nLab future cone

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 $y$.

Related concepts

• light cone

• conal manifold