Apologies if I have this completely wrong (and for the general long-windedness). I've searched online but can't find anything helpful/relevant.
I'm trying to use the geodesic equation
$$\frac{D^{2}\xi^{\mu}}{D\lambda^{2}}+R_{\phantom{\mu}\beta\alpha\gamma}^{\mu}\xi^{\alpha}\frac{dx^{\beta}}{d\lambda}\frac{dx^{\gamma}}{d\lambda}=0$$ to find the geodesic deviation on the surface of a unit 2-sphere. My question is are the following calculations correct?
I start with the Riemann tensor components: $R_{\phantom{\theta}\phi\theta\phi}^{\theta}=\sin^{2}\theta$, $R_{\phantom{\theta}\phi\phi\theta}^{\theta}=-\sin^{2}\theta$, $R_{\phantom{\theta}\theta\theta\phi}^{\phi}=-1$, $R_{\phantom{\theta}\theta\phi\theta}^{\phi}=1$.
Let $u^{\sigma}\equiv\frac{dx^{\sigma}}{d\lambda}$. Then expand out the Riemann components to get: $$\left(R_{\phantom{\mu}\theta\theta\theta}^{\mu}u^{\theta}u^{\theta}+R_{\phantom{\mu}\theta\theta\phi}^{\mu}u^{\theta}u^{\phi}+R_{\phantom{\mu}\phi\theta\theta}^{\mu}u^{\phi}u^{\theta}+R_{\phantom{\mu}\phi\theta\phi}^{\mu}u^{\phi}u^{\phi}\right)\xi^{\theta}+\left(R_{\phantom{\mu}\theta\phi\theta}^{\mu}u^{\theta}u^{\theta}+R_{\phantom{\mu}\theta\phi\phi}^{\mu}u^{\theta}u^{\phi}+R_{\phantom{\mu}\phi\phi\theta}^{\mu}u^{\phi}u^{\theta}+R_{\phantom{\mu}\phi\phi\phi}^{\mu}u^{\phi}u^{\phi}\right)\xi^{\phi}.$$
Set $\mu=\theta$ to give $$\frac{D^{2}\xi^{\theta}}{D\lambda^{2}}+\left(R_{\phantom{\mu}\theta\theta\theta}^{\theta}u^{\theta}u^{\theta}+R_{\phantom{\mu}\theta\theta\phi}^{\theta}u^{\theta}u^{\phi}+R_{\phantom{\mu}\phi\theta\theta}^{\theta}u^{\phi}u^{\theta}+R_{\phantom{\mu}\phi\theta\phi}^{\theta}u^{\phi}u^{\phi}\right)\xi^{\theta}+\left(R_{\phantom{\mu}\theta\phi\theta}^{\theta}u^{\theta}u^{\theta}+R_{\phantom{\mu}\theta\phi\phi}^{\theta}u^{\theta}u^{\phi}+R_{\phantom{\mu}\phi\phi\theta}^{\theta}u^{\phi}u^{\theta}+R_{\phantom{\mu}\phi\phi\phi}^{\theta}u^{\phi}u^{\phi}\right)\xi^{\phi}=0$$
$$\frac{D^{2}\xi^{\theta}}{D\lambda^{2}}+\left(\sin^{2}\theta\right)\left(u^{\phi}u^{\phi}\right)\xi^{\theta}-\left(\sin^{2}\theta\right)\left(u^{\phi}u^{\theta}\right)\xi^{\phi}=0$$ $$\frac{D^{2}\xi^{\theta}}{D\lambda^{2}}=\left(\sin^{2}\theta\right)\left(u^{\phi}u^{\theta}\right)\xi^{\phi}-\left(\sin^{2}\theta\right)\left(u^{\phi}u^{\phi}\right)\xi^{\theta}$$
Set $\mu=\phi$ to give $$\frac{D^{2}\xi^{\phi}}{D\lambda^{2}}+\left(R_{\phantom{\mu}\theta\theta\theta}^{\phi}u^{\theta}u^{\theta}+R_{\phantom{\mu}\theta\theta\phi}^{\phi}u^{\theta}u^{\phi}+R_{\phantom{\mu}\phi\theta\theta}^{\phi}u^{\phi}u^{\theta}+R_{\phantom{\mu}\phi\theta\phi}^{\phi}u^{\phi}u^{\phi}\right)\xi^{\theta}+\left(R_{\phantom{\mu}\theta\phi\theta}^{\phi}u^{\theta}u^{\theta}+R_{\phantom{\mu}\theta\phi\phi}^{\phi}u^{\theta}u^{\phi}+R_{\phantom{\mu}\phi\phi\theta}^{\phi}u^{\phi}u^{\theta}+R_{\phantom{\mu}\phi\phi\phi}^{\phi}u^{\phi}u^{\phi}\right)\xi^{\phi}=0$$
$$\frac{D^{2}\xi^{\phi}}{D\lambda^{2}}+\left(-1\right)\left(u^{\theta}u^{\phi}\right)\xi^{\theta}+\left(1\right)\left(u^{\theta}u^{\theta}\right)\xi^{\phi}=0$$
$$\frac{D^{2}\xi^{\phi}}{D\lambda^{2}}-\xi^{\theta}\left(u^{\theta}u^{\phi}\right)+\xi^{\phi}\left(u^{\theta}u^{\theta}\right)=0$$ $$\frac{D^{2}\xi^{\phi}}{D\lambda^{2}}=\xi^{\theta}\left(u^{\theta}u^{\phi}\right)-\xi^{\phi}\left(u^{\theta}u^{\theta}\right).$$
My reason for asking is that Misner et al give the geodesic deviation equation (the one I use in my question) as $$\frac{D^{2}\xi^{\mu}}{D\lambda^{2}}+R_{\phantom{\mu}\beta\alpha\gamma}^{\mu}\xi^{\alpha}\frac{dx^{\beta}}{d\lambda}\frac{dx^{\gamma}}{d\lambda}=0,$$ whilst in another of my textbooks it's (spot the difference!) $$\frac{D^{2}\xi^{\mu}}{D\lambda^{2}}+R_{\phantom{\mu}\alpha\beta\gamma}^{\mu}\xi^{\alpha}\frac{dx^{\beta}}{d\lambda}\frac{dx^{\gamma}}{d\lambda}=0,$$ with $\alpha$ being the first lower index on the Riemann tensor and the upper index on the right-hand-side $\xi$. When I use the second equation in my 2-sphere calculations everything seems to come out to zero, which doesn't seem right. So I'm wondering if the second equation is wrong.
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