The vector field $Z\colon=E^0+x_2E^1$ is a time like vector field on the subset $U\colon=\{(x_0,x_1,x_2):|x_2| < 1\}$ of Minkowski space $\R_1^3$. Show that there is no function $f:U\rar\R$ such that $Z$ is normal to the submanifold $[f=0]$.
$Z$ and $\nabla f=-\pa_0f\,E^0+\pa_1f\,E^1+\pa_2f\,E^2$ must be parallel, i.e. for some function $g$:
$$
g=-\pa_0f,\quad
gx_2=\pa_1f,\quad
0=\pa_2f~.
$$
The third equation implies that $f$ does not depend on $x_2$; the first equation then implies that $g$ does not depend on $x_2$ either; finally the second equation shows that $g=\pa_2(gx_2)=\pa_2\pa_1f=0$.