[1] The way that Lanczos introduced the tensor originally was as a Lagrange multiplier[2][5] on constraint terms studied in the variational approach to general relativity.
[6] Under any definition, the Lanczos tensor H exhibits the following symmetries: The Lanczos tensor always exists in four dimensions[7] but does not generalize to higher dimensions.
The Curtright field has a gauge-transformation dynamics similar to that of Lanczos tensor.
But Curtright field exists in arbitrary dimensions > 4D.
is the Weyl tensor, the semicolon denotes the covariant derivative, and the subscripted parentheses indicate symmetrization.
Although the above equations can be used to define the Lanczos tensor, they also show that it is not unique but rather has gauge freedom under an affine group.
is an arbitrary vector field, then the Weyl–Lanczos equations are invariant under the gauge transformation where the subscripted brackets indicate antisymmetrization.
An often convenient choice is the Lanczos algebraic gauge,
These gauge choices reduce the Weyl–Lanczos equations to the simpler form The Lanczos potential tensor satisfies a wave equation[13] where
Since the Cotton tensor depends only on covariant derivatives of the Ricci tensor, it can perhaps be interpreted as a kind of matter current.
[14] The additional self-coupling terms have no direct electromagnetic equivalent.
These self-coupling terms, however, do not affect the vacuum solutions, where the Ricci tensor vanishes and the curvature is described entirely by the Weyl tensor.
in perfect analogy to the vacuum wave equation
[4] The simplest, explicit component representation in natural units for the Lanczos tensor in this case is with all other components vanishing up to symmetries.
[11] Similar calculations have been used to construct arbitrary Petrov type D solutions.