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.
Let be a spacetime, that is a Lorentzian manifold 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 be a point in this spacetime. Then:
Let be a more general subset of this spacetimes. Then:
the future of is the subset defined as the union of for all ;
the past of is the subset defined as the union of for all .
A Cauchy surface in is a minimal subset of with the property that is the union of the future and past of . (Does this suffice to define Cauchy surfaces in the case of a Lorentzian manifold that admits a time-orientation?)
Sometimes one wants to remove itself from and (or more precisely, to include only in the case of a closed timelike curve through ). However, the operations and are not quite as mathematically well-behaved in this case. (Note that may still intersect , or even contain all of , even in the absence of closed timelike curves.)