Bach tensor

{{Short description|Rank-2 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}}.<ref>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].</ref> Before 1968, it was the only known conformally invariant tensor that is algebraically independent of the Weyl tensor.<ref>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</ref> In abstract indices the Bach tensor is given by :<math>B_{ab} = P_{cd}{{{W_a}^c}_b}^d+\nabla^c\nabla_cP_{ab}-\nabla^c\nabla_aP_{bc}</math> where ''<math>W</math>'' is the Weyl tensor, and ''<math>P</math>'' the Schouten tensor given in terms of the Ricci tensor ''<math>R_{ab}</math>'' and scalar curvature ''<math>R</math>'' by

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

==See also== *Cotton tensor *Obstruction tensor

== References == {{Reflist}}

==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.

Category:Tensors Category:Tensors in general relativity