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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?

user12262
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  • 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? – yuggib Apr 28 '15 at 19:07
  • @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". – user12262 Apr 28 '15 at 19:19
  • 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).
    • – yuggib Apr 28 '15 at 19:38
  • @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. – user12262 Apr 28 '15 at 19:55
  • 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... – yuggib Apr 28 '15 at 20:10
  • Even I cannot do much more else if I do not give additional structure, I have at least the identification of my points with the tuples (coordinates). – yuggib Apr 28 '15 at 20:11
  • @yuggib: "[... Why] putting the metric as a necessary requirement to define "coordinates"" -- Again: to define "coordinate system". Clearly there's already terminology available for "manifold" or "coordinate patch", to denote "set-isomorphic" (homeomorphic?) mapping. "at least the identification of my points with the tuples (coordinates)." -- Surely $n$-tuples of reals are useful for distinctive labelling. (By Ockham's razor, this presumes that what's being distinctly labelled was in (physical) fact established as being distinct.) But there seems not much "Co-Ordination" in this. – user12262 Apr 28 '15 at 20:39
  • an homeomorphism is an isomorphism that preserves the topological structure; a set-isomorphism is simply a bijective map between elements of two sets. Anyways, if you want to get rid of coordinates maybe the synthetic geometry approach may be interesting for you. – yuggib Apr 28 '15 at 20:59
  • Should we assume that the inverse is a left inverse, a right inverse, both? Is there a reason that we want a single coordinate system for the entire space? That seems very unusual. Can the suitably general metric space be degenerate (e.g. by the map that sends all pairs to zero)? If so then it doesn't really put any constraints. And are we to assume that $n$ is finite? Hence our space has to be small. Whoops I said space, which to me means I'm implying a topology. And I'm not actually seeing the physics here, can we make that be part of the question somehow? – Timaeus May 07 '15 at 05:53
  • @Timaeus: "Should we assume that the inverse is a left inverse, a right inverse, both? [...]" -- My intention was to ask about exactly one "coordinate system"; therefore, AFAIU, about one suitable set (or "patch"?) $\mathcal S$ constituting a suitable reference system $(\mathcal S, s)$, and a suitable set of (real-valued) n-tuples $C_n \subseteq \mathbb R^n$ as coordinates, where $$\phi : \mathcal S \leftrightarrow C_n, \qquad \phi^{-1} : C_n \leftrightarrow \mathcal S,$$ $$\phi^{-1} \circ \phi \equiv \mathbf I_{(\mathcal S)}, \qquad \phi \circ \phi^{-1} \equiv \mathbf I_{(C_n)}$$. – user12262 May 07 '15 at 18:52
  • @Timaeus: "Can the suitably general metric space be degenerate (e.g. [...])?" -- Interesting and even relevant case (e.g. "events on one light ray"). Does the mere distinctive labelling of elements of such a degenerate space lead to a "coordinate system"? (Or a "degenerate coordinate system"?) Hence my question. "$n$ is finite?" -- Does it matter? Even $n = 1$ seems to gives us plenty of distinctive labels ... "[...] topology." -- Well, I like to re-examine why I presumed the underlying reference system as (generalized) "metric space", rather than "topological space". (May take a while.) – user12262 May 07 '15 at 18:53
  • @user12262 How is one coordinate system useful for anything at all?!? Just work with a subset of Rn if that's what you want. – Timaeus May 07 '15 at 21:38
  • @user12262 Regarding whether n is finite. You asked for generality, and I notice you immediately exclude almost every single use of coordinates I've ever seen. Some systems are infinite dimensional. Usually you want your coordinates to cooperate with some structure that imposes limits that might require an infinite basis. And often you want something more general than a metric space. It badges no sense to cut out huge uses of coordinates and then ask for the most generality. – Timaeus May 07 '15 at 21:43
  • @Timaeus: "You asked for generality, and [...] Usually you want your coordinates to cooperate with some structure" -- I admit that my question suffers from this ambiguity: On one hand asking for "the most general structure" by which to characterize some (given) set $\mathcal S$ as constituting a "space $(\mathcal S, \text{ structure})$", intrinsically. (That's interesting in its own right.) But on the other hand: asking what the minimal requirements (if any, apart from sheer distinctive labelling) are on the "cooperation" you mentioned. My focus/intention is the latter aspect:[...contd.] – user12262 May 08 '15 at 05:54
  • Namely to demonstrate that in order to consider and evaluate any such possible "cooperation" we, as physicists, must be able to consider and evaluate the relevant (intrinsically meaningful) "$\text{structure}$" separately, in the first place; beginning with the sheer distinctiveness of all the elements of set $\mathcal S$. Such intrinsic "$\text{structure}$" must not be confused with whatever "incidental coordination" might be super-imposed by any particular assignment of coordinates, or the other. – user12262 May 08 '15 at 05:56
  • p.s. @Timaeus: "[...] often you want something more general than a metric space." -- I always wrote "suitably generalized metric space"; and "$(\mathcal S, s)$", rather than e.g. "$(\mathcal S, d)$". Now, you may find my concrete OP suggestion of "$s$" ... undercomplex. But it can of course be further generalized, e.g. as $$s : \text{Powerset}[~\mathcal S~] \times \text{Powerset}[~\mathcal S~] \rightarrow \mathbb R,$$ (which can include "topological space" as special case), etc. Are there even (infinite dimensional, complex ...) coordinates "to cooperate with this much structure" ?? – user12262 May 09 '15 at 04:52