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 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

${\displaystyle B_{ab}=P_{cd}{{{W_{a}}^{c}}_{b}}^{d}+\nabla ^{c}\nabla _{c}P_{ab}-\nabla ^{c}\nabla _{a}P_{bc}}$

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

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