Appendix 3 – Wiskundige uitwerking van Schwarzschild
Hier zullen we de Christoffel-symbolen uitwerken voor de metrische tensor van de Schwarzschild-configuratie.
Schwarzschild in \(r, \theta, \phi\)-coördinaten:
De definitie van de Christoffel-symbolen:
\[ \Gamma^{\rho}{}_{\mu\nu} = \frac{1}{2} g^{\rho\alpha} \left( \frac{\partial g_{\nu\alpha}}{\partial x^{\mu}} + \frac{\partial g_{\mu\alpha}}{\partial x^{\nu}} - \frac{\partial g_{\mu\nu}}{\partial x^{\alpha}} \right) \]Voor de Schwarzschild-metriek in \((t, r, \theta, \phi)\) krijgen we onder meer:
\[ \Gamma^{0}{}_{10} = \Gamma^{0}{}_{01} = \frac{1}{2} g^{00} \frac{\partial g_{00}}{\partial r} \] \[ \Gamma^{1}{}_{00} = \frac{1}{2} g^{11} \left(- \frac{\partial g_{00}}{\partial r}\right) \] \[ \Gamma^{1}{}_{11} = \frac{1}{2} g^{11} \frac{\partial g_{11}}{\partial r} \] \[ \Gamma^{1}{}_{22} = \frac{1}{2} g^{11} \left(-\frac{\partial g_{22}}{\partial r}\right) \] \[ \Gamma^{1}{}_{33} = \frac{1}{2} g^{11} \left(-\frac{\partial g_{33}}{\partial r}\right) \] \[ \Gamma^{2}{}_{21} = \Gamma^{2}{}_{12} = \frac{1}{2} g^{22} \frac{\partial g_{22}}{\partial r} \] \[ \Gamma^{3}{}_{31} = \Gamma^{3}{}_{13} = \frac{1}{2} g^{33} \frac{\partial g_{33}}{\partial r} \] \[ \Gamma^{2}{}_{33} = \frac{1}{2} g^{22} \left(- \frac{\partial g_{33}}{\partial \theta}\right) \] \[ \Gamma^{3}{}_{23} = \Gamma^{3}{}_{32} = \frac{1}{2} g^{33} \frac{\partial g_{33}}{\partial \theta} \]Alle elementen in de metrische tensor zijn nul, behalve de elementen op de diagonaal. Dit betekent dat de contravariante elementen de directe inverse zijn van de covariante componenten. Dus bijvoorbeeld:
\[ g^{00} = \frac{1}{g_{00}}, \quad g^{11} = \frac{1}{g_{11}}, \quad g^{22} = \frac{1}{g_{22}}, \quad g^{33} = \frac{1}{g_{33}}. \]Voor \(r, \theta, \phi\)-coördinaten:
Afgeleiden van \(\Gamma\) naar \(x^1 = r\):
\[ R_{0011} = R_{0101} = \frac{\partial \Gamma^{0}{}_{10}}{\partial r} = \frac{\partial \Gamma^{1}{}_{00}}{\partial r} = \frac{1}{2}\left( - \frac{1}{g_{00}^{2}}\left(\frac{\partial g_{00}}{\partial r}\right)^{2} + \frac{1}{g_{00}} \frac{\partial^{2} g_{00}}{\partial r^{2}} \right) = \frac{1}{2} g_{00}^{-1}\left( - g_{00}^{-1} (g_{00}')^{2} + g_{00}'' \right) \] \[ R_{1001} = \frac{\partial \Gamma^{1}{}_{00}}{\partial r} = -\frac{1}{2}\left( - \frac{1}{g_{11}^{2}} \frac{\partial g_{11}}{\partial r} \frac{\partial g_{00}}{\partial r} + \frac{1}{g_{11}} \frac{\partial^{2} g_{00}}{\partial