One has that $ds^{2} = g_{ij}(x)dx^{i}dx^{j}$. I often see that the interval is re-expressed with a time "seperation" of the form: $$ ds^{2} = g_{00}(x)dt^{2} + \tilde{g}_{ab}dx^{a}dx^{b} \;\; a,b = 1,2,3 $$ When can this be done?
Why can the proper time infinitesimal always be written in the form (according to Wiki "Proper time"): $ d\tau = \sqrt{g_{00}(x)}dt \; ? $
Thank you in advance for your answers
1)
I believe this can be done always, but I am not sure. You need to get rid of the cross terms $g_{0a}$ and you have 4 coordinate transformations at your disposal to get rid of the 3 metric functions, while keeping $g_{00}$ positive (assuming the signature is (+,-,-,-)). This looks like a problem that has a solution.
2)
It cannot. The proper time is always attached to some worldline. For any worldline whatsoever, the general formula is: $$d\tau=c^{-1}ds=c^{-1}\sqrt{g_{\mu\nu}dx^\mu dx^\nu}$$ This reduces to your formula only along worldlines that keep $x^1$,$x^2$ and $x^3$ constant.
