Riemannscher Krümmungstensor: Unterschied zwischen den Versionen

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Version vom 9. April 2009, 14:58 Uhr

${R^{\color {Violet}\alpha }}_{{\color {WildStrawberry}\beta }{\color {Orange}\mu }{\color {Red}\nu }}=\Gamma _{{\color {WildStrawberry}\beta }{\color {Orange}\mu },\color {Red}\nu }^{\color {Violet}\alpha }-\Gamma _{{\color {WildStrawberry}\beta }{\color {Red}\nu },{\color {Orange}\mu }}^{\color {Violet}\alpha }+\Gamma _{{\color {WildStrawberry}\beta }{\color {Orange}\mu }}^{\sigma }\Gamma _{\sigma {\color {Red}\nu }}^{\color {Violet}\alpha }-\Gamma _{{\color {WildStrawberry}\beta }{\color {Red}\nu }}^{\sigma }\Gamma _{\sigma {\color {Orange}\mu }}^{\color {Violet}\alpha }$

Herleitung

${A^{\color {Violet}\alpha }}_{;{\color {Orange}\mu }{\color {Red}\nu }}-{A^{\color {Violet}\alpha }}_{;{\color {Red}\nu }{\color {Orange}\mu }}={R^{\color {Violet}\alpha }}_{{\color {WildStrawberry}\beta }{\color {Orange}\mu }{\color {Red}\nu }}{A^{\color {WildStrawberry}\beta }}$

Symmetrien

*  $R_{{\color {Violet}\alpha }{\color {WildStrawberry}\beta }{\color {Orange}\mu }{\color {Red}\nu }}=R_{{\color {Orange}\mu }{\color {Red}\nu }{\color {Violet}\alpha }{\color {WildStrawberry}\beta }}$ 
*  $R_{{\color {Violet}\alpha }{\color {WildStrawberry}\beta }{\color {Orange}\mu }{\color {Red}\nu }}=-R_{{\color {WildStrawberry}\beta }{\color {Violet}\alpha }{\color {Orange}\mu }{\color {Red}\nu }}=-R_{{\color {Violet}\alpha }{\color {WildStrawberry}\beta }{\color {Red}\nu }{\color {Orange}\mu }}$ 
*  ${R^{\color {Violet}\alpha }}_{<{\color {WildStrawberry}\beta }{\color {Orange}\mu }{\color {Red}\nu }>}={R^{\color {Violet}\alpha }}_{{\color {WildStrawberry}\beta }{\color {Orange}\mu }{\color {Red}\nu }}+{R^{\color {Violet}\alpha }}_{{\color {Red}\nu }{\color {WildStrawberry}\beta }{\color {Orange}\mu }}+{R^{\color {Violet}\alpha }}_{{\color {Red}\nu }{\color {WildStrawberry}\beta }{\color {Orange}\mu }}=0$

Bianchi Identität

${R^{{\color {Violet}\alpha }{\color {WildStrawberry}\beta }}}_{<{\color {Orange}\mu }{\color {Red}\nu };\sigma >}={R^{{\color {Violet}\alpha }{\color {WildStrawberry}\beta }}}_{{\color {Orange}\mu }{\color {Red}\nu };\sigma }+{R^{{\color {Violet}\alpha }{\color {WildStrawberry}\beta }}}_{\sigma {\color {Orange}\mu };{\color {Red}\nu }}+{R^{{\color {Violet}\alpha }{\color {WildStrawberry}\beta }}}_{{\color {Red}\nu }\sigma ;{\color {Orange}\mu }}=0$

@@ Zeile 50: / Zeile 50: @@
 === Bianchi Identität ===
-<math>{R^{{\color{Violet}\alpha} {\color{WildStrawberry}\beta}}_{<  {\color{Orange}\mu} {\color{Red}\nu};\sigma>}
+<math>{R^{{\color{Violet}\alpha} {\color{WildStrawberry}\beta}}}_{<  {\color{Orange}\mu} {\color{Red}\nu};\sigma>}
 =
-<math>{R^{{\color{Violet}\alpha} {\color{WildStrawberry}\beta}}_{{\color{Orange}\mu} {\color{Red}\nu};\sigma}
+{R^{{\color{Violet}\alpha} {\color{WildStrawberry}\beta}}}_{{\color{Orange}\mu} {\color{Red}\nu};\sigma}
 +
-<math>{R^{{\color{Violet}\alpha} {\color{WildStrawberry}\beta}}_{\sigma {\color{Orange}\mu}; {\color{Red}\nu}}
+{R^{{\color{Violet}\alpha} {\color{WildStrawberry}\beta}}}_{\sigma {\color{Orange}\mu}; {\color{Red}\nu}}
 +
-<math>{R^{{\color{Violet}\alpha} {\color{WildStrawberry}\beta}}_{{\color{Red}\nu} \sigma ;{\color{Orange}\mu} }
+{R^{{\color{Violet}\alpha} {\color{WildStrawberry}\beta}}}_{{\color{Red}\nu} \sigma ;{\color{Orange}\mu} }
 =0
 </math>
 [[Kategorie:Tensor 4. Stufe]]

Riemannscher Krümmungstensor: Unterschied zwischen den Versionen

Version vom 9. April 2009, 14:58 Uhr

Herleitung

Symmetrien

Bianchi Identität

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