"Semi" Synchronous coordinates

[merav]

I understand that one can always construct a set Synchronous coordinates (or Gaussian normal coordinates) on a neighborhood of a point in spacetime.

My question is:
Does one can construct a metric with only $g_{0i}=0$ such that
$dS^2=g_{00}dt^2 + g_{ij}dx^{i} dx^{j}$ (where $i=1,,,,D$ and $j=1,,,,D$) not just in a neighborhood of a point but to all spacetime in general?

In this case one should use

 

[merav]

Yes...
One can use

g'_{0i}=\frac{\partial x^{\alpha}}{\partial x'^{0}}\frac{\partial x^{\\beta}}{\partial x'^{i}}g_{\alpha\beta}=0

so it seems we have enough PDE...