Because the virtual photons between the plates can have only the resonant wavelengths, there will be slightly fewer of them than in the region outside the plates where virtual photons can have any wavelength. Thus there will be slightly fewer virtual photons hitting the inside surfaces of the plates than the outside surfaces. One would therefore expect a force on the plates, pushing them toward each other. This force has actually been detected and has the predicted value. Thus we have experimental evidence that virtual particles exist and have real effects.
The fact that there are fewer virtual photons between the plates means that their energy density will be less than elsewhere. But the total energy density in “empty” space far away from the plates must be zero, because otherwise the energy density would warp the space and it would not be almost flat. So, if the energy density between the plates is less than the energy density far away, it must be negative.
We thus have experimental evidence both that space-time can be warped (from the bending of light during eclipses) and that it can be curved in the way necessary to allow time travel (from the Casimir effect). One might hope therefore that as we advance in science and technology, we would eventually manage to build a time machine. But if so, why hasn’t anyone come back from the future and told us how to do it? There might be good reasons why it would be unwise to give us the secret of time travel at our present primitive state of development, but unless human nature changes radically, it is difficult to believe that some visitor from the future wouldn’t spill the beans. Of course, some people would claim that sightings of UFOs are evidence that we are being visited either by aliens or by people from the future. (If the aliens were to get here in reasonable time, they would need faster-than-light travel, so the two possibilities may be equivalent.)
However, I think that any visit by aliens or people from the future would be much more obvious and, probably, much more unpleasant. If they are going to reveal themselves at all, why do so only to those who are not regarded as reliable witnesses? If they are trying to warn us of some great danger, they are not being very effective.
A possible way to explain the absence of visitors from the future would be to say that the past is fixed because we have observed it and seen that it does not have the kind of warping needed to allow travel back from the future. On the other hand, the future is unknown and open, so it might well have the curvature required. This would mean that any time travel would be confined to the future. There would be no chance of Captain Kirk and the Starship Enterprise turning up at the present time.
This might explain why we have not yet been overrun by tourists from the future, but it would not avoid the problems that would arise if one were able to go back and change history. Suppose, for example, you went back and killed your great-great-grandfather while he was still a child. There are many versions of this paradox but they are essentially equivalent: one would get contradictions if one were free to change the past.
There seem to be two possible resolutions to the paradoxes posed by time travel. One I shall call the consistent histories approach. It says that even if space-time is warped so that it would be possible to travel into the past, what happens in space-time must be a consistent solution of the laws of physics. According to this viewpoint, you could not go back in time unless history showed that you had already arrived in the past and, while there, had not killed your great-great-grandfather or committed any other acts that would conflict with your current situation in the present. Moreover, when you did go back, you wouldn’t be able to change recorded history. That means you wouldn’t have free will to do what you wanted. Of course, one could say that free will is an illusion anyway. If there really is a complete unified theory that governs everything, it presumably also determines your actions. But it does so in a way that is impossible to calculate for an organism that is as complicated as a human being. The reason we say that humans have free will is because we can’t predict what they will do. However, if the human then goes off in a rocket ship and comes back before he or she set off, we will be able to predict what he or she will do because it will be part of recorded history. Thus, in that situation, the time traveler would have no free will.