# Gauge transformation in general relativity

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.

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.