Frames vs Coordinates: Mapping Points in R4 to Events in Manifold

[Dale]

I am generally pretty sloppy in my terminology on this point and use "reference frame" almost synonymously with "coordinate system". Is this a correct distinction between them:

A coordinate system is a mapping from points in R4 to events in the manifold

A reference frame is an orthonormal basis in the tangent space at some event

 

[bcrowell]

A coordinate system is a mapping from points in R4 to events in the manifold

Spelling this out in a little more detail, I'd say that it's a one-to-one, onto map between an open set in R4 and an open set on the manifold.

A reference frame is an orthonormal basis in the tangent space at some event

The notion of orthogonality in a semi-Riemannian space always gets me tied up in knots, and IIRC you've helped me get untied in the past :-)

Your definition makes me uneasy because it seems to assume a space that's locally Minkowski, but I'd prefer to have a definition that works equally well in the case of Galilean relativity. In Galilean relativity, there is no clear way to define the fact that the basis is normalized (because there's no connection between the time and distance scales), and no clear way to define the idea that a spatial basis vector is orthogonal to a temporal vector. (In Minkowski space, I would say that a spacelike vector and a timelike vector were orthogonal if points along the line of the spacelike vector were simultaneous as defined by Einstein synchronization carried out by an observer whose world-line was along the line of the timelike vector.)

 

[Fredrik]

I am generally pretty sloppy in my terminology on this point and use "reference frame" almost synonymously with "coordinate system". Is this a correct distinction between them:

A coordinate system is a mapping from points in R4 to events in the manifold

A reference frame is an orthonormal basis in the tangent space at some event

The word "frame" can be used both the way you just did and as a synonym for coordinate system. A basis doesn't actually have to be orthonormal to be called a "frame", so when you're defining things, an orthonormal basis at p should be called an "orthonormal frame at p".

We can also define a frame bundle as $\mathcal F=\bigcup_{p\in M} \mathcal F_p$ where $\mathcal F_p$ is the set of frames at p, and define a "frame field" as a function $f:M\rightarrow\mathcal F$ such that $f(p)\in F_p$ for all p. Another way of saying that: A frame field is a section of the frame bundle.

You did of course leave out some details. If you want to understand the full definition of a coordinate system, you should read the stuff about smooth structures in Lee's "Introduction to smooth manifolds".

...I'd prefer to have a definition that works equally well in the case of Galilean relativity. In Galilean relativity...

Hm...I think that's a little too ambitious, considering that the spacetime of Galilean relativity doesn't have a metric. Each slice of constant time can be thought of as a 3-dimensional manifold with the Euclidean metric. Spacetime can be thought of as a 4-dimensional manifold, but not as a 4-dimensional manifold with a metric.

 

[bcrowell]

The word "frame" can be used both the way you just did and as a synonym for coordinate system.

I think they're used as synonyms only by authors from the very early era of relativity, ca. 1920 (or by later authors when they're being sloppy).

A basis doesn't actually have to be orthonormal to be called a "frame"

Probably the word is overloaded. I think there are two different goals we could have in mind if we want to define the idea of a frame: (1) I want to be able to write down components of tensors, and (2) I also want to specify an observer's frame of reference. #2 is what's usually meant by a frame in Galilean relativity and SR, and if we want to make contact with that, we need to specify the observer's state of motion. I don't think that's specified unless you require something more than linear independence. For example, I could make a basis out of two lightlike vectors and two spacelike vectors, and that would take care of #1, but not #2.

Hm...I think that's a little too ambitious, considering that the spacetime of Galilean relativity doesn't have a metric. Each slice of constant time can be thought of as a 3-dimensional manifold with the Euclidean metric. Spacetime can be thought of as a 4-dimensional manifold, but not as a 4-dimensional manifold with a metric.

We do have a notion of frames in Galilean relativity, and we also have such a notion in GR. It seems to me that there must be a way to make a definition that is general enough to encompass both.

