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| 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
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| Vol. 304, No. 1476 (Apr. 2, 1968), pp. [http://www.jstor.org/pss/2416002 113]–122</ref> In [[abstract index notation|abstract indices]] the Bach tensor is given by
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| :<math>B_{ab} = P_{cd}{{{W_a}^c}_b}^d+\nabla^c\nabla_aP_{bc}-\nabla^c\nabla_cP_{ab}</math>
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| 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
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| :<math>P_{ab}=\frac{1}{n-2}\left(R_{ab}-\frac{R}{2(n-1)}g_{ab}\right).</math>
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| ==See also==
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| *[[Cotton tensor]]
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| *[[Obstruction tensor]]
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| == References ==
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| {{Reflist}}
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| ==Further reading==
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| * Arthur L. Besse, ''Einstein Manifolds''. Springer-Verlag, 2007. See Ch.4, §H "Quadratic Functionals".
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| * 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".
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| * 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".
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| * Thomas W. Baumgarte, Stuart L. Shapiro, ''Numerical Relativity: Solving Einstein's Equations on the Computer''. Cambridge University Press, 2010. See Ch.3.
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| [[Category:Tensors]]
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| [[Category:Tensors in general relativity]]
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| {{differential-geometry-stub}}
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| {{relativity-stub}}
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Greetings! I am Myrtle Shroyer. Doing ceramics is what her family members and her enjoy. Managing people has been his working day occupation for a while. Puerto Rico is exactly where he's been residing for many years and he will by no means transfer.
Feel free to surf to my webpage; std home test [Highly recommended Online site]