Timelike

Conformal geometry

Looking at the definitions of which tangent vectors are timelike, null and spacelike we see they remain unchanged if we use ${\displaystyle \,g}$ or ${\displaystyle {\hat {g}}}$. As an example suppose ${\displaystyle X}$ is a timelike tangent vector with respect to the ${\displaystyle \,g}$ metric. This means that ${\displaystyle \,g(X,X)<0}$. We then have that ${\displaystyle {\hat {g}}(X,X)=\Omega ^{2}g(X,X)<0}$ so ${\displaystyle X}$ is a timelike tangent vector with respect to the ${\displaystyle {\hat {g}}}$ too.