Kahn-Penrose metric the following line element defines the Kahn-Penrose metric with coordinates $(u,v,x,y)$ and constraints  $u \geq 0$, $v < 0$  
$$ds^2=-2dudv+(1-u)^2dx^2+(1+u)^2dy^2$$
If we restrict ourselves to the following convention, in which the Minkowski metric $\eta$ is mostly plus $(-,+++)$ and $ds^2>0$ for space like intervals.
The reasoning goes as follows: in order to determine the nature of a coordinate, we produce some small but otherwise arbitrary variation of that coordinate while keeping the rest constant (which can be done if we assign algebraically the valor $0$ to the $x^μ$ not being varied. Then the sign of $ds^2$ will tell the space-time-null character of the coordinate in question.
My question is if the following statements are correct:


*

*$x$ and $y$ are space like coordinates.

*$v$ is a space like coordinate

*$u$ is a time like coordinate

 A: The rule is : With a diagonal metrics, the character of the coordinate $x^i$, is given by the sign of $g_{ii}$ 
Setting $t= \frac{u + v}{2}$ and $z= \frac{u - v}{2}$, your metrics becomes (with the constraint $z >0$) : 
$$ds^2= -dt^2 + dz^2+(1-t-z)^2dx^2+(1+t+z)^2dy^2$$
So, the character of $t$ as a time-like coordinate and $z$ as a space-like coordinate (with the constraint $z >0$) appears clearly. And $x$ and $y$ are space-like coordinates too.
The variables $u = t  + z$ and $v = t - z$ are called light-cone coordinates, you could say that they are light-like coordinates. They are very often used in general relativity and string theory. They are also called null coordinates, because for instance, if $dx=dy=0$, then you have $ds^2 = - 2dudv$, so each particle with $U=$Constant or $V=$Constant corresponds to $ds^2=0$, that is a light-like interval (a light ray).
But you cannot say that one of the coordinates $u, v$ is a time-like coordinate, and the other a space-like coordinate.
In Schwarzchild metric, it is not correct to say that $r$ and $t$ could change their sign. The Schwarzchild metric describes a gravitational field only for $r>r_S$. It is a limited description (which does not cover the entire manifold), and you have to use Kruskal-Szekeres coordinates 
to have a glbal view of the black hole. Mathematically, "These coordinates have the advantage that they cover the entire spacetime manifold of the maximally extended Schwarzschild solution and are well-behaved everywhere outside the physical singularity."
