Lanczos
Potential / Tensor |

**In General** > s.a. riemann
tensor.

* __Idea__: A tensor *L*_{abc} proposed
as a potential for the Weyl curvature tensor; It plays the same role in gravity
as the vector potential plays in electromagnetism; Under some conditions it has a superpotential.

$ __Def__: In spinor notation, the tensor such that the Weyl spinor can
be expressed as

Ψ_{ABCD} = 2 ∇_{(A}^{A '}
*L*_{BCD) A'} .

* __Conditions__: The Weyl
spinor has Lanczos potentials in all spacetimes; The Weyl tensor has Lanczos
potentials in all four-dimensional spaces, irrespective of signature, but does not exist in more than 4D.

> __Online resources__:
see Wikipedia page.

**Dynamics** > s.a. higher-order gravity [Lanczos Lagrangian].

* __Riemann-Lanczos equations__:
A system of linear first-order partial differential equations that arise
in general relativity, whereby the Riemann curvature tensor is generated by
an unknown third-order Lanczos tensor potential field.

@ __References__: Dolan & Kim PRS(94)
[wave equation]; Cartin gq/99 [and
linearized general relativity], ht/03 [as
spin-2 field, Born-Infeld type]; Dolan & Gerber JMP(08)
[integrability of Riemann-Lanczos equations].

**References** > s.a. perturbations
in general relativity; spin coefficients.

@ __General__: Lanczos RMP(62);
Roberts MPLA(89),
NCB(95)gq/99 [interpretation];
Dolan & Muratori
JMP(98)
[with Ernst potential].

@ __Existence__: Edgar & Höglund PRS(97)gq/96,
GRG(00)gq/97 [in
4D only]; Andersson & Edgar CQG(01)
[for Weyl spinor, superpotentials]; Edgar & Höglund
GRG(02)gq [non-existence
in *n* ≥ 7];
Edgar JMP(03)gq [conditions].

@ __Specific spacetimes__: Gaftoi et al NCB(98),
Acevedo et al G&C(04)
[Kerr metric]; O'Donnell NCB(04)
[conformally flat]; Mena & Tod CQG(07)gq [perturbed
FLRW spacetime, and gravitational entropy].

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