  A more subtle way to exclude Deutsch-Politzer spacetime focuses on  the generators of \(H^+ (\Sigma_0)\). The  stipulations laid down so far for Thornian time machines imply that  the generators of \(H^+ (\Sigma_0)\) cannot  intersect \(\Sigma_0\). But in addition it can be required  that these generators do not “emerge from a singularity”  and do not “come from infinity,” and this would suffice  to rule out Deutsch-Politzer spacetime and its conformal cousins as  legitimate candidates for time machine spacetimes. More precisely, we can impose what Stephen Hawking (1992a,b) calls the requirement that  \(H^+ (\Sigma_0)\) be compactly  generated; namely, the past endless null geodesics that generate \(H^+ (\Sigma_0)\) must, if extended far  enough into past, fall into and remain in a compact subset of  spacetime. Obviously the spacetime of   Figure 1  fulfills Hawking’s requirement—since in this case  \(H^+ (\Sigma_0)\) is itself  compact—but just as obviously the spacetime of   Figure 2  (conformally doctored or not) does not.