What is a Frame of Reference in Physics?

[leo.]

I'm studying special relativity for the second time and there's something I think I didn't get since I studied classical mechanics: the idea of a frame of reference. I think the underlying idea is that of a point of view, so that we want to study some phenomenon, but we have to observe it so that a natural question should be: we are observing it from which point of view? That "point of view" should be a frame of reference.

The point, however, is that I don't understand how to transition this to mathematics. Many books seems to correspond frames of references and coordinate systems, but in the same moment they refer as a collection of axes. But if we are explicitly talking about axes, we are assuming that the coordinate system is cartesian.

On the other hand, we usually talk about the motion of a reference frame, so that it would mean the movement of a coordinate system.

All of this would work well in euclidean three-space. In that manifold, there's a natural identification between points and coordinates. Indeed if we move from point to point, we could still draw axes on the other point and get the same system and the notion of movement of such system. But on other manifolds, it doesn't work well, because there would be no natural notion of moving a coordinate system.

So, my questions are: is my intuitive idea of a reference frame correct? What, mathematically, really is a reference frame? And I'm asking those questions in a general setting, so that I'm also thinking about classical mechanics (with no idea of spacetime).

 

[thankz]

cgs units maybe?

 

[ChrisVer]

Why "must" it be Cartesian? You can as well have spherical coordinates or other weird choices of coordinates.

 

[leo.]

That's exactly my question. I have the notion of a coordinate system, which is a well defined mathematical idea: a coordinate system on a smooth manifold (that could be euclidean three-space, or even spacetime) is a homeomorphism that assigns to points it's coordinates. Now, when discussing relativity, books usually talk about reference frames as if it were cartesian coordinates that could travel through space.

Now, in ℝ³ this is easy: given a certain point, we can imagine we are fixing axes there (since vectors can roam around freely), then we can allow this point to move around, defining a certain path on ℝ³. In that case, we can write any other point relative to this new moving origin, and this is how we do things usually.

This idea of moving a coordinate system doesn't seem to have much meaning on a more general setting though. In a more general manifold, a coordinate system is for a particular subset, so it cannot start going around freely. In that case, a reference frame is simply a moving set of cartesian axes that we use exclusively on euclidean spaces?

 

[PeterDonis]

I have the notion of a coordinate system, which is a well defined mathematical idea: a coordinate system on a smooth manifold (that could be euclidean three-space, or even spacetime) is a homeomorphism that assigns to points it's coordinates.

This is the correct definition, and yes, it works for spacetime just as it does for Euclidean 3-space.

when discussing relativity, books usually talk about reference frames as if it were cartesian coordinates that could travel through space.

Yes, they do, and this is very unfortunate, because, as you suspect, it's not correct. Coordinates do not "travel"; they just label points in the manifold. And one set of coordinates does not "move" relative to another; they are just two different labelings of points in the manifold, which may have some particular relationship between them.

The correct way to deal with this is to stop saying "reference frame" when one really means "coordinate chart". Unfortunately, I doubt this will happen any time soon; the sloppy terminology just seems to be too easy to give up, because, as you note, in flat manifolds (like Euclidean space and the flat spacetime of SR), you can get away with it and it (appears to) make things easier conceptually. (Books on GR tend to be more careful about this than books on SR, because in GR, when spacetime can be curved, you can no longer rely on the special properties of flat spacetime, just as, when dealing with curved manifolds in ordinary geometry, you can no longer rely on the special properties of Euclidean spaces.)

Even when books and papers talk about a "reference frame" as something distinct from a coordinate chart, it's not always clear what they mean. For what it's worth, here is the best definition I can give:

A "reference frame" (the more correct technical term would be "frame field"--google it if you want to find out more about what I'm describing) is an assignment of a set of 4 orthonormal vectors, one timelike and three spacelike, to each point of spacetime. The assignment must have certain properties (for example, continuity). The timelike vector at a given point can be thought of as the 4-velocity of some "reference" observer passing through that point. The spacelike vectors can be thought of as three mutually perpendicular axes that the observer carries with him to define three reference directions in space.