# Bach tensor

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{{Short description|Rank-2 tensor}}
In [differential geometry](/source/differential_geometry) and [general relativity](/source/general_relativity), the '''Bach tensor''' is a trace-free [tensor](/source/tensor) of rank 2 which is [conformally invariant](/source/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](/source/algebraically_independent) of the [Weyl tensor](/source/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](/source/abstract_index_notation) 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](/source/Weyl_tensor), and ''<math>P</math>'' the [Schouten tensor](/source/Schouten_tensor) given in terms of the [Ricci tensor](/source/Ricci_tensor) ''<math>R_{ab}</math>'' and [scalar curvature](/source/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](/source/Cotton_tensor)
*[Obstruction tensor](/source/Obstruction_tensor)

== References ==
{{Reflist}}

==Further reading==
* Arthur L. Besse, ''Einstein Manifolds''. Springer-Verlag, 2007. See Ch.4, §H "Quadratic Functionals".
* [Demetrios Christodoulou](/source/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](/source/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

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