  Thus, if the operation of a Thornian time machine is to be a live  possibility, some condition weaker than causal determinism must be  used to capture the sense in which the state on \(\Sigma_0\)  can be deemed to be responsible for the subsequent development of  CTCs. Given the failure of causal determinism, it seems the next best thing to demand that the region \(V\) is “adjacent”  to the future domain of dependence  \(D^+ (\Sigma_0)\). Here is an initial stab  at such an adjacency condition. Consider causal curves which have a  future endpoint in the time travel region \(V\) and no past  endpoint. Such a curve may never leave \(V\); but if it does,  require that it intersects \(\Sigma_0\). But this requirement  is too strong because it rules out Thornian time machines altogether. For a curve of the type in question to reach \(\Sigma_0\) it  must intersect \(H^+ (\Sigma_0)\), but once  it reaches \(H^+ (\Sigma_0)\) it can be  continued endlessly into the past without meeting \(\Sigma_0\) because the generators of \(H^+ (\Sigma_0)\) are past endless null geodesics that never meet   \(\Sigma_0\).[14]  This difficulty can be overcome by weakening the requirement at issue by rephrasing it in terms of timelike curves rather than causal  curves. Now the set of candidate time machine spacetimes satisfying  the weakened requirement is non-empty—as illustrated, once  again, by the spacetime of   Figure 1.  But the weakened requirement is too weak, as illustrated by the \((1 +  1)\)-dimensional version of Deutsch-Politzer   spacetime[15]  (see   Figure 2),  which is constructed from two-dimensional Minkowski spacetime by  deleting the points  \(p_1\)–\(p_4\) and then gluing  together the strips as shown. Every past endless timelike curve that  emerges from the time travel region \(V\) of Deutsch-Politzer  spacetime does meet \(\Sigma_0\). But this spacetime is not a  plausible candidate for a time machine spacetime. Up to and including the time \(\Sigma_0\) (which can be placed as close to  \(V\) as desired) this spacetime is identical with empty  Minkowski spacetime. If the state of the corresponding portion of  Minkowski spacetime is not responsible for the development of  CTCs—and it certainly is not since there are no CTCs in  Minkowski spacetime—how can the state on the portion of  Deutsch-Politzer spacetime up to and including the time  \(\Sigma_0\) be held responsible for the CTCs that appear in  the future?