Reference frames versus coordinate systems

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?

• Related: physics.stackexchange.com/q/458854 . Even without worrying about relativity, a coordinate system is not the same as a frame of reference. For example, you could use Cartesian coordinates or spherical coordinates.
– user4552
Aug 21, 2019 at 13:27
• Are you Italian? If yes, you can download my lecture notes on analytical mechanics where that distinction is crucial. Aug 21, 2019 at 18:19

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.)

• Just to be clear, you are describing normal coordinates, correct? Aug 21, 2019 at 15:42
• @WillO Sorry about digging an old post. In your definition, how would you define what's an inertial frame and what's a non-inertial frame? Also is it still true that you can get from a coordinate system to a frame when the coordinate system is curvilinear? Nov 1, 2021 at 23:06
• @RuiLiu: First, just to clarify, this is not "my definition"; it's the standard definition. An inertial frame is an orthogonal frame. But sometimes people say "frame" when what they really mean is "section of the frame bundle" (that is, a collection of frames, one for each point along some worldline, or for each point in some region of spacetime). In this case, "inertial" seems to usually mean "generated from an inertial frame at one point via parallel transport". At other times, people seem to use it to mean "all derived from a single normal coordinate system". Nov 2, 2021 at 0:25
• @WillO What about non-inertial frame? Is collection of frames (frame field) introduced in order to handle it? Nov 2, 2021 at 11:30

The problem is that the term "reference frame" is a little bit ambiguous and is frequently used inconsistently. I don't think that there is a hard and fast rule that you can apply always when someone refers to "reference frame".

The unambiguous term "coordinate chart" or "coordinate system" refers to a smooth and one-to-one mapping between events in spacetime and points in R4. That is, it associates a time and place with a set of four numbers and vice versa.

There is another unambiguous term called a "tetrad", "frame field", or "vierbein". This refers to a set of four vector fields covering a region of spacetime where the vectors are orthonormal and one is timelike and the rest are spacelike. In other words, it associates a time and place with a set of four vectors.

Some people seem to use "reference frame" to refer to "coordinate chart" and some use it to refer to "tetrad" and some, like me, are a bit insane and switch between the two depending on the situation, usually with no warning or explanation.

Note that a tetrad, since it is a set of four vectors spanning the tangent space at each event, can be used as a basis. Note also that the derivatives of the coordinates form a set of four vectors at each point that can also be used as a basis. As such, you might think that there should be an easy mapping between tetrads and coordinate charts. Unfortunately, that is not the case. Sometimes, the basis formed from the derivatives of the coordinates is not orthonormal or there may be null vectors or multiple timelike vectors. In those cases the set of vectors cannot be used as a tetrad. On the other hand, sometimes you can have a perfectly valid tetrad, like the tetrad formed by observers on a rotating disk, but the integral curves of the spacelike vectors cannot be combined into a global surface of simultaneity.

I think of a coordinate system as being the natural coordinate system defined by the differential structure on the manifold - from which you can transform to any arbitary coordinate system.

And I think of a reference frame as being two worldlines - one for the object being observed and one for the observer - using any arbitrary coordinate system.