When employing the Schwarzschild metric, I understand there are nine non-vanishing Christoffel symbols:
$ \Gamma^t_{rt} = -\Gamma^r_{rr} = \frac{r_{\rm s}}{2r(r – r_{\rm s})} \\[3pt]
\Gamma^r_{tt} = \frac{r_{\rm s}(r – r_{\rm s})}{2r^3} \\[3pt]
\Gamma^r_{\phi\phi} = (r_{\rm s} – r)\sin^2(\theta) \\[3pt]
\Gamma^r_{\theta\theta} = r_{\rm s} – r \\[3pt]
\Gamma^\theta_{r\theta} = \Gamma^\phi_{r\phi} = \frac{1}{r} \\[3pt]
\Gamma^\theta_{\phi\phi} = -\sin(\theta)\cos(\theta) \\[3pt]
\Gamma^\phi_{\theta\phi} = \cot(\theta) $
What I am confused about, is that all textbooks appear to be "selective" about which Christoffel symbols to use, when deriving the Geodesic equations via:
$\frac{d^2x^{\lambda}}{d q^2} + \Gamma^{\lambda}_{\mu\nu} \frac{dx^{\mu}}{d q} \frac{dx^{\nu}}{dq} = 0$
While $\Gamma^{\lambda}_{\mu\nu}$ apparently can be substituted with each one of the nine Christoffel Symbols, the sources I encounter always go on to obtain only four Geodesic equations, with $\lambda$ (the upper index of the Christoffel symbol) being equal to: $t,r,\theta,\phi$.
This is understandable as far as being sufficient to find a solution, and yet I am not very clear on whether:
- Is it correct that an additional five geodesic equations can be derived but are just redundant / over-complicated?
- The other five equations for some reason are not interesting, such as being trivial (in the sense $0=0$)?
Best Answer
The expression $$\frac{d^2 x^\lambda}{dq^2} + \Gamma^\lambda_{\mu \nu} \frac{dx^\mu}{dq}\frac{dx^\nu}{dq} = 0$$ yields one equation for each value of $\lambda$. The dummy indices $\mu$ and $\nu$ are repeated, and therefore summed over. Performing the sums explicitly leads to the following expanded form of the geodesic equation:
$$\ddot{x}^\color{red}{\lambda} + \Gamma^\color{red}{\lambda}_{tt} \dot t \dot t+ \Gamma^\color{red}{\lambda}_{tr} \dot t \dot r+ \Gamma^\color{red}{\lambda}_{t\theta} \dot t \dot \theta+ \Gamma^\color{red}{\lambda}_{t\phi} \dot t \dot \phi$$ $$+ \Gamma^\color{red}{\lambda}_{rt} \dot r \dot t+ \Gamma^\color{red}{\lambda}_{rr} \dot r \dot r+ \Gamma^\color{red}{\lambda}_{r\theta} \dot r \dot \theta+ \Gamma^\color{red}{\lambda}_{r\phi} \dot r \dot \phi$$ $$+ \Gamma^\color{red}{\lambda}_{\theta t} \dot \theta \dot t+ \Gamma^\color{red}{\lambda}_{\theta r} \dot \theta \dot r+ \Gamma^\color{red}{\lambda}_{\theta \theta} \dot \theta \dot \theta+ \Gamma^\color{red}{\lambda}_{\theta \phi} \dot \theta \dot \phi$$ $$+ \Gamma^\color{red}{\lambda}_{\phi t} \dot \phi \dot t+ \Gamma^\color{red}{\lambda}_{\phi r} \dot \phi \dot r+ \Gamma^\color{red}{\lambda}_{\phi \theta} \dot \phi \dot \theta+ \Gamma^\color{red}{\lambda}_{\phi \phi} \dot \phi \dot \phi = 0$$
where the dot denotes differentiation with respect to $q$. This is obviously a mess, but the point I mean to emphasize is that there is one geodesic equation for each value of the upper index $\lambda$, and every nonzero $\Gamma$ appears in one of those four.