Einstein’s General Relativity

Derivations, Applications, and Reflections – by Albert Prins

Appendix 3 – Mathematical derivation of Schwarzschild

Here we will work out the Christoffel symbols for the metric tensor of the Schwarzschild configuration.

Schwarzschild in \(r, \theta, \phi\) coordinates:

The definition of the Christoffel symbols:

\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}

For the Schwarzschild metric in \((t, r, \theta, \phi)\) we obtain, among others:

\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}

All elements of the metric tensor are zero, except for the diagonal elements. This means that the contravariant elements are the direct inverses of the covariant components. Thus, for example:

\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}

For \(r, \theta, \phi\)-coordinates:

Derivatives of \(\Gamma\) with respect to \(x_1 = r\):

\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}

Derivatives of \(\Gamma\) with respect to \(x_2 = \theta\):

\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}

Here, a prime (\('\)) denotes differentiation with respect to \(r\), and \(\dfrac{\partial}{\partial\theta}\) denotes differentiation with respect to \(\theta\).

For \(r, \theta, \phi\) coordinates:

Derivatives of \(\Gamma\) with respect to \(x_1 = r\):

\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}

Derivatives of \(\Gamma\) with respect to \(x_2 = \theta\):

Here, a prime (\('\)) denotes differentiation with respect to \(r\), and \(\dfrac{\partial}{\partial\theta}\) denotes differentiation with respect to \(\theta\).