I am working on a problem in the textbook Gravity: An Introduction to Einstein's General Relativity by Hartle. The problem is to show that an observer moving radially through the throat of the wormhole metric $$ds^2=-dt^2+dr^2+(b^2+r^2)(d\theta^2+\sin^2\theta\,d\phi^2)$$ will experience tidal gravitational forces as viewed by a stationary observer. This is simple enough to do by calculating the Riemann tensor and using the equation of geodesic deviation.
However, when I look at the solution, I find that this calculation is not performed in the coordinate basis (as I did) but rather in an orthonormal basis. I do not understand why. For reference, the problem is stated here as Prob 21.12 along with a sketch of the solution.
From what I have read, it seems that the orthonormal frame somehow corresponds to a "natural observer" (whatever that means). Is there an a priori way to determine whether I should be doing a calculation in an orthonormal basis? What is wrong with performing this kind of calculation in the coordinate basis?