Time and Space separation in General Relativity

TL;DR

This paper explores the mathematical relationship between separating time and space in general relativity and the role of line bundles and orientation in this context.

Contribution

It establishes an equivalence between time-space separation and the introduction of a (possibly non-smooth) Riemann metric, linking orientability to topological invariants.

Findings

Time-space separation corresponds to a Riemann metric h.

Time orientability relates to the triviality of a line bundle.

Defines a partial time orientation as a section of the line bundle.

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 $M$.