Lanczos tensor

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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 Hbd 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.

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