## Tuesday, 24 September 2019

### Question

Consider a 2-sphere with coordinates $\left(\theta,\phi\right)$ and a metric
\begin{align}
{ds}^2={d\theta}^2+\sin^2{\theta}{d\phi}^2&\phantom {10000}(1)\nonumber
\end{align}a) Show that lines of constant latitude ($\phi=\rm{constant}$) are geodesics, and that the only line of constant latitude ($\theta=\rm{constant}$) that is a geodesic is the equator  ($\theta=\pi/2$).

b) Take a vector with components $V^\mu=\left(1,0\right)$ and parallel transport it once round a circle of constant latitude. What are the components of the resulting vector, as a function of $\theta$?

Part (a) was quite easy because I have laboured over it before. If I had known that the general solution to $$\frac{d^2V}{d\phi^2}+kV=0$$was$$V=A\cos{\left(k\phi\right)}+B\sin{\left(k\phi\right)}$$then I could have done the second part on my own. I needed some help from Prof Anthony Aguirre at UC Santa Cruz for that differential equation.