# nLab causal cone

### Context

#### Riemannian geometry

Riemannian geometry

## Basic definitions

• Riemannian manifold

• moduli space of Riemannian metrics

• pseudo-Riemannian manifold

• geodesic

• Levi-Civita connection

• ## Theorems

• Poincaré conjecture-theorem
• ## Applications

• gravity

• # Contents

## 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.

## 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.