We often want to be able to talk about inertial frames and noninertial frames. The definition in DaleSpam's #1 clearly isn't capable of expressing that distinction.

There is a lengthy discussion of this kind of stuff here: http://en.wikipedia.org/wiki/Frame_of_reference That might make a good starting point so that we don't reinvent the wheel.

[EDIT] MTW has a good discussion of this on p. 164, box 6.2. Essentially the concept seems to be something like this. Define a timelike world-line, and define an orthonormal frame field only on the world-line itself, with the 0th basis vector always being the velocity vector. The distinction between rotating and nonrotating frames is defined in terms of Fermi-Walker transport. You can define Gaussian normal coordinates that fill space out to some distance L from the world-line. These normal coordinates have goofy behavior of relative size aL, where a is the proper acceleration.

 

[Fredrik]

I think I'd rather avoid a lengthy discussion. (But I also don't see anything in your post that I disagree with).

 

[Dale]

Thanks to both of you.

Ben, if you ever get a good definition that works for Galilean relativity also the please let me know.

 

[Fredrik]

Hm, now I can't think of a single reason why we shouldn't just define Galilean spacetime to be the set ℝ⁴ with the standard manifold structure and the Euclidean metric tensor. I should probably have thought about this some more earlier, instead of saying that Galilean spacetime doesn't have a metric.

The geodesics of the Euclidean metric are the world lines of non-accelerating particles, and I bet its isometries correspond to Galilean transformations the same way that the isometries of the Minkowski metric correspond to Poincaré gransformations.

Even if we don't define a metric on Galilean spacetime, we can still define frames at an event p in essentially the same way. We would just have to say that the 0th basis vector is always in the direction of the common 0 axis of all the inertial coordinate systems, and that the other three basis vectors are mutually orthogonal in the 3-dimensional hypersurface that all the inertial coordinate systems assign the same time coordinate as p. But we only have to do it this way if we can't just use the Euclidean metric on ℝ⁴, and I now I think we can.

 

[bcrowell]

Ben, if you ever get a good definition that works for Galilean relativity also the please let me know.

The MTW treatment is phrased in a (relatively nonmathematical) way that seems to cover the Galilean case. My shortened paraphrase of it loses that.

 

[Landau]

Recently a thread about this was started: click.

 

[Fredrik]

Now I remember that I forgot make a post in this thread saying that I realize that what I said in #7 was wrong. One of the reasons why we can't define Galilean spacetime to be ℝ⁴ with the Euclidean metric, is that the isometries of that metric include rotations in spacetime, but Galilean transformations preserve simultaneity.

 

[Ben Niehoff]

I would expect a "metric" for Galilean relativity to have a zero eigenvalue, so it is degenerate. This is probably another way of saying that 't' is just a parameter, rather than a coordinate.

 

[DrGreg]

For what it's worth, Roger Penrose in The Road to Reality defines Galiliean spacetime to be the fibre bundle with base space $\mathbb{E}^1$ (time) and fibre $\mathbb{E}^3$ (space). ($\mathbb{E}^N$ denotes N-dimensional Euclidean space.)

He also defines a pre-Galilean-Relativity space that he calls "Aristotelian spacetime" to be the simple product $\mathbb{E}^1 \, \times \, \mathbb{E}^3$.

 

[lavinia]

I am generally pretty sloppy in my terminology on this point and use "reference frame" almost synonymously with "coordinate system". Is this a correct distinction between them:

A coordinate system is a mapping from points in R4 to events in the manifold

A reference frame is an orthonormal basis in the tangent space at some event

Wouldn't the mappings have to be smooth and wouldn't the orthonormal basis have to vary smoothly over the domain?

 

[Fredrik]

Wouldn't the mappings have to be smooth and wouldn't the orthonormal basis have to vary smoothly over the domain?

When we're dealing with smooth manifolds, yes. But we might want to try using something else to represent space and time, and then we might want to require that our coordinate systems and frames satisfy some other technical condition.