Hello everybody,
I just realized that I was talking nonsense a week ago, claiming the projective metric would be given by
$$\cosh d_{pr} (P,Q) = \frac{1}{2} \ln \cr(Q,X,P,Y)$$.
The correct formula is
$$d_{pr} (P,Q) = \frac{1}{2} \ln \cr(Q,X,P,Y)$$.
I hope you realized that for yourself in the meantime 😉
Have a nice week-end,
Emanuel