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    $\begingroup$ I thought that two manifolds that are related by a diffeomorphism represent the same physical situation though? I.e. If the universe is described by some manifold $M$ with metric $g_{\mu\nu}$ and matter fields $\psi$, and $\phi :M\rightarrow M$ is a diffeomorphism, then the sets $(M, g_{\mu\nu}, \psi)$ and $(M, (\phi^{\ast}g)_{\mu\nu}, \phi^{\ast}\psi)$ describe the same physical situation?! ... $\endgroup$
    – user35305
    Commented Jan 11, 2017 at 22:33
  • $\begingroup$ @user35305 Yes, that is what I said, a diffeomorphism still describes the same manifold. My point is it is not background independent in the sense you can go from $M$ with $g_{\mu\nu}$ to a $\Sigma$ with $h_{\mu\nu}$ not related by some $\phi$. $\endgroup$
    – JamalS
    Commented Jan 11, 2017 at 22:36
  • $\begingroup$ Ah, apologies, I was a bit too liberal in my wording there, by "background independence" I had meant in the case of related by diffeomorphisms. Regarding the gauge vs diffeo stuff, what confuses me is that in the literature a clear distinction seems to be made between diffeomorphism invariance and gauge invariance without really stating what this is? For example, I have seen papers constructing gravity as a gauge theory, which to me seems to suggest that it wasn't before (although it clear was already diffeomorphism invariant before)?! $\endgroup$
    – user35305
    Commented Jan 11, 2017 at 22:37
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    $\begingroup$ @user35305 It's simply a different, more QFT-like way of viewing gravity. $\endgroup$
    – JamalS
    Commented Jan 11, 2017 at 22:52
  • $\begingroup$ Ah ok. I guess they naively seem like different "entities" since one seems to take place on the base manifold, which induces changes in the scalar/vector/tensor fields involved in the theory (diffeomorphisms), whereas the the other (gauge transformations) seems to take place in the fibre bundle over some base manifold (that is kept fixed) in which the vector/tensor fields themselves are directly transformed with the base space kept fixed. $\endgroup$
    – user35305
    Commented Jan 11, 2017 at 23:00