nLab causal cone

Redirected from "future causal cone".
Contents

Contents

1. 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.

3. 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 12:16:27. See the history of this page for a list of all contributions to it.