box H_{abc}&=R_{ca;b}-R_{cb;a}-{\frac {1}{6}}\left(g_{ca}R_{;b}-g_{cb}R_{;a}\right)\\&\,\,\,-2{R_{c}}^{d}H_{abd}+{R_{a}}^{d}H_{bcd}+{R_{b}}^{d}H_{acd}\\&\,\,\,+\left(H_{dbe}g_{ac}-H_{dae}g_{bc}\right)R^{de}+{\frac {1}{2}}RH_{abc},\end{aligned}}$

where $\Box$ is the d'Alembert operator. The first four terms, which depend only on covariant derivatives of the Ricci tensor and Ricci scalar, is known as the Cotton tensor and can be interpreted as a kind of matter current. The additional self-coupling terms, however, have no electromagnetic equivalent. For the vacuum solutions in particular, where the Ricci tensor vanishes and the curvature is described entirely by the Weyl tensor, the Einstein field equations are equivalent to the homogeneous wave equation $\Box H_{abc}=0.$ This shows a formal similarity between gravitational waves and electromagnetic waves.

In the weak field approximation where $g_{ab}=\eta _{ab}+h_{ab}$, a convenient form for the Lanczos tensor in the Lanczos gauge is

$4H_{abc}\approx h_{ac,b}-h_{bc,a}-{\frac {1}{6}}(\eta _{ac}{h^{d}}_{d,b}-\eta _{bc}{h^{d}}_{d,a})$

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

$L_{010}={\frac {GM}{r^{2}}}$

and all other components vanishing. This form, however, is not in the Lanczos gauge. The nonvanishing terms of the Lanczos tensor in the Lanczos gauge are

$L_{010}={\frac {2GM}{3r^{2}}}$

$L_{122}={\frac {-GM}{3(1-2GM/r^{2})}}$

$L_{133}={\frac {-GM\sin ^{2}\theta }{3(1-2GM/r^{2})}}$

It is further possible to show, even in this simple case, that the Lanczos tensor cannot in general be reduced to the spin coefficients of the Newman–Penrose formalism, which attests to its fundamental nature.

External links

• [1] Introduction to 2-spinors in general relativity
• gr-qc/9904006