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I'm in trouble with the definition of reference system in the context of General Relativity intended as coordinate chart (i.e. no frame field). Various sources define it as a one-to-one smooth mapping from an open region of spacetime into an open region of $\mathbb R^4$ endowed with the standard Euclidean topology.

Now my point is: in any specific circumstance, e.g. Schwarzschild spacetime, in order to assign coordinates to events a sort of whatsoever "rule" is actually needed. Following for instance the point made by Landau & Lifshitz in The classic Theory of Fields - $\S82$:

This result essentially changes the very concept of a system of reference in the general theory of relativity, as compared to its meaning in the special theory. In the latter we meant by a reference system a set of bodies at rest relative to one another in unchanging relative positions. Such systems of bodies do not exist in the presence of a variable gravitational field, and for the exact determination of the position of a particle in space we must, strictly speaking, have an infinite number of bodies which fill all the space like some sort of "medium". Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity.

So, a coordinate chart isn't just a mathematical abstract tools, it (at least) tacitly includes/implies the rules used to map events to points in the map's image (an open set of $\mathbb R^4$).

Edit: as adviced in the comments I'd ask for feedbacks on the following.

Let's take Schwarzschild spacetime for instance. To derive the metric tensor field $g$ as solution of the Einstein Field Equations (EFE), one begins by assuming spherical symmetry for spacelike hypersurfaces of constant coordinate time $t$. Therefore one picks spherical coordinates $(r,\theta, \phi)$ although at this stage doesn't know yet what will be the physical interpretation of the radial coordinate $r$. Basically at the very beginning one assumes that, whatever will be the interpretation of $r$, the metric $g$ will be the same at all events on each spacelike hypersurface sharing the same $r$ value.

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    $\begingroup$ It seems that you are mixing two ralted but seperate consepts, coordinate charts and frames. It is also not clear what your question is. $\endgroup$ Commented 16 hours ago
  • $\begingroup$ Yes, the definition from Landau&Lifshitz is about coordinate charts, I think, right ? $\endgroup$ Commented 16 hours ago
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    $\begingroup$ What is the question? $\endgroup$ Commented 16 hours ago
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    $\begingroup$ So are you making a point and not asking a question? If so, that isn’t what this site is for. This is a question and answer site. Not a discussion forum. You might consider physicsforums.com instead $\endgroup$ Commented 15 hours ago
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    $\begingroup$ Ok, so on this site you need to write a question to which the feedback you want would be the answer. That needs to be edited into the question itself, not the comments. $\endgroup$ Commented 15 hours ago

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There is room for a range of opinion on the relation between mathematics and physics in General Relativity (GR). For example, mathematically one may say that a coordinate chart provides a smooth labelling of points in a manifold and has no other physical significance. But in practice one will usually adopt a chart with further properties which may be useful owing to their physical significance. For example in a static spacetime, and in spacetimes with certain symmetries, there is a natural foliation into spacelike surfaces and one would ordinarily assign coordinates which reflect the symmetries. Also, in the vicinity of any event there is a coordinate choice which makes the metric have the Minkowski form (also called Lorentz form) and those coordinates have both physical and mathematical significance. For, the temporal coordinate then has a simple linear mapping to the physical behaviour of a freely falling clock at the local origin of coordinates. Also, spatial coordinate separations can then be arranged to reflect the physical dimensions of ordinary rods in free fall, etc. So you see now the coordinates are not mere mathematical labels, they are also telling us something about physical behaviour. This is owing to the Strong Principle of Equivalence, which is a statement about physical behaviour not just the spacetime metric and curvature.

Rindler, in his textbook(s), describes a freely falling frame in G.R. by using a physical description: it is an idealized group of solid blocks all in free fall, located near to one another and furnished with clocks (and offering only negligible contributions to local curvature). Other authors may resist that kind of physical specification, and write instead about the metric. But whatever approach is adopted, it will have to agree with the implications of Rindler's picture, because that picture is also correct.

I think the reason for the variety of approaches is partly owing to the fact that the spacetime manifold of GR is already of interest from a purely mathematical point of view, irrespective of all the further aspects of behaviour that interest physicists (such as electromagnetic, strong and weak forces, quantum mechanics etc.) Sometimes the pure mathematician is not interested in something so messy as an actual caesium atomic clock, or a mirror hanging at the end of a long rigid tube. But the physicist wants to talk about those things, otherwise they don't know what the coordinates even mean.

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