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I am studying Weinberg's Cosmology. In chapter 5, he describes 'General theory of Small Fluctuations' where he considered a general coordinate transformation $$ x^{\mu} \to x^{\prime\mu}= x^{\mu} + \epsilon^{\mu}(x)\tag{5.3.1} $$ and how the metric transforms under this transformation. After that he writes, and I quote

Instead of working with such transformations, which affect the coordinates and unperturbed fields as well as the perturbations to the fields, it is more convenient to work with so-called gauge transformations, which affect only the field perturbations. For this purpose, after making the coordinate transformation $(5.3.1)$, we relabel coordinates by dropping the prime on the coordinate argument, and we attribute the whole change in $g_{\mu\nu} (x)$ to a change in the perturbation $h_{\mu\nu}(x)$.

Can anyone please explain what he meant by 'gauge transformation' (the dropping of the prime) here and how and why is it different from the general coordinate transformation?

I will appreciate an elaborate answer and some reference to literature.

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The whole idea is that the division between the "background" and the "perturbation" is arbitrary. Hence, as long as the coordinate form of the metric is affected only linearly by the coordinate transform, we can "reassign" a different part of the metric as the "background" and as the perturbation so that the coordinate form of the metric is always fixed.

For concreteness consider the background to be the Minkowski metric in Cartesian coordinates $\eta = diag(-+++)$ with some perturbation $h$, $g=\eta +h$. Now we make an infinitesimal coordinate transform. This transform generally takes the coordinates away from Cartesian and $\eta$ is now not in the form $diag(-+++)$, it is now $\eta=diag(-+++) + \delta$ with some small $\delta$. However, since $\delta$ is small, we can redefine the division between the background and perturbation so that the background metric is $diag(-+++)$ even after the coordinate transform and the perturbation is redefined as $h' = h+\delta$.

This procedure may seem contrived at this point but it leads to a structure of equations where $h$ behaves exactly as a massless spin-2 field on a curved background. The infinitesimal transformation parameters $\epsilon^\mu$ then play the role of gauge potentials.

I.e. the same way we have a gauge transform of the electromagnetic potential $$A_\mu \to A_\mu + \chi_{,\mu}$$ for some gauge potential $\chi$, the infinitesimal transform along with the "redivision" procedure given above yield the "gauge transform" $$h_{\mu \nu} \to h_{\mu \nu} + \epsilon_{\mu;\nu} + \epsilon_{\nu;\mu}$$ which is exactly the same as a gauge transform for a spin-2 field on a curved background.

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