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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?

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  • $\begingroup$ Minor comment to the post (v2): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. $\endgroup$ – Qmechanic Mar 4 at 18:09
  • $\begingroup$ I've added the homework-and-exercises tag. In the future, please use this tag on this type of question. $\endgroup$ – Ben Crowell Mar 4 at 20:02
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In general, as long as the basis is complete, any basis will work since you can rotate one basis into the another.

The exception to the rule is when the author asks you to construct an orthogonal basis as in part (a) of the problem 21.12.

Can you rotate your basis into the orthogonal basis? Do the answers match?

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