Likewise, in our own spacetime every inextendible, past-directed timelike path is incomplete (and our spacetime is singular): they all “run into the Big Bang.” ❋ Curiel, Erik (2009)
Force of analogy suggests that one define a spacetime to have points missing from it if and only if it contains incomplete, inextendible paths, and then try to use these incomplete paths to construct in some fashion or other new, properly situated points for the spacetime, the addition of which will make the previously inextendible paths extendible. ❋ Curiel, Erik (2009)
However, we need some way of overcoming the fact that non-singular spacetimes include inextendible paths of finite proper length. ❋ Curiel, Erik (2009)
This would follow because, even in flat spacetime, there are timelike paths with unbounded acceleration which have only a finite proper length (proper time, in this case) and are also inextendible. ❋ Curiel, Erik (2009)
While the point of this project may seem at bottom identical to the path incompleteness account discussed in insofar as singular structure will be defined by the presence of incomplete, inextendible paths, there is a crucial semantic and logical difference between the two. ❋ Curiel, Erik (2009)
A maximal spacetime is singular if and only if it contains an inextendible path of finite generalized affine length. ❋ Curiel, Erik (2009)
One need not consider anything so outré as incomplete, inextendible paths, though, in order to produce examples of entities that seem undeniably to exist in some sense of the term or other, and yet which cannot have any even vaguely determined location in time and space predicated of them. ❋ Curiel, Erik (2009)
In this context, an incomplete path in spacetime is one that is both inextendible and of finite proper length, which means that any particle or observer traversing the path would experience only a finite interval of existence that in principle cannot be continued any longer. ❋ Curiel, Erik (2009)
The most obvious option is to define a spacetime as singular if and only if it contains incomplete, inextendible timelike geodesics, i.e., paths representing the trajectories of inertial observers, those in free-fall experiencing no acceleration “other than that due to gravity.” ❋ Curiel, Erik (2009)
That the paths be incomplete and inextendible means, roughly speaking, that, after a finite amount of time, a particle or observer following that path would “run out of world,” as it were ” it would hurtle into the tear in the fabric of spacetime and vanish. ❋ Curiel, Erik (2009)
For example, Geroch (1968) demonstrates that a spacetime can be geodesically complete and yet possess an incomplete timelike path of bounded total acceleration ” that is to say, an inextendible path in spacetime traversable by a rocket with a finite amount of fuel, along which an observer could experience only a finite amount of proper time. ❋ Curiel, Erik (2009)
