 Minkowski simultaneity relative to an inertial observer

In Minkowski spacetime of metric signature $(-,+,+,+)$, let $u$ be a constant future-directed four-velocity with $\eta (u,u)=-c^2$. The time difference assigned to two spacetime events is $\Delta t_u=-\eta (u,Q-P)/c^2$. Thus the observer's simultaneous events form an affine hyperplane orthogonal to $u$. This is an invariant relation of the three arguments $u,P,Q$; the Minkowski metric alone does not single out one $u$. For fixed $u$, it defines an equivalence relation and the hyperplanes form a foliation. Changing $u$ changes this foliation.