No response to this on Physics Stack Exchange, so I'm hoping for better luck here.
My question is, can anyone tell me where I'm going wrong trying to use the equation of geodesic deviation$$\frac{D^{2}\xi^{\mu}}{D\lambda^{2}}+R_{\phantom{\mu}\beta\alpha\gamma}^{\mu}\xi^{\alpha}\frac{dx^{\beta}}{d\lambda}\frac{dx^{\gamma}}{d\lambda}=0$$ to show that on the surface of a unit radius sphere two particles separated by initial distance $d$, starting from the equator and travelling north (ie on lines of constant $\phi$) at equal speed will have a separation $s$ given by$$s=d\sin\theta?$$ This is similar to https://physics.stackexchange.com/questions/107421/geodesic-devation-on-a-two-sphere except that question was solved using simple spherical geometry.
My plan was to first find $\frac{D^{2}\xi^{\mu}}{D\lambda^{2}}$ by calculating the Riemann tensor part$$R_{\phantom{\mu}\beta\alpha\gamma}^{\mu}\xi^{\alpha}\frac{dx^{\beta}}{d\lambda}\frac{dx^{\gamma}}{d\lambda}.$$ And then find $\frac{D^{2}\xi^{\mu}}{D\lambda^{2}}$ by using the absolute derivative $$\frac{DV^{\alpha}}{d\lambda}=\frac{dV^{\alpha}}{d\lambda}+V^{\gamma}\Gamma_{\gamma\beta}^{\alpha}\frac{dx^{\beta}}{d\lambda},$$ and take the second derivative of this. Next I hoped to try to juggle the results to show the separation $s=\xi^{\phi}$ as a function of $\theta$.
The line element for spherical coordinates$$l^{2}=dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta d\phi^{2}$$ for a great circle of constant $\phi$ on a sphere of unit radius reduces to $$dl^{2}=d\theta^{2}$$ giving $\frac{d\theta}{dl}=\frac{d\theta}{d\lambda}=1$ and $\frac{d\phi}{dl}=\frac{d\phi}{d\lambda}=0$.
Expanding out the Reimann tensor components gives:$$\frac{D^{2}\xi^{\theta}}{D\lambda^{2}}=0$$ and$$\frac{D^{2}\xi^{\phi}}{D\lambda^{2}}=-\xi^{\phi}.$$
The absolute derivative for $\Gamma_{\theta\phi}^{\phi}=\Gamma_{\phi\theta}^{\phi}=\frac{\cos\theta}{\mathbf{\mathbf{\sin\theta}}}$ is$$\frac{D\xi^{\phi}}{d\lambda}=\frac{d\xi^{\phi}}{d\lambda}+\xi^{\theta}\Gamma_{\theta\phi}^{\phi}\frac{d\phi}{d\lambda}+\xi^{\phi}\Gamma_{\phi\phi}^{\phi}\frac{d\phi}{d\lambda}+\xi^{\theta}\Gamma_{\theta\theta}^{\phi}\frac{d\theta}{d\lambda}+\xi^{\phi}\Gamma_{\phi\theta}^{\phi}\frac{d\theta}{d\lambda}=\frac{d\xi^{\phi}}{d\lambda}+\xi^{\phi}\frac{\cos\theta}{\mathbf{\mathbf{\sin\theta}}}.$$
And for $\Gamma_{\phi\phi}^{\theta}=\sin\theta\cos\theta$ $$\frac{D\xi^{\theta}}{d\lambda}=\frac{d\xi^{\theta}}{d\lambda}+\xi^{\phi}\Gamma_{\phi\phi}^{\theta}\frac{d\phi}{d\lambda}+\xi^{\phi}\Gamma_{\phi\theta}^{\theta}\frac{d\theta}{d\lambda}+\xi^{\theta}\Gamma_{\theta\phi}^{\theta}\frac{d\phi}{d\lambda}+\xi^{\theta}\Gamma_{\theta\theta}^{\theta}\frac{d\theta}{d\lambda}=\frac{d\xi^{\theta}}{d\lambda}.$$
However, $$\frac{D\xi^{\phi}}{d\lambda}=\frac{d\xi^{\phi}}{d\lambda}+\xi^{\phi}\frac{\cos\theta}{\mathbf{\mathbf{\sin\theta}}}$$ doesn't look right as it blows up when $\theta=0$. Any suggestions where I might be going wrong?
