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| | Hello, my title is Andrew and my wife doesn't like it at all. One of the issues she loves most is canoeing and she's been doing it for fairly a whilst. Mississippi is where his house is. My working day job is an information officer but I've currently utilized for another one.<br><br>My site :: psychics online ([http://mybrandcp.com/xe/board_XmDx25/107997 mybrandcp.com]) |
| In [[semi-Riemannian geometry]], the '''Bel decomposition''', taken with respect to a specific [[timelike congruence]], is a way of breaking up the [[Riemann tensor]] of a [[pseudo-Riemannian manifold]] into lower order tensors with properties similar to the [[electric field]] and [[magnetic field]]. Such a decomposition was partially described by Alphonse Matte in 1953<ref>{{citation|first1=A.|last1=Matte|title=Sur de nouvelles solutions oscillatoires des equations de la gravitation|journal=Canad. J. Math.|volume=5|page=1|year=1953|doi=10.4153/CJM-1953-001-3}}</ref> and by [[Lluis Bel]] in 1958.<ref>{{citation|first1=L.|last1=Bel|title=Définition d'une densité d'énergie et d'un état de radiation totale généralisée|journal=Comptes rendus hebdomadaires des séances de l'Académie des sciences|volume=246|page=3015|year=1958|url=http://gallica.bnf.fr/ark:/12148/bpt6k723q/f965.image.langEN}}</ref>
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| This decomposition is particularly important in [[general relativity]]. This is the case of four dimensional [[Lorentzian manifold]]s, for which there are only three pieces with simple properties and individual physical interpretations.
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| == Decomposition of the Riemann tensor ==
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| In four dimensions the Bel decomposition of the Riemann tensor, with respect to a timelike unit vector field <math>\vec{X}</math>, not necessarily geodesic or hypersurface orthogonal, consists of three pieces
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| #the ''[[electrogravitic tensor]]'' <math>E[\vec{X}]_{ab} = R_{ambn} \, X^m \, X^n</math>
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| #the ''[[magnetogravitic tensor]]'' <math>B[\vec{X}]_{ab} = {{}^\star R}_{ambn} \, X^m \, X^n</math>
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| #the ''[[topogravitic tensor]]'' <math>L[\vec{X}]_{ab} = {{}^\star R^\star}_{ambn} \, X^m \, X^n</math>
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| Because these are all ''transverse'' (i.e. projected to the spatial hyperplane elements orthogonal to our timelike unit vector field), they can be represented as linear operators on three dimensional vectors, or as three by three real matrices. They are respectively symmetric, [[traceless]], and symmetric (6,8,6 linearly independent components, for a total of 20). If we write these operators as '''E''', '''B''', '''L''' respectively, the principal invariants of the Riemann tensor are obtained as follows:
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| *<math>K_1/4</math> is the trace of '''E'''<sup>2</sup> + '''L'''<sup>2</sup> - 2 '''B''' '''B'''<sup>T</sup>,
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| *<math>-K_2/8</math> is the trace of '''B''' ( '''E''' - '''L''' ),
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| *<math>K_3/8</math> is the trace of '''E''' '''L''' - '''B'''<sup>2</sup>.
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| ==See also==
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| *[[Bel–Robinson tensor]]
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| *[[Ricci decomposition]]
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| ==References==
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| <references/>
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| [[Category:Lorentzian manifolds]]
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| [[Category:Tensors in general relativity]]
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| {{relativity-stub}}
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Hello, my title is Andrew and my wife doesn't like it at all. One of the issues she loves most is canoeing and she's been doing it for fairly a whilst. Mississippi is where his house is. My working day job is an information officer but I've currently utilized for another one.
My site :: psychics online (mybrandcp.com)