  This no-go result does not touch the situation illustrated in   Figure 1.  Taub-NUT spacetime is a vacuum solution to Einstein’s field equations so the weak energy condition is automatically satisfied, and  \(H^+ (\Sigma_0)\) is compact and, a  fortiori, compactly generated. Hawking’s theorem is not contradicted  since \(\Sigma_0\) is compact. By the same token the theorem  does not speak to the possibility of operating a Thornian time  machine in a spatially closed universe. To help fill the gap, Hawking proved that when \(\Sigma_0\) is compact and  \(H^+ (\Sigma_0)\) is compactly generated,  the Einstein field equations and the weak energy condition together  guarantee that both the convergence and shear of the null geodesic  generators of \(H^+ (\Sigma_0)\) must  vanish, which he interpreted to imply that no observers can cross  over \(H^+ (\Sigma_0)\) to enter the  chronology violating region \(V\). But rather than showing that  it is physically impossible to operate a Thornian time machine in a  closed universe, this result shows only that, given the correctness  of Hawking’s interpretation, the observers who operate the time  machine cannot take advantage of the CTCs it produces.