Riemannscher Krümmungstensor: Unterschied zwischen den Versionen

Aktuelle Version vom 12. September 2010, 18:29 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

*  $R_{α β μ ν} = R_{μ ν α β}$ 
*  $R_{α β μ ν} = - R_{β α μ ν} = - R_{α β ν μ}$ 
*  ${R^{α}}_{< β μ ν >} = {R^{α}}_{β μ ν} + {R^{α}}_{ν β μ} + {R^{α}}_{ν β μ} = 0$

Bianchi Identität

{R^{α β}}_{< μ ν; σ >} = {R^{α β}}_{μ ν; σ} + {R^{α β}}_{σ μ; ν} + {R^{α β}}_{ν σ; μ} = 0

@@ Zeile 1: / Zeile 1: @@
-<math>{R^{\color{Violet}\alpha}} _{ \color{Red}{\color{WildStrawberry}\beta} {\color{Orange}\mu} {\color{Red}{\color{Red}{\color{Red}\nu}}} } = {\partial _{\color{Red}{\color{Orange}\mu}} }\Gamma _{{\color{Red}{\color{WildStrawberry}\beta}} {\color{Red}{\color{Red}{\color{Red}\nu}}} }^{\color{Violet}\alpha}  - {\partial _{{\color{Orange}\mu} {\color{Red}{\color{Red}{\color{Red}\nu}}} }}\Gamma _{{\color{WildStrawberry}\beta} {\color{Orange}\mu} }^{\color{Violet}\alpha}  + \Gamma _{\sigma {\color{Orange}\mu} }^{\color{Violet}\alpha} \Gamma _{{\color{WildStrawberry}\beta} {\color{Red}{\color{Red}\nu}} }^\sigma  - \Gamma _{\sigma {\color{Red}{\color{Red}{\color{Red}\nu}}} }^{\color{Violet}\alpha} \Gamma _{{\color{WildStrawberry}\beta} {\color{Orange}\mu} }^\sigma
+<math>
+{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}
 </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, 18:29 Uhr

Herleitung

Symmetrien

Bianchi Identität

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