# Arbitrary frames in special relativity

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.

• I find very confusing that paragraph in the book. It is already false for spatial geometry. In the Euclidean 3D space you can use non-Cartesian three-dimensional coordinate systems where the relations between coordinates and rulers are exactly those summarized in the paragraph but it does not imply that the space is curved because it is flat by hypothesis! – Valter Moretti Jan 12 at 17:56
• The point is that if the space is curve there are no coordinate systems where rulers and coordinates "agree". The converse statement is instead false. – Valter Moretti Jan 12 at 17:59
• So what the book should say is that if there are no coordinate systems where globally rulers and clocks agree with the coordinates, then the geometry is curved. This is however a very delicate statement because locally, using geodesical coordinates, one can find coordinates where in a precise mathematical sense locally coirdinates an rulers agree. – Valter Moretti Jan 12 at 18:02
• Sure, I understand all that. Basically this entails to finding a coordinate transformation which makes the metric flat (globally) hence making the space flat as a result. However that's not what Callahan was saying at all, instead he argued the opposite: since we are in a frame where the metric is not flat then the space is not flat (even though we know from GR that this is not a viable reason to say that spacetime is curved and there should exist a coordinate transformation which makes the metric flat globally in minkowski space). – Leonid Jan 12 at 18:23
• Also just to make things clear: This was not just one paragraph, the author devoted an entire chapter to this very idea which makes it extra confusing. – Leonid Jan 12 at 18:23

The thing that i find very confusing in that quote … 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.

That is not what he is saying. After considering “disagreement” between “radar grid” and “ruler–clock grid” and noticing that similar disagreement exists between Earth surface and its maps (and noticing that this disagreement is unavoidable due to Earth's curvature) he makes a hypothesis that such a disagreement would also be inherent for the spacetime (which at this moment is no longer assumed to be a Minkowski spacetime).

Notice his language:

By analogy … may be true … perhaps … speculations …

It is obvious, that there is nothing automatic about curvature of spacetime.

Callahan, from what I can tell, is arguing that non-inertial frames imply spacetime curvature, even in Minkowski space.

Again, that is wrong summary and Callahan says nothing of the sort. He argues that consideration of non-inertial frames together with examples of curved surfaces suggests (not implies, here we make a hypothesis) that the spacetime may not be Minkowski space but instead a more general curved spacetime (and the name Minkowski is not even present in the quote).

• I've re-read the quote many times and I simply can't see how you came to that conclusion. He explicitly states the reason for curvature: " 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". Which is in direct analogy to what happens on the sphere according to him. – Leonid Jan 12 at 21:52
• Also I don't see how the space is anything but Minkowski. The entire chapter is just SR in arbitrary frames, the "curved surfaces" you speak of, are just analogies that he used to link curvature of spacetime with spatial curvature on a sphere/Earth. While the inherent reason for curvature itself is apparently the non-uniform measurements he talks of. The bolded highlighted part also doesn't support your conclusions but rather point to his intent that non-inertial frames directly lead to spacetime curvature (he then links gravity to curvature via the equivalence principle later on). – Leonid Jan 12 at 21:59
• @Leonid: Once more: what the author calls speculations cannot be termed as directly lead, there are additional assumptions being made here (that were absent before). That is how theory building works: you make new assumptions and go with it. And in order to go with it you may reformulate already known, old theory in a language best suited for new assumptions. Once assumptions are made, the objects of the old theory often become different, gain new properties. So here, once the author made new assumption that the spacetime is curved, it ceased to be Minkowski spacetime. – A.V.S. Jan 13 at 15:16