Bach tensor

In differential geometry and general relativity, the Bach tensor is a trace-free tensor of rank 2 which is conformally invariant in dimension {{nowrap|1=n = 4}}.Rudolf Bach, "Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungstensorbegriffs", Mathematische Zeitschrift, 9 (1921) pp. [http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002365812&IDDOC=16903 110]. Before 1968, it was the only known conformally invariant tensor that is algebraically independent of the Weyl tensor.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. [https://www.jstor.org/pss/2416002 113]–122 In abstract indices the Bach tensor is given by

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

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

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

Bach tensor

See also

References

Further reading