future cone


Riemannian geometry


The future cone


The future cone of a point xx is the set consisting of all points in the future of xx. (We usually interpret this condition weakly, so that xx 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)(X,x) equipped with a time orientation then

  • the future cone at xXx \in X is the subspace V x +T xXV_x^+ \subset T_x X of the tangent space of XX consisting of all those tangent vectors which are future pointing;

  • the future of xx consists of all points yy such that there exists a future-directed curve (which may be timelike, lightlike, or a mixture) from xx to yy.

If we may put a global time coordinate tt on the manifold (which is a stronger condition), then the future cone of xx consists of all points yy such that t(y)t(x)t(y) \geq t(x) and xx and yy 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, xx, in a directed topological space is, more-or-less, the directed subspace of those points yy which are greater than xx. 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, yy, for which there is a distinguished path from xx to yy.

Revised on September 6, 2017 03:28:08 by Urs Schreiber (