Bach tensor - Alchetron, The Free Social Encyclopedia

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. Before 1968, it was the only known conformally invariant tensor that is algebraically independent of the Weyl tensor. In abstract indices the Bach tensor is given by

B a b = P c d W a c b d + ∇ c ∇ a P b c − ∇ c ∇ c P a b

where W is the Weyl tensor, and P the Schouten tensor given in terms of the Ricci tensor R a b and scalar curvature R by

P a b = 1 n − 2 ( R a b − R 2 ( n − 1 ) g a b ) .

References

Bach tensor Wikipedia

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