nLab causal cone

Idea

Given a time-oriented Lorentzian manifold $\Sigma$, then for a pont $x \in \Sigma$

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

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

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

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

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

Given a subset $S \subset X$, 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.

Related concepts

• causal locality

• causal additivity

• causal complement, causal closure

• causal perturbation theory

• conal manifold

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)