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];
Roberts a1910 [Bianchi spacetime].

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