Benutzer:MovGP0/Mathematik/Riemannscher Krümmungstensor

MovGP0

Über mich

Hilfen

Artikel

Weblinks

Literatur

Zitate

Notizen

Programmierung

Physik

Physik Mathematische Grundlagen Mathematik für ET Semester 1 • Semester 2 • Semester 3 Mathematik Kombinatorik und Permutationen • Ableitung • Dreiecksnotation • Gradient, Divergenz, Rotation • Metrischer Tensor • Christoffelsymbole • Krümmungstensor • Energie-Impuls-Tensor • Kosmologische Konstante Quantenmechanik mit Matrizen Imaginäre und komplexe Zahlen • Vektoren • Matrizen • Jacobi-Matrizen • Pauli-Matrizen • Hermitesche Matrizen Lineare Algebra und Tensoren Tensornotation • Tensoren • Dirac-Notation • Körper • Vektorraum Gruppentheorie Gruppen, Lie Gruppen, Lie Algebren Riemannscher Krümmungstensor Kommutator $[A,B]=AB-BA$ für Zahlen gilt: $AB-BA=0$ allgemein gilt: ${\begin{array}{rclr}\left[{\frac {\partial }{\partial x}},f(x)\right]\,V&=&{\frac {\partial }{\partial x}}f(x)V-f(x){\frac {\partial }{\partial x}}V&|{\text{Kettenregel:}}\ d(AB)=A\,dB+B\,dA\\&=&V{\frac {\partial f(x)}{\partial x}}+f(x){\frac {\partial V}{\partial x}}-f(x){\frac {\partial V}{\partial x}}\\&=&V{\frac {\partial f(x)}{\partial x}}+{\cancel {f(x){\frac {\partial V}{\partial x}}}}-{\cancel {f(x){\frac {\partial V}{\partial x}}}}\\&=&{\frac {\partial f(x)}{\partial x}}V\end{array}}$ Parallel-Transport eines Vektors im Raum Kommutator Parallel-Transport eines Vektors V vom Punkt A nach B nach C nach A': $\operatorname {diff} \left(\mathrm {d} x_{\mu }\right)=\left({\vec {V}}_{C}-{\vec {V}}_{D}\right)-\left({\vec {V}}_{B}-{\vec {V}}_{A}\right)$ $\operatorname {diff} \left(\mathrm {d} x_{\nu }\right)=\left({\vec {V}}_{C}-{\vec {V}}_{B}\right)-\left({\vec {V}}_{D}-{\vec {V}}_{A'}\right)$ ${\begin{array}{rcl}\operatorname {diff} \left(\mathrm {d} x_{\mu }\right)-\operatorname {diff} \left(\mathrm {d} x_{\nu }\right)&=&\left(\left({\vec {V}}_{C}-{\vec {V}}_{D}\right)-\left({\vec {V}}_{B}-{\vec {V}}_{A}\right)\right)-\left(\left({\vec {V}}_{C}-{\vec {V}}_{B}\right)-\left({\vec {V}}_{D}-{\vec {V}}_{A'}\right)\right)\\&=&\left({\vec {V}}_{C}-{\vec {V}}_{D}-{\vec {V}}_{B}+{\vec {V}}_{A}\right)-\left({\vec {V}}_{C}-{\vec {V}}_{B}-{\vec {V}}_{D}+{\vec {V}}_{A'}\right)\\&=&{\vec {V}}_{C}-{\vec {V}}_{D}-{\vec {V}}_{B}+{\vec {V}}_{A}-{\vec {V}}_{C}+{\vec {V}}_{B}+{\vec {V}}_{D}-{\vec {V}}_{A'}\\&=&{\cancel {{\vec {V}}_{C}}}{\cancel {-{\vec {V}}_{D}}}{\cancel {-{\vec {V}}_{B}}}+{\vec {V}}_{A}{\cancel {-{\vec {V}}_{C}}}{\cancel {+{\vec {V}}_{B}}}{\cancel {+{\vec {V}}_{D}}}-{\vec {V}}_{A'}\\&=&{\vec {V}}_{A}-{\vec {V}}_{A'}\end{array}}$ Die Differenz $\mathrm {d} V={\vec {V}}_{A}-{\vec {V}}_{A'}=0$ bzw. ${\vec {V}}_{A}={\vec {V}}_{A'}$, falls der Raum