r^{2}} \right) = -\frac{1}{2} g_{11}^{-1}\left( - g_{11}^{-1} g_{11}' g_{00}' + g_{00}'' \right) \] \[ R_{1111} = \frac{\partial \Gamma^{1}{}_{11}}{\partial r} = \frac{1}{2}\left( - \frac{1}{g_{11}^{2}}\left(\frac{\partial g_{11}}{\partial r}\right)^{2} + \frac{1}{g_{11}} \frac{\partial^{2} g_{11}}{\partial r^{2}} \right) = \frac{1}{2} g_{11}^{-1}\left( - g_{11}^{-1} (g_{11}')^{2} + g_{11}'' \right) \] \[ R_{1221} = \frac{\partial \Gamma^{1}{}_{22}}{\partial r} = -\frac{1}{2}\left( - \frac{1}{g_{11}^{2}} \frac{\partial g_{11}}{\partial r} \frac{\partial g_{22}}{\partial r} + \frac{1}{g_{11}} \frac{\partial^{2} g_{22}}{\partial r^{2}} \right) = -\frac{1}{2} g_{11}^{-1}\left( - g_{11}^{-1} g_{11}' g_{22}' + g_{22}'' \right) \] \[ R_{1331} = \frac{\partial \Gamma^{1}{}_{33}}{\partial r} = -\frac{1}{2}\left( - \frac{1}{g_{11}^{2}} \frac{\partial g_{11}}{\partial r} \frac{\partial g_{33}}{\partial r} + \frac{1}{g_{11}} \frac{\partial^{2} g_{33}}{\partial r^{2}} \right) = -\frac{1}{2} g_{11}^{-1}\left( - g_{11}^{-1} g_{11}' g_{33}' + g_{33}'' \right) \] \[ R_{2121} = R_{2211} = \frac{\partial \Gamma^{2}{}_{21}}{\partial r} = \frac{\partial \Gamma^{2}{}_{12}}{\partial r} = \frac{1}{2}\left( - \frac{1}{g_{22}^{2}}\left(\frac{\partial g_{22}}{\partial r}\right)^{2} + \frac{1}{g_{22}} \frac{\partial^{2} g_{22}}{\partial r^{2}} \right) = \frac{1}{2} g_{22}^{-1}\left( - g_{22}^{-1} (g_{22}')^{2} + g_{22}'' \right) \] \[ R_{3131} = R_{3311} = \frac{\partial \Gamma^{3}{}_{31}}{\partial r} = \frac{\partial \Gamma^{3}{}_{13}}{\partial r} = \frac{1}{2}\left( - \frac{1}{g_{33}^{2}}\left(\frac{\partial g_{33}}{\partial r}\right)^{2} + \frac{1}{g_{33}} \frac{\partial^{2} g_{33}}{\partial r^{2}} \right) = \frac{1}{2} g_{33}^{-1}\left( - g_{33}^{-1} (g_{33}')^{2} + g_{33}'' \right) \] \[ R_{2331} = \frac{\partial \Gamma^{2}{}_{33}}{\partial r} = -\frac{1}{2}\left( - \frac{1}{g_{22}^{2}} \frac{\partial g_{22}}{\partial r} \frac{\partial g_{33}}{\partial \theta} + \frac{1}{g_{22}} \frac{\partial^{2} g_{33}}{\partial r \partial \theta} \right) = -\frac{1}{2} g_{22}^{-1}\left( - g_{22}^{-1} g_{22}' \frac{\partial g_{33}}{\partial \theta} + \frac{\partial^{2} g_{33}}{\partial r \partial \theta} \right) \] \[ R_{3231} = R_{3321} = \frac{\partial \Gamma^{3}{}_{32}}{\partial r} = \frac{\partial \Gamma^{3}{}_{23}}{\partial r} = \frac{1}{2}\left( - \frac{1}{g_{33}^{2}} \frac{\partial g_{33}}{\partial r} \frac{\partial g_{33}}{\partial \theta} + \frac{1}{g_{33}} \frac{\partial^{2} g_{33}}{\partial r \partial \theta} \right) = \frac{1}{2} g_{33}^{-1}\left( - g_{33}^{-1} g_{33}' \frac{\partial g_{33}}{\partial \theta} + \frac{\partial^{2} g_{33}}{\partial r \partial \theta} \right) \]Afgeleiden van \(\Gamma\) naar \(x^2 = \theta\):
