nLab
future

Context

Riemannian geometry

Gravity

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 September 6, 2017 03:31:34 by Urs Schreiber (77.56.177.247)