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I know that GR must be diffeomorphism invariant, which (in my own words) means that GR, and by extension any observable, should not care about what coordinate system one chooses to use.

Suppose I have a spacetime metric that is fairly complicated when it is written in terms of coordinates ($t,r,\theta,\phi$). If this spacetime metric can be written in a simpler form under a change of coordinates (such as Minkowski, or FLRW), what is the significance this?

If the EFE are easier to solve and work with in these new coordinates that would be obviously useful, but I would still think that I need to convert back to the original coordinate system to describe what is going on in the universe described by the ($t,r,\theta,\phi$) directions.

I think my question is mainly (descriptively) how should I think about diffeomorphism invariance, when is it useful, and when is it not useful, and what does it tell us.

The second answer from this post helped, but I think that I do not really appreciating the utility of diffeomorphism invariance. The universe shouldn't care what coordinate system I choose to work with, even though I want to write predictions for observables in terms of coordinates ($t,r,\theta,\phi$).

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Yes, you have to convert it back.

This does, however, not change the physics, only our description! (That is the essential Point). Maybe it's easier to get the grasp on something simpler, like spherical coordinates: Given any Object, let's say a vector, you can represent it in the coordinate system of choice, maybe the cartesian one. But I can choose a different system, the spherical one. Now, in my coordinates the vector does look different from your's but the Object itself, the vector, is still the same! Just as you are the same Human, no matter in what coordinate system one might describe you! (In ordinary $\mathbb{R}^3$)

This does also apply to GR. The reason you want it diffeomorphism invariant is that the physics have to remain the same, but we can describe them in the best possible way, and since the physics do remain the same, there wouldn't be a reason to convert it back to the complicated system in the first place.

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    $\begingroup$ if diffeomorphism invariance is only about coordinate invariance, then it is trivial and all physical laws and forces must be "diffeomorphic invariant". True or False? $\endgroup$
    – lurscher
    Jun 29 '18 at 16:15
  • $\begingroup$ So if I have a funky metric (in terms of $t,r,\theta\phi$) that can be written as Minkowski after a change of coordinates ($t\rightarrow \tilde{t}$, etc.) , would the proper way forward to work with this metric be to change coordinates to $\tilde{t},\tilde{r},\tilde{\theta},\tilde{\phi}$, solve the Einstein Field Equations in terms of those tilde'd coordinates where the metric is Minkowski, and then switch back to the usual $t,r,\theta,\phi$ coordinates to connect predictions from my original funky metric with observation? $\endgroup$
    – Bob
    Jun 29 '18 at 16:32
  • $\begingroup$ I'm by no means an expert, but for example in Yang Mills theory, the laws are only preserved by gauge transformations, which are a special kind of diffeomorphisms on the bundle used to describe the theory, so its more a question of in what model you what to impose which kind of symmetry(?) So I'd say false in general, it should hold for theorys described by tensorial equations. $\endgroup$
    – Creo
    Jun 29 '18 at 16:33
  • $\begingroup$ @lurscher for example Newton's first law is not valid if you use accelerating coordinates. However the geodesic equation (probably the closest to Newton's first law in GR) is valid for all coordinate systems, accelerating, curved or whatever. $\endgroup$ Jun 29 '18 at 16:46
  • $\begingroup$ well @JohnRennie , arguably one can extend Newton's first law to include Coriolis and Centripetal terms to account for noninertial frames. Probably something similar is true of electromagnetism $\endgroup$
    – lurscher
    Jun 29 '18 at 18:54

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