\[ R_{1222} = \frac{\partial \Gamma^{1}{}_{22}}{\partial \theta} = -\frac{1}{2} g_{11}^{-1} \frac{\partial^{2} g_{22}}{\partial r \partial \theta} \] \[ R_{1332} = \frac{\partial \Gamma^{1}{}_{33}}{\partial \theta} = -\frac{1}{2} g_{11}^{-1} \frac{\partial^{2} g_{33}}{\partial r \partial \theta} \] \[ R_{2332} = \frac{\partial \Gamma^{2}{}_{33}}{\partial \theta} = -\frac{1}{2}\left( - \frac{1}{g_{22}^{2}} \frac{\partial g_{22}}{\partial \theta} \frac{\partial g_{33}}{\partial \theta} + \frac{1}{g_{22}} \frac{\partial^{2} g_{33}}{\partial \theta^{2}} \right) = -\frac{1}{2} g_{22}^{-1}\left( - g_{22}^{-1} \frac{\partial g_{22}}{\partial \theta} \frac{\partial g_{33}}{\partial \theta} + \frac{\partial^{2} g_{33}}{\partial \theta^{2}} \right) \] \[ R_{2222} = \frac{\partial \Gamma^{2}{}_{22}}{\partial \theta} = \frac{1}{2}\left( - \frac{1}{g_{22}^{2}}\left(\frac{\partial g_{22}}{\partial \theta}\right)^{2} + \frac{1}{g_{22}} \frac{\partial^{2} g_{22}}{\partial \theta^{2}} \right) = \frac{1}{2} g_{22}^{-1}\left( - g_{22}^{-1}\left(\frac{\partial g_{22}}{\partial \theta}\right)^{2} + \frac{\partial^{2} g_{22}}{\partial \theta^{2}} \right) \] \[ R_{3312} = R_{3132} = \frac{\partial \Gamma^{3}{}_{13}}{\partial \theta} = \frac{\partial \Gamma^{3}{}_{31}}{\partial \theta} = \frac{1}{2}\left( - \frac{1}{g_{33}^{2}} \frac{\partial g_{33}}{\partial r} \frac{\partial g_{33}}{\partial \theta} + \frac{1}{g_{33}} \frac{\partial^{2} g_{33}}{\partial r \partial \theta} \right) = \frac{1}{2} g_{33}^{-1}\left( - g_{33}^{-1} g_{33}' \frac{\partial g_{33}}{\partial \theta} + \frac{\partial^{2} g_{33}}{\partial r \partial \theta} \right) \] \[ R_{3232} = R_{3322} = \frac{\partial \Gamma^{3}{}_{32}}{\partial \theta} = \frac{\partial \Gamma^{3}{}_{23}}{\partial \theta} = \frac{1}{2}\left( - \frac{1}{g_{33}^{2}}\left(\frac{\partial g_{33}}{\partial \theta}\right)^{2} + \frac{1}{g_{33}} \frac{\partial^{2} g_{33}}{\partial \theta^{2}} \right) = \frac{1}{2} g_{33}^{-1}\left( - g_{33}^{-1}\left(\frac{\partial g_{33}}{\partial \theta}\right)^{2} + \frac{\partial^{2} g_{33}}{\partial \theta^{2}} \right) \]Hierbij betekent een prime (\('\)) afgeleide naar \(r\), en \(\dfrac{\partial}{\partial\theta}\) de afgeleide naar \(\theta\).