Definition
Time orientation
A continuous choice of one component of the timelike cone on a Lorentzian manifold.
Definition
A time orientation on a Lorentzian manifold is a continuous choice, at every point, of one of the two connected components of the timelike cone. Equivalently, it is represented by a continuous timelike vector field; timelike vectors in the chosen component are future-directed, as are causal vectors in its closure. A manifold admitting such a choice is time-orientable.
For Minkowski space with
the standard choice declares a timelike vector future-directed when . Its closure contains the future-directed null cone and determines the direction used by future-directed causal curves.
Transformations
A Lorentz transformation is orthochronous when it preserves the chosen future cone. This property is independent of spatial orientation. Requiring both determinant and preservation of time orientation selects the proper orthochronous Lorentz group.
Time orientation is additional structure: orientability of the underlying manifold does not imply time-orientability, nor conversely.
References
- Barrett O'Neill, Semi-Riemannian Geometry With Applications to Relativity, Academic Press, 1983, §5. Publisher record.
- Robert M. Wald, General Relativity, University of Chicago Press, 1984, §8.1. Publisher record.