# nLab future cone

The future cone

### Context

#### Riemannian geometry

Riemannian geometry

## Applications

#### Gravity

gravity, supergravity

# 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 $(X,x)$ equipped with a time orientation then

• the future cone at $x \in X$ is the subspace $V_x^+ \subset T_x X$ of the tangent space of $X$ consisting of all those tangent vectors which are future pointing;

• 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.

For more see at causal cone.

## 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 $y$.

Last revised on November 10, 2017 at 16:45:05. See the history of this page for a list of all contributions to it.