I am wondering whether the region $H:=\{(t,x) : x^2-t^2<1\}$ of $(1+1)$-dimensional Minkowski spacetime, equipped with the restriction $g_H$ of the standard Minkowski metric $g=-\mathrm{d}t \otimes \mathrm{d}t + \mathrm{d}x \otimes \mathrm{d}x$, is globally conformally equivalent to the vertical strip $S:=\{(t,x):|x|<1\}$, again equipped with the restriction $g_S$ of the Minkowski metric. To further clarify: by "globally conformally equivalent" I mean that there should be a diffeomorphism $\phi:H \to S$ with $\phi_*(g_H)=\Omega^2 g_S$ for some smooth, strictly positive function $\Omega$.
I have the feeling that there might be an "obvious" reason why this is false, but the proof is escaping me at the moment. Thanks for your help.