  The most obvious move is to construe “responsible for” in the sense of causal determinism. But in the present setting this move quickly runs into a dead end. For if CTCs exist to the future of  \(\Sigma_0\) they are not causally determined by the state on  \(\Sigma_0\) since the time travel region \(V\), if it is non-null, lies outside the future domain of dependence  \(D^+ (\Sigma_0)\) of \(\Sigma_0\),  the portion of spacetime where the field equations of relativistic  physics uniquely determine the state of things from the state on   \(\Sigma_0\).[12]  The point is illustrated by the toy model of   Figure 1.  The surface labeled \(H^+ (\Sigma_0)\) is  called the future Cauchy horizon of \(\Sigma_0\). It  is the future boundary of   \(D^+ (\Sigma_0)\),[13]  and it separates the portion of spacetime where conditions are  causally determined by the state on \(\Sigma_0\) from the  portion where conditions are not so determined. And, as advertised,  the CTCs in the model of   Figure 1  lie beyond \(H^+ (\Sigma_0)\).