nLab
causal cone

Contents

Idea

Given a time-oriented Lorentzian manifold Σ\Sigma, then for a pont xΣx \in \Sigma

  1. its open future cone is the set of all points yy distinct from xx such that there is a future-directed time-like curve from xx to yy;

  2. its closed future cone is the set of all points yy such that there is a future-directed time-like or light-like curve from xx to yy;

  3. its open past cone is the set of all points yy distinct from xx such that there is a past-directed time-like curve from xx to yy;

  4. its closed past cone is the set of all points yy such that there is a past-directed time-like or light-like curve from xx to yy.

The boundary of the union of the past and future closed cone is the light cone of the point.

Given a subset SXS \subset X, then its future/past open/closed cone is the union of that of all its points. The open cones above are conical spaces.

The complement of the (closed) causal cone is the causal complement.

References

  • Christian Bär, section 1 of Green-hyperbolic operators on globally hyperbolic spacetimes, Communications in Mathematical Physics 333, 1585-1615 (2014) (doi, arXiv:1310.0738)

Last revised on August 1, 2018 at 08:16:27. See the history of this page for a list of all contributions to it.