Hellenica World

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In differential geometry and general relativity, the Bach tensor is a trace-free tensor of rank 2 which is conformally invariant in dimension n = 4.[1] Before 1968, it was the only known conformally invariant tensor that is algebraically independent of the Weyl tensor.[2] In abstract indices the Bach tensor is given by

$$B_{ab} = P_{cd}{{{W_a}^c}_b}^d+\nabla^c\nabla_aP_{bc}-\nabla^c\nabla_cP_{ab}$$

where W is the Weyl tensor, and P the Schouten tensor given in terms of the Ricci tensor $$R_{ab}$$ and scalar curvature R by

$$P_{ab}=\frac{1}{n-2}\left(R_{ab}-\frac{R}{2(n-1)}g_{ab}\right).$$

Cotton tensor
Obstruction tensor

References

Rudolf Bach, "Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungstensorbegriffs", Mathematische Zeitschrift, 9 (1921) pp. 110.

P. Szekeres, Conformal Tensors. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 304, No. 1476 (Apr. 2, 1968), pp. 113–122