the following line element defines the Kahn-Penrose metric with coordinates $(u,v,x,y)$ and constraints $u \geq 0$, $v < 0$
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