# Difference between space of reference and system of coordinates in special relativity

In the book "The meaning of the relativity" by A. Einstein, it is referring to two different concepts: space of reference and system of coordinates. What it is the difference?

It says:

"we cannot speak of space in the abstract, but only of the "space belonging to a body A" (i.e. space of reference)

What book recommend to study both concepts?

• A good high school physics textbook contains all you need to know. – CuriousOne May 31 '16 at 8:04
• If you really wish to learn SR, leave everything and start playing with Lorentz Transformations for a while. Just remember the formulas, and try out some calculations. Remember space/length and time are just measurable quantities of events, just like torque/momentum are of motion and their values can change from observer to observer. – Isomorphic May 31 '16 at 8:09
• @Isomorphic: I am not sure the OP is ready for that, yet, I think he/she is still struggling with the fundamental questions of physical ontology (which is experimental and matter based). Einstein tried to make sure that nobody thought that he was talking about some abstract/philosophical notions of space that were unverifiable but obviously messing with people's minds. I honestly hope that our high school textbooks get this sorted out right away... even before introducing Newtonian physics. – CuriousOne May 31 '16 at 8:12
• Fortunately I'm already familiar with special relativity and consequently with the transformations of co-ordinates, tensor calculus, etc. But in the first part of the book that I quoted, I do not understand the difference between this concepts. – FUUNK1000 May 31 '16 at 8:19
• "Space in the abstract" is not a physical concept, at all. Nothing in the abstract is. All physical concepts are based on matter and radiation, i.e. "stuff" that has mass-energy, momentum, angular momentum, a length and a time scale, maybe even a temperature. In mathematics you aren't studying abstracts, either. If you ever take a class in mathematical logic, one of the first things that the teacher will (or at least should) tell you is to check that your axioms define objects that can be satisfied by something else than the empty set. – CuriousOne May 31 '16 at 8:22

The "system of coordinates" is just a fancy way to describe particular "coordinates", which you said to now what it means, with the focus on the "choices" which were made to choose some way of defining and measuring coordinates and not others.

"System of reference" was just a particular phrase used by Einstein once, it is not routinely used, and what he meant was nothing else than a "system of coordinates". In special relativity, the phrase automatically meant an "inertial system of coordinates", in general relativity, he allowed other systems of coordinates, too.

Also, "space of reference" only meant a "system of coordinates" which allows to identify points/locations by numbers, and this "imagined grid" in the empty space is thought of as a "space".

The only point of that paragraph by Einstein was to express the principle of relativity i.e. say that the space isn't an absolute object in the sense that the vacuum would be filled with some metallic grids. Instead, the empty space is really empty and doesn't even justify our talking about its objective location, let alone the velocity of the non-existing (metallic) grid. All velocities are equally good for formulating the laws of physics – the principle of relativity. If we want to talk about well-defined locations of objects with respect to the "space", we can't assume that the "space" makes some objective sense. Instead, we must use a "space we construct" by actually adding rulers or metallic grids into space.

In this way, the coordinates of objects are really expressed relatively to the positions of solid objects such as the metallic rods and grids and rulers, and the abstract space that was completely empty but supposed to behave as a piece of matter must be replaced by actual matter objects embedded into space. The speed of these actual matter objects – rulers, rods, grids – may be arbitrary. And indeed, the laws of physics must carefully distinguish the effect of the speed of these reference objects. That's why the time depends on the reference frame – it (and the simultaneity of events) depends on the speed of the "metallic grids" that replace the previous, ill-conceived idea of a "universal grid".

If you understand relativity and how it differs from classical physics by making many things (such as the simultaneity of events) dependent on the chosen inertial system, you shouldn't spend time with the two phrases and this paragraph, let alone search for whole "books" to clarify this particular awkward phrase (there are no "whole books" dedicated to this phrase). The paragraph says nothing else than the usual wisdom about the need to describe physics relatively to a particular chosen coordinate system which isn't unique.

There's a plain distinction between a reference system (to use a more contemporary designation) and a coordinate system: a coordinate system is a reference system together with an (one-to-one) assignment of a coordinate value ($n$-tuple) to each element.

Expressed more formally, a reference system to begin with is constituted by a set of distinguishable, identifiable elements (such as set $\mathcal P$ of "material points", or set $\mathcal S$ of "events") together with a geometric characterization, such as

• distance values between (suitable) pairs of points: $d : \mathcal P \times \mathcal P \rightarrow \mathbb R$, or

• spacetime interval values between event pairs (in a suitably flat region): $s^2 : \mathcal S \times \mathcal S \rightarrow \mathbb R$.

Specific pairs $(\mathcal P, d)$ or $(\mathcal S, s^2)$ are examples of reference systems.

A corresponding coordinate system then requires some additional coordinate assignment, e.g.

• $c_{\mathcal P} : \mathcal P \longleftrightarrow \mathbb R^n$, or

• $c_{\mathcal S} : \mathcal S \longleftrightarrow \mathbb R^n$,

where $n \ge 1$, and some values of $n$ may of course be regarded more suitable than others, depending on properties of the specific geometric characterization.

Specific triples $(\mathcal P, d, c_{\mathcal P})$ or $(\mathcal S, s^2, c_{\mathcal S})$ are then examples of coordinate systems.

In addition to any coordinate assignment being one-to-one, so that each element of a given reference system is uniquely identified by a coordinate tuple value, it may have to satisfy further requirements; for instance such that smoothness is guaranteed for the induced metric function

$\mathfrak g : \mathbb R^n \times \mathbb R^n \rightarrow \mathbb R, \qquad \forall A, B \in \mathcal P : \mathfrak g[~c_{\mathcal P}[~A~], c_{\mathcal P}[~B~]~] \mapsto d[~A, B~].$

There is an issue with metallic rods and clocks. Metallic rods cannot deform unless there is well defined dynamical forces in action. In inertial frames in relative motion, no dynamic forces are identified due to merely relative motion. This is the cause for confusion on the notion of length contraction and time dilation. These effects are observations form far away observer by no means of measuring time and space. The only means of measurement is via EM waves; i.e. Doppler effects.

• This does not seem like an answer to this question. Did you mean to post it elsewhere? – Codename 47 Mar 15 at 9:30
• @Codename47 it may be that was intended to be a comment on Lubos' post – Kyle Kanos Mar 15 at 9:59