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What is the most general definition of a coordinate system?

Specificly:

  • given a suitably general metric space $(\mathcal S, s)$ consisting of a set $\mathcal S$ of elements
    (for instance: a set of events) together with a function $s : \mathcal S \times \mathcal S \rightarrow \mathbb R$, and

  • given a function $\varphi : \mathcal S \rightarrow \mathbb R^n$, for a natural number $n \ge 1$,
    such that there exists the inverse function $\varphi^{-1}$
    (namely any such function $\varphi$, without any specific further conditions),

do $\mathcal S$, $s$, and $\varphi$ together constitute a coordinate system?
Or which additional conditions would have to be imposed on function $\varphi$, to this end?

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  • $\begingroup$ why do you want a metric space? more specifically, your system seems to be a global coordinate system (a unique map from all the space to all $\mathbb{R}^n$; which topological/metrical properties of the latter do you want to be preserved by the inverse map? $\endgroup$
    – yuggib
    Commented Apr 28, 2015 at 19:07
  • $\begingroup$ @yuggib: "why do you want a metric space?" -- Any suitably generalized metric space. So we may speak of some sort of "system" at all. "your system seems to be a global coord. system; a unique map from all the space to all $\mathbb R^n$" -- That's not the intended meaning. I mean $\varphi$ to be a unique map from all the (non-empty) space to some (non-empty) subset of $\mathbb R^n$, or all $\mathbb R^n$. "which topological/metrical properties of the latter do you want to be preserved by the inverse map?" -- None which aren't outright required for speaking of a "coordinate system". $\endgroup$
    – user12262
    Commented Apr 28, 2015 at 19:19
  • $\begingroup$ 1) Well, the least requirement possible would be a collection of points without any additional structure. Or you may think of a space with topology but without a metric, for example. 2) the coordinate system is global because maps all $S$ into the subset of $\mathbb{R}^n$; in manifolds usually the mapping is local (in the sense that $S$ is only locally (in a neighborood of each point) isomorphic to a subset of $\mathbb{R}^n$). 3) The requirements of a cooridnate system depend, in my opinion, to what you need them for; as I said above you may only need them to be a set (a collection of points). $\endgroup$
    – yuggib
    Commented Apr 28, 2015 at 19:38
  • $\begingroup$ @yuggib: "1 [...] the least requirement possible would be a collection of points without any additional structure." -- Presumably no "structure" in addition to $$\mathfrak g : \mathbb R^n \times \mathbb R^n \rightarrow \mathbb R, \qquad \mathfrak g[~\mathbf x_a, \mathbf x_b~] \mapsto s[~\varphi^{-1}[~\mathbf x_a~], \varphi^{-1}[~\mathbf x_b~]~].$$ "[...] 3) The requirements of a coordinate system depend, in my opinion, to what you need them for" -- Well, the larger point of my question is to establish that, in Physics, there is no genuine need for coordinates. $\endgroup$
    – user12262
    Commented Apr 28, 2015 at 19:55
  • $\begingroup$ Even if there is no necessity of coordinates, they are quite useful; also in relativity it is postulated that the (coordinate free) space-time is locally isomorphic to the minkowski space-time (and thus local coordinates emerge quite naturally). Anyways I do not see why do you insist in putting the metric as a necessary requirement to define "coordinates"; if I define the coordinates as a subset of $\mathbb{R}^n$ set-isomorphic to my given set (or a part of it), I am defining an identification of the points of my set with the $n$-tuples of reals, that I may call coordinates... $\endgroup$
    – yuggib
    Commented Apr 28, 2015 at 20:10

1 Answer 1

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In principle, yes, $\varphi$ is a coordinate system, in that it tags every member of $\mathcal S$ with an appropriate $n$-tuple of real numbers. If you know a set of coordinates $(c_1,\ldots,c_n)$, you can use $\varphi^{-1}$ to find the corresponding point in $\mathcal S$.

In practice, this will not be useful at all, mostly because the amount of real numbers you can actually "know" is very, very limited. The situation you have described includes all invertible functions $\varphi:\mathbb R\to\mathbb R$, and this is a very big, very scary set. (To see the sort of stuff you allow by doing this, grab a copy of Counterexamples in analysis, in print or online.)

To provide a minimally useful coordinate system, we usually require at the very least that $\varphi$ and its inverse be continuous. This permits one to permit finite knowledge about a point with only finite knowledge about its coordinates, and vice versa, which is the very minimum necessary to perform anything that looks like an experiment.

In the end, though, this is strictly a semantics issue. Different authors will mean different things by the term, which is why we prefer to use unambiguous terms like homeomorphism and diffeomorphism. Authors which use the term coordinate system in any rigorous environment will define what they mean by the term; if they don't, it is assumed to be as well-behaved as necessary for the purpose at hand.

There are in the end no hard answers to your question. "Coordinate system" means what you want it to mean. However, you do need to be aware that this is distinct from the more subtle notion of "useful coordinate system", which depends on what you're trying to do; provide too little structure for your purpose and you will fail. Ask yourself "what do I want this coordinate system for?" and that will tell you what other conditions you need to impose on it.

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  • $\begingroup$ Emilio Pisanty: "In principle, yes" -- That's by itself an acceptable answer. "the more subtle notion of "useful coordinate system", which depends on what you're trying to do;" -- That's a stunningly imprecise formulation; and perhaps the main point of contention; No: "usefulness" depends apparently on 1. having some "structure", such as prototypically $(\mathcal S, s)$, given ("through measurement") and 2. the "goodness" of mapping to whatever intrinsic structure of $\mathbb R^n$ ("natural topology", "natural vector space"). "Coordinate usefulness" is never original/genuine. $\endgroup$
    – user12262
    Commented Apr 29, 2015 at 5:38
  • $\begingroup$ Yeah, it's an imprecise notion - that's just the way things are. Coordinate systems are in the end human tools to understand the world (crucial tools, but tools nevertheless). As such their usefulness depends on the task you're using them for. Is there a topological or metric structure of S that you're probing? Then you need continuous tools to understand it. Do you want to take derivatives with respect to done natural measure of rates of change in S? Then you should make sure your charts are smooth enough. $\endgroup$ Commented Apr 29, 2015 at 6:44
  • $\begingroup$ Unfortunately, though, I don't really have more time to discuss semantics. $\endgroup$ Commented Apr 29, 2015 at 6:44

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