Christoffel symbols for Schwarzschild metric

Question

So, I am doing coursework for maths, and I'm trying to find out how I could calculate the Christoffel symbols for the Schwarzschild metric; however, I have realized that calculating them is too difficult for the level of maths I currently have, as I haven't even finished high school. So I was wondering if anyone knows where I could find them. So far this is the data I have:

The Schwarzschild metric $$\begin{pmatrix} c^2-\frac{2GM}{r} & 0 & 0 & 0 \\ 0 & \frac{-1}{1-\frac{2GM}{rc^2}} & 0 & 0 \\ 0 & 0 & -r^2 & 0 \\ 0 & 0 & 0 & -r^2\sin^2\theta \end{pmatrix}$$ and some of the Christoffel symbols $$\begin{align} \Gamma^t_{tt} &= 0 \\ \Gamma^t_{tr} &= \frac{GM}{r^2c^2}\left(\frac{-1}{1-\frac{2GM}{rc^2}}\right) \\ \Gamma^t_{rt} &= \frac{GM}{r^2c^2}\left(\frac{-1}{1-\frac{2GM}{rc^2}}\right) \\ \Gamma^t_{rr} &= 0 \\ \Gamma^r_{tt} &= \frac{GM}{r^2}\left(1-\frac{2GM}{rc^2}\right) \\ \Gamma^r_{tr} &= 0 \\ \Gamma^r_{rt} &= 0 \\ \Gamma^r_{rr} &= \frac{-GM}{r^2c^2}\left(\frac{-1}{1-\frac{2GM}{rc^2}}\right) \end{align}$$ I want to use some Christoffel symbols I couldn't find. Also, if anything in the formulas is wrong, could you tell me?

Answer 1

INTRODUCTION

First of all, let's focus on the notation used in the question. The Schwarzschild metric is, in $(−+++)$ sign convention and units of $c=G=1$: $$ d{s}^2= -\left(1-\dfrac{2M}{r} \right) d{t}^2 + \left(1-\dfrac{2M}{r} \right)^{-1} d{r}^2 +r^2(d{\theta}^2+\sin^2{\theta}d{\phi}^2) $$ The reasons why it is convenient to put both the speed of light and Newton gravitational constant equal to $1$ are various and you can find an exhaustive list of the possible choices of the natural units in the following site.

Moreover, the convention of choosing the signature $(-+++)$ instead of the alternative $(+---)$ relies on the fact that it's the most used in modern textbooks on General Relativity and personally I find myself more comfortable with it. I'm sure you will be able to adjust this notation to your specific needs.

In our case, the covariant metric components form a diagonal matrix, namely: $$ g_{\mu\nu}=\begin{pmatrix} -\left(1-\dfrac{2M}{r} \right) & 0 & 0 & 0\\ 0 & \left(1-\dfrac{2M}{r} \right)^{-1}& 0 & 0\\ 0 & 0 & r^2 & 0\\ 0 & 0 & 0 & r^2\sin^2{\theta} \end{pmatrix} $$

CHRISTOFFEL SYMBOLS

In general, Christoffel symbols are related to the metric in the following way: $$ \Gamma^\sigma_{\mu \nu}=\dfrac{1}{2}g^{\sigma\rho}\left(\partial_\mu g_{\nu\rho}+\partial_\nu g_{\rho\mu}-\partial_\rho g_{\mu\nu} \right)\,\text{,} \qquad \qquad \partial_\mu \equiv \dfrac{\partial}{\partial x^\mu} $$

So, as I suggested in the comments below your question, instead of blindly calculating all $64$ symbols, we can use their symmetry properties and cleverly noting that some of them are identically equal to zero. For example, the Schwarzschild metric doesn't depend on $t$ or $\phi$.

Thus, by setting appropriately the index $\sigma =t$ from the equation above, also $\rho =t$ must be valid because the metric $g^{\sigma \rho}$ doesn't have off-diagonal elements. For the same reason, the last term $\partial_\rho g_{\mu\nu}$ is always zero for every value of $\mu$ and $\nu$ since nothing depends explicitly on time $t$. You have to then force $\mu=t$ in order to have diagonal elements of the metric and since this term only depends on $r$, there's only one possible choice for $\nu=r$. Doing this, you can compute $\Gamma^{t}_{rt}=\Gamma^{t}_{tr}$ and understand that all the other symbols for $\sigma=t$ are vanishing.

This might seem tricky and overcomplicated, but it allows us to discard by simple inspection all the symbols which are trivially equal to zero. An equivalent reasoning might be also implemented for $\sigma =\phi$ with the difference that now the element $g_{\phi\phi}$ depends both on $r$ and $\theta$, so we have two different non-zero symbols to consider.

After that, with a little work it's possible to identify the nonzero Riemann-Christoffel symbols: \begin{equation} \begin{matrix} \Gamma^r_{tt}=\dfrac{M(r-2M)}{r^3}\,\text{,} & ~ & \Gamma^r_{rr}=-\dfrac{M}{r(r-2M)}\,\text{,} & ~ & \Gamma^r_{\theta \theta}=-(r-2M)\,\text{,}\\ ~ & ~ & ~ & ~ & ~\\ \Gamma^r_{\phi \phi}=-(r-2M)\sin^2{\theta}\,\text{,} & ~ & \Gamma^t_{rt} = \dfrac{M}{r(r-2M)}\,\text{,} & ~ & \Gamma^\theta_{r\theta}=\dfrac{1}{r}\,\text{,}\\ ~ & ~ & ~ & ~ & ~\\ \Gamma^\theta_{\phi\phi}=-\sin{\theta}\cos{\theta}\,\text{,} & ~ & \Gamma^\phi_{r\phi}=\dfrac{1}{r}\,\text{,} & ~ & \Gamma^\phi_{\theta\phi} = \dfrac{\cos{\theta}}{\sin{\theta}}\,\text{.}\\ ~ & ~ & \end{matrix} \end{equation}