I have the following distinction clear in my mind:
Reference frame → state of motion of the observer
Coordinate system → set of numbers used to map the space points within a reference frame
So for any given reference frame, multiple coordinate systems are possible (e.g. Cartesian, spherical, etc)
This distinction is in my opinion fundamental. For example: work of a force (a scalar) is invariant with respect to coordinate transformations within the same reference frame. But if we use a difference reference frame (in relative motion with respect to the first one) the same work will be different → this scalar is not invariant anymore!
My problem is, I have not found so far a physics textbook which clearly states this difference between these two entities (reference frame and coordinate system), and develop its results taking this difference into account. The two concepts are often used interchangeably → I find this confusing and frustrating, since I can't appreciate what exactly the author means.
This is especially true in relativity theory, whose tensorial analysis require a deep understand of these concepts.
So my question is: can anybody suggest some relativity books (or at least some general physics book) in which this distinction is made clear from the beginning, and in which the results are carried on under this assumption?
Best Answer
A frame (at an event $E$) is an ordered basis for the tangent space to spacetime at $E$. A coordinate system is a diffeomorphism from an open subset of spacetime to an open subset of ${\mathbb R}^{3+1}$.
(More commonly, such a diffeomorphism is called a "chart" and its inverse is called a coordinate system, but I'll use the slightly less common language.)
A frame at $E$ induces a coordinate system on the tangent space at $E$ (call it $T_E$) in the obvious way --- given a frame $(v_1,v_2,v_3,v_4)$, map the point $\Sigma a_iv_i$ to $(a_1,a_2,a_3,a_4)$.
Let $U_E$ be the image of the exponental map from $T_E$. Then composing with the inverse of the exponential map gives a coordinate system on $U_E$.
So every frame yields a coordinate system.
Conversely, given a coordinate system $(\phi_1,\ldots,\phi_4)$ on any open set containing $E$, we get a frame $(\partial/\partial\phi_1,\ldots,\partial/\partial\phi_4)$ at $E$. So every coordinate system yields a frame.
The composition $$\hbox{Frames}\rightarrow\hbox{Coordinate Systems}\rightarrow\hbox{Frames}$$ is clearly the identity. The composition in the other direction is clearly not the identity (think of a polar coordinate system, for example).
The coordinate systems that come from frames are called normal, so there is a one-one correspondence between frames and normal coordinate systems. Sometimes in informal language, a frame and the corresponding coordinate system are identified.
(There's also a version of this where the frames are required to be orthonormal, which is sometimes tacitly assumed.)