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

B a b = P c d W a c b d + ∇ c ∇ c P a b − ∇ c ∇ a P b c {\displaystyle B_{ab}=P_{cd}{{{W_{a}}^{c}}_{b}}^{d}+\nabla ^{c}\nabla _{c}P_{ab}-\nabla ^{c}\nabla _{a}P_{bc}}

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

P a b = 1 n − 2 ( R a b − R 2 ( n − 1 ) g a b ) . {\displaystyle P_{ab}={\frac {1}{n-2}}\left(R_{ab}-{\frac {R}{2(n-1)}}g_{ab}\right).}

• Cotton tensor
• Obstruction tensor

• Arthur L. Besse, Einstein Manifolds. Springer-Verlag, 2007. See Ch.4, §H "Quadratic Functionals".
• Demetrios Christodoulou, Mathematical Problems of General Relativity I. European Mathematical Society, 2008. Ch.4 §2 "Sketch of the proof of the global stability of Minkowski spacetime".
• Yvonne Choquet-Bruhat, General Relativity and the Einstein Equations. Oxford University Press, 2011. See Ch.XV §5 "Christodoulou-Klainerman theorem" which notes the Bach tensor is the "dual of the Coton tensor which vanishes for conformally flat metrics".
• Thomas W. Baumgarte, Stuart L. Shapiro, Numerical Relativity: Solving Einstein's Equations on the Computer. Cambridge University Press, 2010. See Ch.3.