I am reading Tong's notes about string theory, the second chapter, and I encountered this part that I don't know how is derived. We are considering the worldsheet $(\tau,\sigma)$ whose gauge we set to the flat metric $\eta_{\mu\nu}$. Then, Tong introduces lightcone coordinates $$\sigma^\pm = \tau\pm\sigma,\tag{2.9}$$ under which the metric is $$ds^2 = -d\sigma^+d\sigma^-.\tag{p.32}$$ Then, Tong states that any transformation of the form $$\sigma^+\to\tilde{\sigma}^-(\sigma^-),\sigma^-\to\tilde{\sigma}^-(\sigma^-),\tag{2.10}$$ which is equation (2.10), can be undone by a corresponding Weyl transformation. So this is a residual symmetry of the worldsheet metric.
What I have trouble understanding is this statement above eq. (2.12):
... we claimed that we could go to static gauge $X^0=R\tau$ for some dimensionful parameter $R$. It is easy to check that this is simple to do using reparameterizations of the form (2.10).
I guess I don't really understand what the reparametrization invariance has to do with it, or how these transformations would explicitly play out. I'm not sure how to write arbitrary $\tau,\sigma$ in terms of $t$ and the spatial coordinates $x^i$. I guess we would have arbitrary $\tau(t,x^i)$ and $\sigma(t,x^i)$ and then find some transformation $f(\tau+\sigma)$,$g(\tau-\sigma)$ to reparametrize $\sigma^\pm$, but I can't see how this is guaranteed to make $\tau$ proportional to $t$.
What would be these reparametrizations of the form (2.10) to get from an arbitrary $(\sigma,\tau)$ to static gauge?
The question seems very simple and probably has a simple solution I'm not getting.
My attempt to solve: We have the worldsheet metric $\eta_{ab}$ embedded inside a higher-dimensional metric $g_{ab}$, and we set $\eta$ to be flat.
$$\eta_{ab} = \partial_aX^\mu\partial_bX^\nu g_{ab}$$
so with $(\sigma_-,\sigma_+)$ coordinates on the worldsheet metric,
$$\partial_-X^\mu\partial_+X^\nu g_{\mu\nu} = -1/2$$ and $$\partial_-X^\mu\partial_-X^\nu g_{\mu\nu} = \partial_+X^\mu\partial_+X^\nu g_{\mu\nu} = 0.$$
We would like $\sigma_\pm \to \tilde{\sigma}_\pm$ such that $\tau = Rt$, which means $\tilde{\sigma}_+ + \tilde{\sigma}_- \propto X^0$. Here I'm not sure how to proceed.
Is the relevant factor/explanation that both the $\sigma_\pm$ and the $\tilde{\sigma}_\pm$ parametrizations have the same off-diagonal metric, so $\partial\tilde{\sigma}^\pm/\partial_\sigma^\pm = 0$?
