Global conformally flat coordinates in 2d spacetimes Let $(M,g)$ be a 2 dimensional pseudo-Riemannian manifold that is topologically a disc. Is it possible to construct a global coordinate system in which the metric is conformally flat? I.e. coordinates $(t,x)$ which cover the whole manifold such that the line element takes the form
$ds^2=\Omega^2(t,x)(-dt^2 + dx^2)$
for some conformal factor $\Omega$.

This is an old question but it deserves a correct answer. As it turns out, the open 2-dimensional disk admits continuum of conformally inequivalent pseudo Riemannian metrics, see 
Uncountably many $C^0$ conformally distinct Lorentz surfaces and a finiteness theorem, by Robert W. Smyth, Proc. Amer. Math. Soc. 124 (1996), 1559-1566.  
Edit. Furthermore, there are Lorentz metrics on the open 2-disk which do not embed conformally in the Lorentzian plane, see p. 117 of 
T. Weinstein, An Introduction to Lorentz Surfaces, de Gruyter, 1996.
Last thing: In his answer Luboš Motl  confused the Riemannian and pseudo Riemannian cases.
