Example
The most basic nontrivial case for expressing the Lanczos tensor is, of course, for the Schwarzschild metric. The simplest, explicit component representation in natural units for the Lanczos tensor in this case is
with all other components vanishing up to symmetries. This form, however, is not in the Lanczos gauge. The nonvanishing terms of the Lanczos tensor in the Lanczos gauge are
It is further possible to show, even in this simple case, that the Lanczos tensor cannot in general be reduced to a linear combination of the spin coefficients of the Newman–Penrose formalism, which attests to the Lanczos tensor's fundamental nature. Similar calculations have been used to construct arbitrary Petrov type D solutions.
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