Consider this quote by James J. Callahan in his book The Geometry of Spacetime where he summarizes the conclusions of his chapter on arbitrary frames in SR (page 165):
Once again we find that the radar grid and the rulers-and clocks grid disagree. We have further evidence that in the noninertial frame of an accelerated observer G, no coordinates simultaneously give measurements of a single ruler and clock-as they naturally do in an inertial frame. A map of the earth suffers the same defect: Measurements on the map cannot all be made proportional to measurements on the surface of the earth. No accurate map of (a substantial portion of) the earth can be made with just a single scale. On the earth we ascribe this defect to curvature-more precisely, to the fact that the earth is curved but the map is not. By analogy, we consider that the same may be true for spacetime: Since measurements within the accelerated frames that we have considered are not proportional to measurements of the corresponding spacetime intervals, perhaps spacetime itself is curved. Our speculations can be summarized this way: accelerated motions ==> noninertial frames ==> curved spacetime
The thing that i find very confusing in that quote -- and in that whole chapter actually -- is that Callahan is apparently saying that by virtue of being in a non-inertial frame (while still being in a Minkowski space) spacetime is automatically curved. I've already taken a course in GR and i know that can't be right because Minkowski space has a flat metric. On the other hand Callahan's argument seems reasonable. He's basically saying that, for instance, in a rotating frame you can't measure time and space uniformly like you do in an inertial frame (since v is a function of r and hence time dilation will be a function of r i.e. you can't synchronize the clocks in your frame no matter how hard you tried; a similar effect to how you can't uniformly measure distances on a sphere). A similar question arises in the case of a rotating disc, where the rotating observer apparently experiences non-euclidean geometry. But how can that be? We're still in Minkowski space, the riemann curvature tensor must vanish so why is non-Euclidean geometry present?
My question can be summarized as follows: Callahan, from what I can tell, is arguing that non-inertial frames imply spacetime curvature, even in Minkowski space which completely contradicts what I've learned before. More specifically I need clarification on the part of Callahan's quote which i highlighted with bold.