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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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  • $\begingroup$ Is my question controversial? It's been tagged as "Homework", but I'm in my sixth decade and haven't formally studied physics for forty years. I'm just curious as to whether I'm on the right track. Thanks. $\endgroup$ – Peter4075 Aug 27 '14 at 19:00
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    $\begingroup$ @Peter4075 Dear Peter, no don't worry, your question is perfectly valid and I'm sure someone will soon help you (specially since you've shown effort and written out your attempt at the question). The discussion between Dilaton and KyleKanos is just regarding the tags...It is just between them. $\endgroup$ – Phonon Aug 27 '14 at 19:03
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    $\begingroup$ I think the question is badly phrased - there's not even an explicit question here. It's a bunch of calculations and then a rather precise confusion about different statements of the geodesic equations. The latter part is not off-topic, but the former (if the implied question is Am I right?) is off-topic as per the homework policy, no matter its technical level. I would ask @Peter4075 to remove or rephrase everything up to "My reason for asking" (and then notify me so I can retract the close vote I am going to cast now). $\endgroup$ – ACuriousMind Aug 27 '14 at 20:19
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    $\begingroup$ This looks fine to me now, but it seems to me that you can cut down on a lot of the calculations and simply state the final paragraph. $\endgroup$ – Emilio Pisanty Aug 28 '14 at 12:41
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    $\begingroup$ Maybe I should have simply asked which of the two textbook equations of geodesic deviation is correct. Instead I tried to show effort by attempting an answer myself in the form of calculating the geodesic deviation on the surface of a unit 2-sphere. I wasn't confident about these calculation, hence my request for someone to take a look at them. I thus seem to have fallen into the “check my work” trap. As a self-studier, I can't say I feel encouraged by the negative responses to my question. Anyway, I'll try again with a rephrased question. $\endgroup$ – Peter4075 Aug 28 '14 at 13:20

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