nLab
future

Past and 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)(X,g,o) be a spacetime, that is a Lorentzian manifold (X,g)(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 xx be a point in this spacetime. Then:

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

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

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

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

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

Properties

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

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

  • the future of AA contains AA;
  • the future of the future of AA is simply the future of AA;
  • if AA contains BB, then the future of AA contains the future of BB.

Variations

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

Revised on July 9, 2011 23:01:05 by Toby Bartels (76.85.192.183)