Riemannscher Krümmungstensor: Unterschied zwischen den Versionen

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Version vom 9. April 2009, 13:52 Uhr

${R^{α}}_{β μ ν} = Γ_{β μ, ν}^{α} - Γ_{β ν, μ}^{α} + Γ_{β μ}^{σ} Γ_{σ ν}^{α} - Γ_{β ν}^{σ} Γ_{σ μ}^{α}$

Herleitung

${A^{α}}_{; μ ν} - {A^{α}}_{; ν μ} = {R^{α}}_{β μ ν} A^{β}$

Symmetrien

*  $R_{α β μ ν} = R_{μ ν α β}$ 
*  $R_{α β μ ν} = - R_{β α μ ν} = - R_{α β ν μ}$ 
*  ${R^{α}}_{< β μ ν >} = {R^{α}}_{β μ ν} + {R^{α}}_{ν β μ} + {R^{α}}_{ν β μ}$

@@ Zeile 26: / Zeile 26: @@
 == Symmetrien ==
-<math>R_{{\color{Violet}\alpha}{\color{WildStrawberry}\beta} {\color{Orange}\mu} {\color{Red}\nu}}
+ * <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>
+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} }
+</math>
-[[Kategorie:Tensor_4._Stufe]]
+[[Kategorie:Tensor 4. Stufe]]

Riemannscher Krümmungstensor: Unterschied zwischen den Versionen

Version vom 9. April 2009, 13:52 Uhr

Herleitung

Symmetrien

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