Cauchy surface

Intuitively, a Cauchy surface is a plane in space-time which is like an instant of time; its significance is that giving the initial conditions on this plane determines the future (and the past) uniquely. Intuitively, a Cauchy surface is a plane in space-time which is like an instant of time; its significance is that giving the initial conditions on this plane determines the future (and the past) uniquely. More precisely, a Cauchy surface is any subset of space-time which is intersected by every inextensible, non-spacelike (i.e. causal) curve exactly once. A partial Cauchy surface is a hypersurface which is intersected by any causal curve at most once. It is named for French mathematician Augustin Louis Cauchy. Given a Lorentzian manifold M {displaystyle {mathcal {M}}} , if S {displaystyle {mathcal {S}}} is a space-like surface (i.e., a collection of points such that every pair is space-like separated), then D + ( S ) {displaystyle D^{+}({mathcal {S}})} is the future of S {displaystyle {mathcal {S}}} , i. e.: D + ( S ) := { p ∈ M     such that every inextensible, past-directed, non-spacelike curve through  p  intersects  S } {displaystyle D^{+}({mathcal {S}}):={pin {mathcal {M}} { ext{such that every inextensible, past-directed, non-spacelike curve through }}p{ ext{ intersects }}{mathcal {S}}}} Similarly D − ( S ) {displaystyle D^{-}({mathcal {S}})} , the past of S {displaystyle {mathcal {S}}} , is the same thing going forward in time. When there are no closed timelike curves, D + {displaystyle D^{+}} and D − {displaystyle D^{-}} are two different regions. When the time dimension closes up on itself everywhere so that it makes a circle, the future and the past of S {displaystyle {mathcal {S}}} are the same and both include S {displaystyle {mathcal {S}}} . The Cauchy surface is defined rigorously in terms of intersections with inextensible curves in order to deal with this case of circular time. An inextensible curve is a curve with no ends: either it goes on forever, remaining timelike or null, or it closes in on itself to make a circle, a closed non-spacelike curve.