Some “No Hole” Spacetime Properties are Unstable

Abstract

We show a sense in which the spacetime property of effective completenessa type of "local hole-freeness" or "local inextendibility"-is not stable.

Some Other “No Hole” Spacetimes Properties Are Unstable Too

“…These two observations (in conjunction with the crucial observation of [1] concerning neighborhoods of null compactified Minkowski spacetime in the F -topology) imply that a geodesically complete, effectively complete, epistemically hole free (g) and not future nakedly singular spacetime can be arbitrarily close in the fine topology to a spacetime which is not effectively complete, fails to be epistemically hole free (g), and which is future nakedly singular. This is bad news.…”

“…Recently, [1] constructed an example of a spacetime (M, g ab ) (isometric to a twodimensional Minkowski spacetime compactified along one null direction) which is geodesically complete (therefore also effectively complete), and a sequence of spacetimes (M, g ab (n)) such that in every neighborhood of (M, g ab ) in the F topology there is a spacetime (M, g ab (n)) which is isometric to a portion of maximally extended Misner spacetime (which, in turn, is not effectively complete). The full details of that construction (essential in what follows) are to be found in [1]. Since the null compactified Minkowski spacetime (M, g ab ) of [1] is geodesically complete, it is not future nakedly singular.…”

“…The full details of that construction (essential in what follows) are to be found in [1]. Since the null compactified Minkowski spacetime (M, g ab ) of [1] is geodesically complete, it is not future nakedly singular. It is also easy to see that since Minkowski spacetime is epistemically hole free (g), so is Minkowski spacetime compactified along one null direction.…”

“…First, note that every (M, g ab (n)) is isometric to a portion of Misner spacetime (again, see [1] for the justification of this claim). So it is sufficient to argue that (this portion of) Misner spacetime has epistemic hole.…”

“…Recently [1] has shown that a certain "no hole" spacetime property, effective completeness, is unstable in the fine topology (so a "hole free" spacetime can be arbitrarily close to spacetimes which have "holes"). In this note I point out that the very same example can be used to show that (a) another "no hole" property, being epistemically hole free in the geodesic sense, is F -unstable, and that (b) the spacetime property of not being future nakedly singular is F -unstable as well.…”