Time and Space separation in General Relativity

Abstract: Let $(M,g)$ be a spacetime. That is, $M$ is a real manifold of dimension $4$ equipped with a Lorentzian metric $g$. We show that any separation of time and space in $M$ is equivalent to introducing a (non-smooth) Riemann metric $h$. If $h$ is smooth, it induces a smooth line bundle $T_p\rightarrow M$, whose any fiber is generated by a time-like vector, called the time bundle. Whether $(M,g,h)$ is time orientable or not corresponds to whether this line bundle is trivial or not. As well-known, the last condition is characterized by the first Stiefel-Whitney class $w^1(T_p)\in H^1(M,\mathbb{Z}/2)$. We then define a partial time orientation of $M$ as a section of the line bundle $T\rightarrow M$. As applications, we discuss time and space differentiations on $M$.