Difference between reference frame and coordinate system

In wikipedia, regarding this topic, different examples are given as to why a reference frame and coordinate system are NOT the same thing (https://en.wikipedia.org/wiki/Frame_of_reference). Example from J.D Norton:

More recently, to negotiate the obvious ambiguities of Einstein’s treatment, the notion of frame of reference has reappeared as a structure distinct from a coordinate system.

I am trying to understand the different definition and I have come up with the following understanding, for which I want to know whether is correct or faulty. First off all I consider that a difference between the two exists. For once a coordinate system is a mathematical/abstract concept while the reference frame is something grounded in physics/or physical. I would consider a coordinate system of whatever kind as "tool" (for lack of better words), which assigns a set of numbers to each point in space. The reference frame is an "entity" (for lack of better words again), which is correlated or related to a physical location/body etc, to which a coordinate system of whatever kind is attached to, giving to the observer, in this physical location (one can say the observer is at rest in this reference frame) the possibility to assign a location and a time to an event E.

reference-frames
coordinate-systems
definition

208k 48 48 gold badges 571 571 silver badges 2.3k 2.3k bronze badges asked Jan 8, 2022 at 17:28 1,434 8 8 silver badges 25 25 bronze badges Commented Jan 9, 2022 at 22:26 $\begingroup$ Does this answer your question? Reference frames versus coordinate systems $\endgroup$ Commented Apr 3, 2022 at 4:49

5 Answers 5

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First off all I consider that a difference between the two exists. For once a coordinate system is a mathematical/abstract concept while the reference frame is something grounded in physics/or physical

It is very important to recognize that different authors may use the same term differently. I have seen at least three different meanings of the term “reference frame” in different sources:

some authors treat the term “reference frame” as a synonym for “coordinate system”, sometimes with the implication that it is restricted to coordinate systems with one timelike coordinate three spacelike coordinates.
other authors use the term “reference frame” to refer to a tetrad. Again usually that is restricted to a tetrad with one timelike vector field and three spacelike vector fields, and often with the further requirement that the vector fields be orthogonal and normalized.
yet other authors use the term “reference frame” to refer to a collection of rods and clocks or other physical devices used for measuring time and position. These are automatically timelike and spacelike respectively, and often orthogonal.

I would not try to take a very rigid stance on the meaning of the term. Simply find out how an individual author is using it. Also, if there is any ambiguity, then simply clarify how you are using it. It is simply a choice of definition, so there is no particular right or wrong answer. But when people don’t clarify their usage it can lead to confusion

answered Jan 8, 2022 at 18:53 105k 11 11 gold badges 156 156 silver badges 309 309 bronze badges $\begingroup$

Indeed, it is helpful to distinguish these two concepts.

Coordinate System

For whatever phenomenon you want to describe, be that a particle or a field, a set of coordinates $(x_1. x_n)$ must be chosen. Once you've done that, there is an unlimited number of arbitrary transformations $(x_1. x_n)\rightarrow(y_1. y_n)$ you could do to represent evertyhing in another coordinate system. You don't need to invoke any laws of physics to do this, it's simply a mathematical construct. A typical example is the transformation from Cartesian Coordinates to Polar Coordinates.

Reference Frame

It's customary for physicists, since Einstein, to define something by defining its relations to something else. That's why, to define what a reference frame is, we define the transformations between different reference frames.

A reference frame transformation is a coordinate transformation, this simply means that there is a mathematical transformation which can translate someone's observation to another person, at some other position and time. However, it's a special transformation that cares about physics.

The way to include physics in reference frame transformation is that We usually make someone (perhaps unwillingly) ride a spaceship away from you, at very high speed, and compare our observation of the same stuff. The game is for you to observe the phenomenon, and at the same time guess what the other person will observe. This is a coordinate transformation, but you cannot just make up any coordinate transformation, it's a special one related to where the other person is, how fast he is moving, and so on. In this way, if you correctly guess what the other person observes, you can confirm your believe about certain physical laws( Energy, momentum, how to add speed, is Newton's law right. ), or perhaps change your laws such that it satisfies this strange coordinate transformation, which is based on our believes and experiments on real physics. This is what special relativity is doing, in a nutshell.