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

In summary, a coordinate system is a mapping from points in R4 to events in the manifold, while a reference frame is an orthonormal basis in the tangent space at some event. However, the term "frame" can be used interchangeably with coordinate system in some contexts. A frame field can be defined as a section of the frame bundle, and a frame bundle can be defined as a collection of frames at each point on the manifold. There are also different goals for defining a frame, such as writing down components of tensors or specifying an observer's frame of reference. This distinction is important in discussing inertial and non-inertial frames. A timelike world-line can also be used as a basis for defining a frame in
  • #1
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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
 
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  • #2
DaleSpam said:
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.

DaleSpam said:
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.)
 
  • #3
DaleSpam said:
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 [itex]\mathcal F=\bigcup_{p\in M} \mathcal F_p[/itex] where [itex]\mathcal F_p[/itex] is the set of frames at p, and define a "frame field" as a function [itex]f:M\rightarrow\mathcal F[/itex] such that [itex]f(p)\in F_p[/itex] 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".

bcrowell said:
...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.
 
  • #4
Fredrik said:
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).

Fredrik said:
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.

Fredrik said:
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.
 
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  • #5
I think I'd rather avoid a lengthy discussion. :smile: (But I also don't see anything in your post that I disagree with).
 
  • #6
Thanks to both of you.

Ben, if you ever get a good definition that works for Galilean relativity also the please let me know.
 
  • #7
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.
 
  • #8
DaleSpam said:
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.
 
  • #9
Recently a thread about this was started: click.
 
  • #10
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.
 
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  • #11
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.
 
  • #12
For what it's worth, Roger Penrose in The Road to Reality defines Galiliean spacetime to be the fibre bundle with base space [itex]\mathbb{E}^1[/itex] (time) and fibre [itex]\mathbb{E}^3[/itex] (space). ([itex]\mathbb{E}^N[/itex] denotes N-dimensional Euclidean space.)

He also defines a pre-Galilean-Relativity space that he calls "Aristotelian spacetime" to be the simple product [itex]\mathbb{E}^1 \, \times \, \mathbb{E}^3[/itex].
 
  • #13
DaleSpam said:
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?
 
  • #14
lavinia said:
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.
 

1. What is the difference between frames and coordinates in the context of mapping points in R4 to events in a manifold?

Frames and coordinates are two different ways of representing points in R4 (four-dimensional space) and mapping them to events in a manifold (a mathematical space that can have any number of dimensions). A frame is a set of basis vectors that define a coordinate system, while coordinates are the values assigned to each of these basis vectors. In simpler terms, frames are the building blocks of coordinates.

2. How do frames and coordinates help us map points in R4 to events in a manifold?

Frames and coordinates provide a mathematical framework for mapping points between two spaces. In R4, a point is represented by four coordinates (x, y, z, t), and in a manifold, an event is represented by its own set of coordinates. By using frames and coordinates, we can create a transformation that maps a point in R4 to an event in the manifold, allowing us to study the relationship between these two spaces.

3. Can frames and coordinates be used interchangeably?

No, frames and coordinates cannot be used interchangeably. Frames are the basis for coordinates and are unique to each coordinate system, while coordinates are the values assigned to each basis vector in a frame. Changing the frame would result in different coordinate values, even though the point in R4 remains the same. Therefore, frames and coordinates are not interchangeable.

4. What are some real-world applications of frames and coordinates in mapping points in R4 to events in a manifold?

Frames and coordinates have various applications in science and engineering, such as in the fields of physics, astronomy, and robotics. For example, in physics, frames and coordinates are used to study the motion of objects in four-dimensional space-time. In astronomy, they are used to map the positions of celestial objects in the sky. In robotics, frames and coordinates are used to program the movements of robots in three-dimensional space.

5. Is it possible to map points in R4 to events in a manifold without using frames and coordinates?

No, it is not possible to map points in R4 to events in a manifold without using frames and coordinates. Frames and coordinates provide a mathematical framework for mapping points between two spaces and are essential for accurately representing and studying the relationship between these spaces. Without frames and coordinates, it would be challenging to make meaningful connections between points in R4 and events in a manifold.

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