ThePlebanski tensor is anorder 4 tensor ingeneral relativity constructed from thetrace-free Ricci tensor. It was first defined byJerzy Plebański in 1964.^[1]

Let${\displaystyle S_{ab}}$ be the trace-free Ricci tensor:

${\displaystyle S_{ab}=R_{ab}-\frac{1}{4}R{g}_{ab}.}$

Then the Plebanski tensor is defined as

${\displaystyle P^{ab}{}_{cd}=S^{[a}{}_{[c}{S}^{b]}{}_{d]}+{\delta }^{[a}{}_{[c}{S}^{b]e}{S}_{d]e}-\frac{1}{6}{\delta }^{[a}{}_{[c}{\delta }^{b]}{}_{d]}{S}^{ef}{S}_{ef}.}$

The advantage of the Plebanski tensor is that it shares the same symmetries as theWeyl tensor. It therefore becomes possible to classify differentspacetimes based on additional algebraic symmetries of the Plebanski tensor in a manner analogous to thePetrov classification.^[2]

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