flach ist. $V_{C}-V_{D}=\underbrace {\frac {\partial V}{\partial x^{\mu }}} _{\text{Gradient}}\underbrace {\mathrm {d} x^{\mu }} _{\text{Distance}}V=\nabla _{\mu }\mathrm {d} x^{\mu }V$ $\left({\vec {V}}_{C}-{\vec {V}}_{D}\right)-\left({\vec {V}}_{B}-{\vec {V}}_{A}\right)=\nabla _{\nu }\mathrm {d} x^{\nu }\,\nabla _{\mu }\mathrm {d} x^{\mu }\,V$ $\left({\vec {V}}_{C}-{\vec {V}}_{B}\right)-\left({\vec {V}}_{D}-{\vec {V}}_{A'}\right)=\nabla _{\mu }\mathrm {d} x^{\mu }\,\nabla _{\nu }\mathrm {d} x^{\nu }\,V$ einsetzen ${\begin{array}{rcl}\mathrm {d} V&=&\nabla _{\nu }\nabla _{\mu }\mathrm {d} x^{\nu }\mathrm {d} x^{\mu }V-\nabla _{\mu }\nabla _{\nu }\mathrm {d} x^{\mu }\mathrm {d} x^{\nu }V\\&=&\mathrm {d} x^{\mu }\mathrm {d} x^{\nu }V\,\underbrace {\left(\nabla _{\nu }\nabla _{\mu }-\nabla _{\mu }\nabla _{\nu }\right)} _{\text{Kommutator}}\\&=&\mathrm {d} x^{\mu }\mathrm {d} x^{\nu }\,V\,\left[\nabla _{\nu },\nabla _{\mu }\right]\end{array}}$ mit Gl. 7: $\nabla _{\nu }V_{\mu }={\frac {\mathrm {d} V_{\mu }}{\mathrm {d} x^{\nu }}}+\Gamma _{\nu \mu }^{\rho }V_{\rho }=\left(\mathrm {d} _{\nu }+\Gamma _{\nu }\right)V_{\mu }$ Herleitung des Ricci-Tensors ${\begin{array}{rcl}\left[\nabla _{\nu },\nabla _{\mu }\right]&=&\left(\mathrm {d} _{\nu }+\Gamma _{\nu }\right)\,\left(\mathrm {d} _{\mu }+\Gamma _{\mu }\right)-\left(\mathrm {d} _{\mu }+\Gamma _{\mu }\right)\,\left(\mathrm {d} _{\nu }+\Gamma _{\nu }\right)\\&=&+\left(\mathrm {d} _{\nu }\mathrm {d} _{\mu }+\Gamma _{\nu }\mathrm {d} _{\mu }+\mathrm {d} _{\nu }\Gamma _{\mu }+\Gamma _{\nu }\Gamma _{\mu }\right)\\&&-\left(\mathrm {d} _{\mu }\mathrm {d} _{\nu }+\Gamma _{\mu }\mathrm {d} _{\nu }+\mathrm {d} _{\mu }\Gamma _{\nu }+\Gamma _{\mu }\Gamma _{\nu }\right)\\&=&+\mathrm {d} _{\nu }\mathrm {d} _{\mu }\color {Brown}+\Gamma _{\nu }\mathrm {d} _{\mu }\color {PineGreen}+\mathrm {d} _{\nu }\Gamma _{\mu }\color {Violet}+\Gamma _{\nu }\Gamma _{\mu }\\&&-\mathrm {d} _{\mu }\mathrm {d} _{\nu }\color {PineGreen}-\Gamma _{\mu }\mathrm {d} _{\nu }\color {Brown}-\mathrm {d} _{\mu }\Gamma _{\nu }\color {Violet}-\Gamma _{\mu }\Gamma _{\nu }\\&=&0\color {Brown}-\underbrace {\left[d_{\mu },\Gamma _{\nu }\right]} _{\frac {\mathrm {d} \Gamma _{\nu }}{\mathrm {d} x^{\mu }}}\color {PineGreen}+\underbrace {\left[d_{\nu },\Gamma _{\mu }\right]} _{\frac {\mathrm {d} \Gamma _{\mu }}{\mathrm {d} x^{\nu }}}\color {Violet}+\left[\Gamma _{\nu },\Gamma _{\mu }\right]\color {Black}=\underbrace {R_{\mu \nu }} _{\text{Ricci-T.}}\end{array}}$ $\mathrm {d} V=\mathrm {d} x^{\mu }\mathrm {d} x^{\nu }\,V\,\left[\nabla _{\nu },\nabla _{\mu }\right]=\mathrm {d} x^{\mu }\mathrm {d} x^{\nu }\,V\,R_{\mu \nu }$ Herleitung des Krümmungsskalars $g^{\mu \nu }R_{\mu \nu }=R$