References: Anthony Zee, Einstein Gravity in a Nutshell, (Princeton University Press, 2013) – Chapter I.3, Problem 5.
In this problem, Zee gives us a 3-d vector and asks us to find the quantity averaged over the direction of . I wasn’t entirely clear what this question was asking, since once you’ve defined , surely its direction is fixed so how can you average over a fixed direction? I think what he means is: suppose you rotate through all possible angles in 3-d while keeping its magnitude fixed. What then is the average of over all these rotations?
There is also a mistake in the formula he gives for the average. If and are the usual spherical angles, then the average of is given by
(not in the integral).
We can work out the integral by writing in rectangular coordinates in the usual way:
From symmetry, if . If this isn’t obvious, suppose we’re looking at , and we pick one direction specified by angles and . Then the initial coordinates are
If we then rotate by keeping constant and rotating by so that then
Thus this position for cancels the original position in the average, so the overall average . A similar argument applies to and .
We could also work out these averages by direct integration. For example
For the other terms, we have