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 . 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

where ' is the Weyl tensor, and ' the Schouten tensor given in terms of the Ricci tensor ' and scalar curvature ' by

See also

• Cotton tensor
• Obstruction tensor

References

Further reading

• 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.