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Let us consider a classical field theory on flat background spacetime. The action is $$S[\Phi] = \int d^nx \mathcal{L}(\Phi,\partial_\mu\Phi).$$ Why shouldn't this action be independent of the chosen coordinates ? It doesn't depend on the inertial observer that we choose since it's invariant under Lorentz transformations but why shouldn't is be under more general coordinates ? What about spherical coordinates and other systems of coordinates that don't corresponds to another observer ?

In general, I don't understand how any physical theory can depend on the chosen coordinate system.

xpsf
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  • This question and its answers might interested you and likely contain the answer to your question. 2. As written, the question is a bit confusing - it first talks about "choosing coordinates", then it talks about Lorentz transformations, and then it talks about isometries - but in flat spacetime, the Lorentz transformations are the isometries, and so the claim that "only GR is invariant under isometries" is definitely false - any special relativistic theory certainly is also invariant under its isometries.
    • – ACuriousMind Jun 05 '21 at 20:02
  • I admit it was unclear, thank you for the comment. I removed the second part. – xpsf Jun 05 '21 at 20:08
  • See also this answer for a similar confusion - the moment the expression on the r.h.s. of $S[\phi]$ is well-defined, it is automatically coordinate-independent. – ACuriousMind Jun 05 '21 at 20:11