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The idea of a reference frame as discussed in this question is that of a viewpoint. So that we have some phenomenon, we want to describe be able to predict things and we must specify the viewpoint from which the observations are made.

In basic Physics, however, and in texts about Special Relativity we usually see reference frames being depicted as certain sets of cartesian axes (in the second case, with clocks included). This usually is not a "mathematical construction", but rather just the way we interpret the equations.

What I mean is the following: we have position vectors $\mathbf{r}$ labeling points and force fields $\mathbf{F}$ as functions of those points. The description is not complete without specifying wich is the origin and how the axes are constructed to build those position vectors. In that case, to specify a reference frame relative to another, we simply specify how the position of the origin of the new frame varies with time, that is, we give a curve $\mathbf{R}(t)$. In that case, when we want to write things in the new frame, we just have to remeber they are with respect to the new origin.

In that case, we imagine axes fixed at the point $\mathbf{R}(t)$ at time $t$ and so we have a moving cartesian coordinate system.

Now, if we go deeper into what a coordinate system is, we find out that it the answer in the theory of Differential Geometry: it is one homeomorphism from some region $U$ of the space under study $M$ onto a space of coordinates $\mathbb{R}^n$. That said, a coordinate system is adapted to that region. In general, coordinate systems are not like the cartesian, that we can move around freely. So from a mathematical standpoint, a coordinate system cannot move.

On the other hand, we have the idea of a frame field in Differential Geometry. Let $\pi : E\to B$ be a smooth vector bundle, then if $p\in B$, $E_p$ is a vector space. An ordered basis $(e_{1|p},\dots, e_{n|p})$ for $E_p$ is said a frame. Such frames can be considered isomorphisms $\varphi :\mathbb{R}^n\to F_p$ with $\varphi(e_i) = e_{i|p}$. Let $F_p$ be the set of all frames at the point $p$. Then the disjoint union:

$$F(E) = \coprod_{p\in B} F_p = \bigcup_{p\in B}\{p\}\times F_p$$

Can be given the structure of a principal bundle with structure group $GL_n(\mathbb{R})$. Given $g\in GL_n(\mathbb{R})$ we have the action on $F_p$ given by $\varphi \cdot g = \varphi \circ g$, composing with the linear transformation $g$.

One section of such bundle is then a mapping $S : B\to F(E)$ with the property that $S\circ \pi = \operatorname{id}_{B}$ and $S(p) = (e_{1|p},\dots,e_{n|p})$. Such maps can be identified with collections of $n$ vector fields witch are linearly independent at each point, that is $(E_1,\dots, E_n)$ with the property that $\{E_1(p),\dots, E_n(p)\}$ is linearly independent.

Given the definitions, it seems that the idea of reference frame is more connected to the idea of a frame field than to the idea of a coordinate system. Indeed, if $\gamma$ is a path, we can make sense of a frame traveling around that path, is just composition $S(\gamma(t))$.

So my questions are:

  1. The real rigorous mathematical formulation for reference frames is really that of frame fields instead of coordinate systems?

  2. Why fixing axes through space like that is a good way to make the idea of "viewpoint to observe a phenomenon" mathematically precise?

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  • $\begingroup$ In physics, we always ignore that we only have local patches on which the coordinates are defined, and we almost always choose "unaccelerated" reference frames which correspond to constant vector fields. So yes, the frame bundle is the correct rigorous generalization of what we usually choose as frames. "Fixing axes through space" is the same as choosing a viewpoint because you are fixing what "direction" appears as the time direction, essentially. $\endgroup$ – ACuriousMind Mar 13 '15 at 1:36
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  1. Yes I'd agree that reference frames are sections in the frame bundle.
  2. For physical intuition, consider the notion of frames in classical GR.

    • An observer, at any given point in spacetime, would measure things in an orthogonal basis. That is, to the observer, the time he observes should be orthogonal to the spacial distances he observes, and spacial directions should be orthogonal to eachother. This is saying the observer would measure components of vector/tensor quantities in a orthonormal basis in the tangent space at his particular point in spacetime. In the language you suggested - his 'viewpoint' would be a frame field.

    • As an example, observe for 4-manifolds with Lorentizian signature, orthonormal bases (with respect to the metric) are related by rotations in flat Minkovski space, i.e. Lorentz transformations. This is very intuitive - it says at each point in spacetime different frames are related by lorentz transformations.

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