nLab
future

Idea

The past of any physical event (object, system, etc) consists of everything that might (by the principles of causality?) have potentially influenced that event; while its future consists of everything that it might potentially influence.

Definitions

Let $(X,g,o)$ be a spacetime, that is a Lorentzian manifold $(X,g)$ equipped with time-orientation?. This time-orientation consists precisely of specification of which timelike and lightlike curves are future-directed (and which are complementarily past-directed).

Let $x$ be a point in this spacetime. Then:

• the future of $x$ is the subset $J^{+}(x)$ of all points of $X$ connected to $x$ by a future-directed timelike or lightlike curve starting at $x$;

• the past of $x$ is the subset $J^{-}(x)$ of all points of $X$ connected to $x$ by a future-directed timelike or lightlike curve ending at $x$.

Let $A$ be a more general subset of this spacetimes. Then:

• the future of $A$ is the subset $J^{+}(A)$ defined as the union of $J^{+}(x)$ for all $x\in A$;

• the past of $A$ is the subset $J^{-}(A)$ defined as the union of $J^{-}(x)$ for all $x\in A$.

Properties

A Cauchy surface $\Sigma$ in $(X,g)$ is a minimal subset of $X$ with the property that $X$ is the union of the future and past of $\Sigma$. (Does this suffice to define Cauchy surfaces in the case of a Lorentzian manifold that admits a time-orientation?)

The operations $J^{+}$ and $J^{-}$ are (separately) Moore closures on the power set of $X$. Stated explicitly (for $J^{+}$):

• the future of $A$ contains $A$;
• the future of the future of $A$ is simply the future of $A$;
• if $A$ contains $B$, then the future of $A$ contains the future of $B$.

Variations

Sometimes one wants to remove $x$ itself from $J^{+}(x)$ and $J^{-}(x)$ (or more precisely, to include $x$ only in the case of a closed timelike curve through $x$). However, the operations $J^{+}$ and $J^{-}$ are not quite as mathematically well-behaved in this case. (Note that $J^{+}(A)$ may still intersect $A$, or even contain all of $A$, even in the absence of closed timelike curves.)