Lanczos tensor

There are two different tensors sometime 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:

:$C\_\{abcd\}=H\_\{abc;d\}-H\_\{abd;c\}+H\_\{cda;b\}-H\_\{cdb;a\},$::$-(g\_\{ac\}(H\_\{bd\}+H\_\{db\})-g\_\{ad\}(H\_\{bc\}+H\_\{cb\})+\; g\_\{bd\}(H\_\{ac\}+H\_\{ca\})-g\_\{bc\}(H\_\{ad\}+H\_\{da\}))/2,$::$+2H^\{ef\}\_\{;;;e;f\}(g\_\{ac\}g\_\{bd\}-g\_\{ad\}g\_\{bc\})/3,,$

where the Lanczos tensor has the symmetries:$H\_\{abc\}+H\_\{bac\}=0,,$:$H\_\{abc\}+H\_\{bca\}+H\_\{cab\}=0,,$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 ofthe vector potential "A" for the electromagnetic field.

External links

* [http://www.arXiv.org/abs/hep-th/gr-qc/9904006 gr-qc/9904006]