# 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.

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:

1. 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.

2. 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.

3. 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

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.

• Note that we can only do meaningful coordinate transforms if we have an accurate set of rules about the underlying topology. So a coordinate system, by its nature, cares about physics. Cartesian to polar conversions require understanding the physics of a Euclidean universe, for example. I'm not sure how this answer is different from "coordinates are abstract, frames of reference are physical". And I'm not sure that's right, considering we model frames of reference all the time. Commented Jan 9, 2022 at 10:13
• @MichaelS I concur: coordinate transformation is meaningful only if the coordinate system that you transform to embodies the same underlying metric structure as the starting coordinate system. In order to use newtonian mechanics one assumes the Euclidean metric, combined with assuming that an object released to inertial motion will in equal intervals of time cover equal intervals of distance. In order to do special relativity one assumes the Minkowksi metric. The set of usable coordinate systems is the set of coordinate systems that embodies the metric structure of the theory of motion. Commented Jan 9, 2022 at 14:35
• As other answers point out, there are indeed different definitions. I'm merely drawing my own line here, which I find useful. I find it useful to distinguish the mathematical procedure of a change of variables, and the physical aspect of two observers describing the same phenomenon. I use this to distinguish these two concepts. You may find it trivial or unhelpful, and you may draw the line somewhere else, perhaps associate it with metric or curvature or topology or homotopy or other stuff as you wish, that's your choice. Commented Jan 9, 2022 at 20:36

I concur with the answer by contributor Dale; I upvoted.

Indeed when you read an author it is necessary to reconstruct (from context) how that author is using the expression 'frame of reference'.

Indeed when you use the expression 'frame of reference' yourself it is up to you to make sure that the reader is aware of how you are using it. You must either define explicitly, or you must provide context such that the reader can infer what meaning you are using.

Personally I have stopped using the expression 'frame of reference' altogether. There is nothing intrinsically wrong with that expression, but there is so much variance in how it will be understood that it's just not a dependable intrument for communication.

That leaves the expression 'coordinate system'. Of course, the expression 'coordinate system' is rich with connotations. When using the expression 'coordinate system' in writing I have to be aware of that.

A prominent connotation is that a coordinate system has a zero point. So: when I feel it's necessary to counterplay that connotation I will refer to 'the equivalence class of inertial coordinate systems', instead of to 'an inertial coordinate system'. The members of the equivalence class of inertial coordinate systems all have a different zero point; at the same time all those class members stand in a well defined relation to each other; it's quite an exclusive class.

My assessment is that by not using the expression 'frame of reference' I put myself in a position that allows me to express myself with more clarity.

Reference:
I acquired the expression 'equivalence class of inertial coordinate systems' from Kevin Brown's website Mathpages.com, from the article A primer on special relativity

I personally think of a reference frame as being a velocity and position in space from which to observe things, and a coordinate system as being the measuring system by which we locate things at other positions. It matches the general usage of the terms better.

Essentially, the reference frame describes here, while the coordinate system describes everywhere else with respect to here.

Note phrases like this (from IsaacPhysics.org) that are quite common:

The different observations occur because the two observers are in different frames of reference.

You can't be in two different coordinate systems. You can be at different locations, or have different velocities, in a single coordinate system. You can use different coordinate systems. But both observers are in locations, at velocities, described by both coordinate systems.

That snippet is followed by the definition:

A frame of reference is a set of coordinates that can be used to determine positions and velocities of objects in that frame; different frames of reference move relative to one another.

This essentially defines a reference frame as a coordinate system, but that doesn't make sense in context. It says a reference frame is something that can move. A coordinate system spans all of spacetime, but something that moves needs a finite space it occupies so we can locate it and therefore determine it's moving. The thing that's moving is the origin of the coordinate system.

1. A reference frame is something that moves. The origin of a coordinate system moves.

2. An object can exist in one reference frame while not existing in another. Two coordinate system origins can be at different locations, moving at different speeds.

The usages of "frame of reference" in the quotes above are satisfied by calling it the origin of a coordinate system. But they're not satisfied by calling it a coordinate system, since an object can't exist in one coordinate system but not the other, and coordinate systems don't really have locations as they span everywhere.

The origin of a coordinate system is a position and velocity in space that we use to calculate how other objects would look from said origin, leading to the simple definition I gave above:

A reference frame is a velocity and position in space from which to observe things.

Side note 1: we don't have to use the origin. We could substitute "origin" with "specific location", since all the locations within a coordinate system can move like the origin, and an object can be in one location in one coordinate system and not be at a specific location in another coordinate system. But it's generally simplest to use coordinate systems whose origins are located at the point from which we want to view things.

Side note 2: "position" here includes rotational orientation as well as linear position.

• This definition is not in line with the normal reference frame definitions. For example, in the normal reference frame definition, two observers are in the same reference frame if they don't move wrt. to each other, even though they are not at the same location. And in the normal usage of "reference frame", all objects are present in all reference frames. Your definition does not allow for those common notions. Commented Jan 10, 2022 at 16:26
• @fishinear: In practice, at a normal people level anyways, the way I see real people use terms like "reference frame" is not synonymous with "coordinate system". You can find thousands of examples of quotes like I provided above that don't fit your "normal" definition. And only some definitions consider the same velocity at different locations to be the same frame of reference. In a purely SR world that works, but not completely in the real world with GR influences. (My definition also happens to coincide with the non-physics definition used in everyday speech.) Commented Jan 10, 2022 at 21:15
• And, as I mentioned above, you don't have to use the origin. If you're just picking a location, then two coordinate systems not moving with respect to each other have locations that coincide, meaning you can consider them the same reference frame. I just prefer to think of a reference frame from an origin because it's more useful to me, but I allowed that others might not. Commented Jan 10, 2022 at 21:16

Reading this topic this is what I could come up with A coordinate system a mathematical means of expressing the projected positions of points on various surfaces.

A reference system is the practical realization of the reference systems through observing the designed monuments to come up with a set of coordinates. This means that with the aid of the reference frames, the geodetic reference systems can be realized and made usable.