  The issue that must be faced now is what further conditions must be  imposed in order that the appearance of CTCs to the future of  \(\Sigma_0\) can be attributed to the operation of a time  machine. Not surprisingly, the answer depends not just on the  structure of the spacetime at issue but also on the physical laws  that govern the evolution of the spacetime structure. If one adopts  the attitude that the label “time machine” is to be  reserved for devices that operate within a finite spatial range for a finite stretch of time, then one will want to impose requirements to  assure that what happens in a compact region of spacetime lying on or to the future of \(\Sigma_0\) is responsible for the CTCs. Or  one could be more liberal and allow the would-be time machine to be  spread over an infinite space. We will adopt the more liberal stance  since it avoids various complications while still sufficing to elicit key points. Again, one could reserve the label “time  machine” for devices that manipulate concentrations of  mass-energy in some specified ways. For example, based on Gödel  spacetime—where matter is everywhere rotating and a CTC passes  through every spacetime point—one might conjecture that setting into sufficiently rapid rotation a finite mass concentration of  appropriate shape will eventuate in CTCs. But with the goal in mind  of proving negative general results, it is better to proceed in a  more abstract fashion. Think of the conditions on the partial Cauchy  surface \(\Sigma_0\) as encoding the instructions for the  operation of the time machine. The details of the operation of the  device—whether it operates in a finite region of spacetime,  whether it operates by setting matter into rotation, etc.—can  be left to the side. What must be addressed, however, is whether the  processes that evolve from the state on \(\Sigma_0\) can be  deemed to be responsible for the subsequent appearance of CTCs.