Riemannscher Krümmungstensor: Unterschied zwischen den Versionen

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Version vom 9. April 2009, 13:58 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 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, 13:58 Uhr

Herleitung

Symmetrien

Bianchi Identität

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