Riemannscher Krümmungstensor: Unterschied zwischen den Versionen

Aktuelle Version vom 12. September 2010, 19:29 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 14: / Zeile 14: @@
 </math>
-[[Kategorie:Tensor_4._Stufe]]
+== Herleitung ==
+:<math>
+{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}}
+</math>
+== Symmetrien ==
+ * <math>
+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} }
+</math>
+ * <math>
+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}}
+</math>
+ * <math>
+{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
+</math>
+=== Bianchi Identität ===
+:<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}
++
+{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
+</math>
+[[Kategorie:Tensor 4. Stufe]]

Riemannscher Krümmungstensor: Unterschied zwischen den Versionen

Aktuelle Version vom 12. September 2010, 19:29 Uhr

Herleitung

Symmetrien

Bianchi Identität

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