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I am studying topological manifolds as a prerequisite to studying General Relativity and although this question is premature since I have not yet begun the latter it is bothering me.

From basic physics I always heard that General Relativity is background independent. Is this the same thing as saying it can be done without coordinate systems or just that it must be done with a coordinate system but the one you use doesn't really matter?

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    $\begingroup$ I mean, consider the statement: "this morning my little kid's height reached 3 feet". That is a highly coordinate-dependent statement which depends on a notion of time, a definition of a height unit, a definition of where the zero of height is, and a definition for which direction you measure height in. Every actual concrete statement you make about the world uses coordinates. Making everything coordinate-free is a fun diversion for mathematicians, but not useful if you actually want to describe specific, real things. $\endgroup$ – knzhou Sep 22 '18 at 13:54
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Pretty much the latter: coordinates are the tools we use to describe what is going on: there are no coordinates in nature and it does not really matter which coordinate system we use, so long as it's 'good': it should provide a suitably differentiable 1-1 map of the manifold (or an open subset of the manifold) into $\mathbb{R}^4$, so that we can talk about what is going on. Of course, in practice the choice of coordinates matters a lot, because you want to choose one that makes calculations practical: this is just the same thing as in other parts of physics: you don't really want to be working on a problem with spherical symmetry in cartesian coordinates, say.

Of course there are also things you can do in GR without using coordinates at all: only some (probably most!) calculations require coordinates.

However there is a slight caveat to this: the way manifolds are defined (at least to a physicist: mathematicians may have more abstract definitions) is by specifying that there are continuous 1-1 maps between open sets of the manifold and open sets of $\mathbb{R}^n$. These collections of maps are, in fact, coordinate systems for the manifold. So you could argue that there are, inherently, coordinate systems. But it does not matter what the coordinate systems are: you can choose any good set you like.

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