Even the Minkowski space is holed

To cure the lack of predictive power of general relativity, Geroch proposed to complete the theory with an additional postulate that only “hole-free” spacetimes are permitted. This postulate (or, rather, that obtained from it by some disambiguating) seems physically well-founded and at the same time appropriately restrictive. I show, however, that it is too strong—it prohibits even the Minkowski space.

Figure 1

(a) The spacetime is obtained by cutting out the shadowed sector from the Minkowski space and gluing together the rays bounding the sector (the rays do not comprise the vertex of the angle). (b) Remove a disk $D$ from a domain $N$, take a copy of this incised spacetime, and identify the upper bank of either cut with the lower bank of the other. If $N$ and $N^{\prime }$ are different spacetimes, the result is a double covering of $N-S$, where $S=\mathrm{Bd}D$ is a sphere of codimension 2. If they are just different regions of the same spacetime, then, depending on the position of the disks, the resulting spacetime is either a time machine [10] or a wormhole [11].

Figure 2

(a) The light-gray area $U$ is a neighborhood of the dark-gray area $D^{+}(\mathcal{S})$. The intersection of $U$ with the hatched strip is $O_2$. (b) The light-gray area is $M^{\prime }$—an extension of $U$ alternative to the Minkowski plane.