Wave Equation
The Lanczos potential tensor satisfies a wave equation
where is the d'Alembert operator and
is known as the Cotton tensor. Since the Cotton tensor depends only on covariant derivatives of the Ricci tensor, it can perhaps be interpreted as a kind of matter current. 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. Thus in vacuum, the Einstein field equations are equivalent to the homogeneous wave equation in perfect analogy to the vacuum wave equation of the electromagnetic four-potential. This shows a formal similarity between gravitational waves and electromagnetic waves, with the Lanczos tensor well-suited for studying gravitational waves.
In the weak field approximation where, a convenient form for the Lanczos tensor in the Lanczos gauge is
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