# The future cone

## Idea

The future cone of a point $x$ is the set consisting of all points in the future of $x$. (We usually interpret this condition weakly, so that $x$ itself belongs to its own future cone.)

Perhaps the terms forward cone resp. backward cone might be used as synonyms for the concept.

The dual concept is the past cone.

## In relativity theory

Given a Lorentzian manifold equipped with a global choice of which timelike curves are future-directed, the future of $x$ consists of all points $y$ such that there exists a future-directed curve (which may be timelike, lightlike, or a mixture) from $x$ to $y$.

If we may put a global time coordinate $t$ on the manifold (which is a stronger condition), then the future cone of $x$ consists of all points $y$ such that $t(y) \geq t(x)$ and $x$ and $y$ are not space-like separated.

## In directed homotopy theory

In the context of directed homotopy theory the future cone of a point, $x$, in a directed topological space is, more-or-less, the directed subspace of those points $y$ which are greater than $x$. In case the space might not be that ‘nice’ or, for instance, we have a local partial order rather than a global one, it is preferable to say that it is the directed space of points, $y$, for which there is a distinguished path from $x$ to $y$.

Revised on October 30, 2013 22:23:16 by Urs Schreiber (82.169.114.243)