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