mean square displacement on the 3-sphere

Question

I would like to compute the mean square displacement (MSD) for a particle moving on the surface of a 3-sphere of radius R.

I see that I could eventually use the polar coordinates and get a polar displacement for each of them, but I would like to have the displacement in units of (arc)length. Is there an easy way to obtain the MSD, provided that I know the Cartesian coordinates of the point at all time t?

Answer 1

There is a bijective correspondence between Cartesian distances and arc lengths on the $n$-sphere. It's independent of $n$, since it only involves the great circle the two points are on. The length of a chord that subtends an angle $\alpha$ in a circle of radius $r$ is $2r\sin\alpha/2$, so you can calculate the arc length $l=\alpha r$ from the Cartesian distance $x$ as

$$l=2r\arcsin\frac x{2r}\;.$$