Lanczos tensor

There are two different tensors sometimes referred to as the Lanczos tensor (both named after Cornelius Lanczos):

A tensor in the theory of quadratic Lagrangians, which vanishes in four dimensions.
The potential tensor H for the Weyl tensor C, this can be expressed as:

$C_{abcd}=H_{abc;d}-H_{abd;c}%2BH_{cda;b}-H_{cdb;a}\,$

$-(g_{ac}(H_{bd}%2BH_{db})-g_{ad}(H_{bc}%2BH_{cb})%2B g_{bd}(H_{ac}%2BH_{ca})-g_{bc}(H_{ad}%2BH_{da}))/2\,$

$%2B2H^{ef}_{\;\;\;e;f}(g_{ac}g_{bd}-g_{ad}g_{bc})/3,\,$

where the Lanczos tensor has the symmetries

$H_{abc}%2BH_{bac}=0,\,$

$H_{abc}%2BH_{bca}%2BH_{cab}=0,\,$

and where $H_{bd}$ is defined by

$H_{bd}\ \stackrel{\mathrm{def}}{=}\ H^{~e}_{b\;\;d;e}-H^{~e}_{b\;\;e;d}\;.$

Thus, the Lanczos potential tensor is a gravitational field analog of the vector potential A for the electromagnetic field.

See also 'Introduction to 2-spinors in general relativity' (World Scientific, 2003) by Peter O'Donnell for a more detailed discussion of the Lanczos tensor and spinor.

External links

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