I can't understand the comment on page 409, Gravitation, by Misner, Thorne, Wheeler
It follows that the ten components $G_{\alpha\beta} =8\pi T_{\alpha\beta}$ of the field equation must not determine completely and uniquely all ten components $g_{\mu\nu}$ of the metric.
On the countrary, $G_{\alpha\beta} =8\pi T_{\alpha\beta}$ must place only six independent constraints on the ten $g_{\mu\nu}(\mathcal{P})$, leaving four arbitrary functions to be adjusted by man's specialization of the four coordinate functions $x^{\alpha}(\mathcal{P})$.
I can't understand it. I think we can always solve the field equation with appropriate initial/boundary conditions to get unique $g_{\mu\nu}$. After all those are just second order differential equations. To be specific, let me try to construct a counter example, the vacuum Einstein equation, $$G_{\mu\nu}=0$$ If we apply the initial conditions $g_{\mu\nu}|_{t=0}=\eta_{\mu\nu}$ and ${\dot{g}_{\mu\nu}}|_{t=0} =0$, obviously the flat spacetime $g_{\mu\nu}=\eta_{\mu\nu}$ should be the solution. If the solution $g_{\mu\nu}$ is unique, what's the alternative solution?
If there does exist an alternative solution, does it come from "specialization of the four coordinate functions"?
Update: user23660 constructed an explicit alternative solution, which is $$ g_{00}=(f'(t))^2,\quad g_{ij}=-\delta_{ij} $$ with other components being zero.
The function $f$ only need to satisfy $f'(0)=1,f''(0)=0$, that makes this metric compatible with the initial data; other than that, it's completely arbitrary! And we see that it does come from the coordinate transformation $t=f(\tau)$
To get the solution to be $\eta_{\mu\nu}$, we need to put further constraints on the metric directly in this coordinate system, like $g_{00}=1,g_{0i}=0$.
This redundant degrees of freedom(gauge) result from the contracted Bianchi identity, as explained in the following paragraph in MTW page 409, $$G^{\alpha\beta}{}_{;\beta}=0$$ is true automatically, and so the equation of motion of the matter fields $T^{\alpha\beta}{}_{;\beta}=0$ doesn't really put restrictions on the evolution of the metric. Therefore, there are only 6 independent equations!