Global properties of physically interesting Lorentzian spacetimes

TL;DR

This paper explores the global topological and geometric properties of Lorentzian spacetimes, emphasizing their physical implications and advocating for a focus on structures like tetrads and complex structures over the metric alone.

Contribution

It revisits and emphasizes the importance of global properties such as orientability, parallelizability, and complex structures in Lorentzian spacetimes from an empirical physics perspective.

Findings

Spacetime should be space-, time-, and spacetime-orientable.

Spacetime should possess a globally defined tetrad or vielbein.

A globally-defined almost complex structure is nearly unavoidable.

Abstract

Under normal circumstances most members of the general relativity community focus almost exclusively on the local properties of spacetime, such as the locally Euclidean structure of the manifold and the Lorentzian signature of the metric tensor. When combined with the classical Einstein field equations this gives an extremely successful empirical model of classical gravity and classical matter --- at least as long as one does not ask too many awkward questions about global issues, (such as global topology and global causal structure). We feel however that this is a tactical error --- even without invoking full-fledged "quantum gravity" we know that the standard model of particle physics is also an extremely good representation of some parts of empirical reality; and we had better be able to carry over all the good features of the standard model of particle physics --- at least into the realm of semi-classical quantum gravity. Doing so gives us some interesting global features that spacetime should possess: On physical grounds spacetime should be space-orientable, time-orientable, and spacetime-orientable, and it should possess a globally defined tetrad (vierbein, or in general a globally defined vielbein/n-bein). So on physical grounds spacetime should be parallelizable. This strongly suggests that the metric is not the fundamental physical quantity; a very good case can be made for the tetrad being more fundamental than the metric. Furthermore, a globally-defined "almost complex structure" is almost unavoidable. Ideas along these lines have previously been mooted, but much is buried in the pre-arXiv literature and is either forgotten or inaccessible. We shall revisit these ideas taking a perspective very much based on empirical physical observation.