Confusion in Wald's GR book: observers correspond to orthonormal basis Book: General Relativity by Robert Wald (pgs: 342-343)
He starts out by suggesting that observers in general relativity correspond to orthonormal basis fields on the manifold, which I am OK with. However, on pg 343 he seems to suggest that a non-orthonormal basis doesn't correspond to any "physically realizable family of observers." Why can't a non-orthonormal basis correspond to an observer?
 A: In SR and GR, orthonormality of time and space vectors is physically equivalent to the fact that your clocks measuring your proper time  and your rulers at rest with you produce the value $c$ (=1) for the speed of light.
At first glance all that means that, if the rest space  $\Sigma$, spanned by spacelike vectors is not normal to your time vector $\partial_t$, it means that you are actually using the rest space of another observer instead of yours. This new observer is the one whose temporal axis $\partial_{t'}$ is normal to $\Sigma$.
Actually, in my view there are some way out since in some sound cases it is physically impossible to construct space and time orthogonal for some extended observers. Think of a rotating platform. Without a more general definition it turns out very difficult to study some well known physical effects as the Sagnac one.
I introduced these topics in a course of mine regarding applications of differential geometry to SR and  GR. You may find some discussion in second part of my lecture notes, especially Sect. 8.5. Please notice that I have had no time to completely correct them so that they sill contain a number of typos (and errors of various nature).
