Lanczos tensor

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]


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