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:
\begin{equation}
\begin{aligned}
\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)
\end{aligned}
\end{equation}
Voor de Schwarzschild-metriek in \((t, r, \theta, \phi)\) krijgen we onder meer:
\begin{equation}
\begin{aligned}
\Gamma^{0}_{10} = \Gamma^{0}_{01}
= \frac{1}{2} g^{00} \frac{\partial g_{00}}{\partial r}
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\Gamma^{1}_{00}
= \frac{1}{2} g^{11} \left(- \frac{\partial g_{00}}{\partial r}\right)
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\Gamma^{1}_{11}
= \frac{1}{2} g^{11} \frac{\partial g_{11}}{\partial r}
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\Gamma^{1}_{22}
= \frac{1}{2} g^{11} \left(-\frac{\partial g_{22}}{\partial r}\right)
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\Gamma^{1}_{33}
= \frac{1}{2} g^{11} \left(-\frac{\partial g_{33}}{\partial r}\right)
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\Gamma^{2}_{21}
= \Gamma^{2}_{12}
= \frac{1}{2} g^{22} \frac{\partial g_{22}}{\partial r}
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\Gamma^{3}_{31}
= \Gamma^{3}_{13}
= \frac{1}{2} g^{33} \frac{\partial g_{33}}{\partial r}
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\Gamma^{2}_{33}
= \frac{1}{2} g^{22} \left(- \frac{\partial g_{33}}{\partial \theta}\right)
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
\Gamma^{3}_{23}
= \Gamma^{3}_{32}
= \frac{1}{2} g^{33} \frac{\partial g_{33}}{\partial \theta}
\end{aligned}
\end{equation}
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:
\begin{equation}
\begin{aligned}
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}}.
\end{aligned}
\end{equation}
Voor \(r, \theta, \phi\)-coördinaten:
Afgeleiden van \(\Gamma\) naar \(x_1 = r\):
\begin{equation}
\begin{aligned}
R_{0011} = R_{0101}
= \frac{\partial \Gamma^{0}_{01}}{\partial r}
= \frac{\partial \Gamma^{0}_{10}}{\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)
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
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)
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
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)
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
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)
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
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)
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
R_{2121} = R_{2211}
= \frac{\partial \Gamma^{2}_{12}}{\partial r}
= \frac{\partial \Gamma^{2}_{21}}{\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)
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
R_{3131} = R_{3311}
= \frac{\partial \Gamma^{3}_{13}}{\partial r}
= \frac{\partial \Gamma^{3}_{31}}{\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)
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
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)
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
R_{3231} = R_{3321}
= \frac{\partial \Gamma^{3}_{23}}{\partial r}
= \frac{\partial \Gamma^{3}_{32}}{\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)
\end{aligned}
\end{equation}
Afgeleiden van \(\Gamma\) naar \(x_2 = \theta\) :
\begin{equation}
\begin{aligned}
R_{1222}
= \frac{\partial \Gamma^{1}_{22}}{\partial \theta}
= -\frac{1}{2} g_{11}^{-1}
\frac{\partial^{2} g_{22}}{\partial r \partial \theta}
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
R_{1332}
= \frac{\partial \Gamma^{1}_{33}}{\partial \theta}
= -\frac{1}{2} g_{11}^{-1}
\frac{\partial^{2} g_{33}}{\partial r \partial \theta}
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
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)
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
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)
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
R_{3312} = R_{3132}
= \frac{\partial \Gamma^{3}_{31}}{\partial \theta}
= \frac{\partial \Gamma^{3}_{13}}{\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)
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
R_{3232} = R_{3322}
= \frac{\partial \Gamma^{3}_{23}}{\partial \theta}
= \frac{\partial \Gamma^{3}_{32}}{\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)
\end{aligned}
\end{equation}
Hierbij betekent een prime (\('\)) afgeleide naar \(r\), en \(\dfrac{\partial}{\partial\theta}\) de afgeleide naar \(\theta\).