Tuesday, 24 September 2019

Exercise 3.05 2-sphere: geodesics and parallel transport

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##?

Answers

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.

It seems that 'parallel transporting' is pretty wonky. Here's how it goes at 70° colatitude (polar angle):
Once the vector has got all round the sphere (at 70°) it's pointing in a new direction, even thought it's length never changes. (It appears to here from perspective effects.) Around equation 3.42 Carroll writes "It follows that the inner product of two parallel-transported vectors is preserved ... This means that parallel transport ... preserves the norm of vectors, the sense of orthogonality and so on.💡💡

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