Recently, the media described a paper by
time machine production at the LHC have the same scaling laws for the cross sections, sizes, and energies as black hole production.
The authors know that it remains very unlikely that a macroscopic time machine could be created or constructed in a foreseeable future. I would certainly bet against such a science-fiction scenario, too. On the other hand, there are people on both extreme sides in this debate. Some people think that time machines are certainly a normal thing and it is just a matter of patience or money to create them. Others think that everything in science that remotely looks like a time machine is an unscientific metaphysical fairy-tale.
Both groups are wrong. There exist strong constraints that probably make many kinds of time machines impossible. At the same moment, there also exist constraints that make it difficult if not impossible to prohibit all exotic phenomena that are used in various constructions of time travel.
Arguments for and hopes: solutions to general relativity
Time machines look bizarre and no one would think about them seriously in the context of the flat spacetime as included in Newtonian physics. However, Einstein's general relativity has made spacetime flexible. Quite suddenly, pieces of spacetime could be glued together in new ways. One can think about wormholes, tubes behaving as shortcuts connecting distant places in space and perhaps time.
They have a long history. Kurt Gödel was the first person to discuss a spacetime in general relativity with closed time-like curves. His rotating Universe was so captivating that Einstein, being convinced that time machines are unacceptably crazy, started to doubt his own general relativity. In string theory, some Gödel-like solutions were shown to be T-dual (equivalent) to Penrose's pp-waves: search for "Hořava" below.
Other physicists have added their solutions later: van Stockum and Tipler; Kerr and Newman; Gott; Morris-Thorne; Ori.
Once again, what makes time machines worth considering is that there exist wormhole-like solutions that satisfy the equations and rules of general relativity (at least some of the rules) in each region. The only unusual aspect of the picture is the way how the regions are glued together.
But locality is one of the principles underlying general relativity. If individual regions (much like individual people) obey the relevant laws locally, nothing can prevent them from being combined together. There arguably exists no "big brother" who would monitor the global properties of the Universe and who could "overrule" the local laws to prevent the Universe from "something bad".
Such an "overruling" would macroscopically violate the known local laws. It would imply that the small pieces of spacetime can't behave independently of others. Locality would be dead. And physicists simply believe that locality can only be violated "infinitesimally", not "macroscopically", and whenever it occurs, new laws and their reconciliation with the old local laws must be written down.
Let me re-emphasize that the previous paragraph is an argument why the exotic solutions to general relativity cannot be thrown away immediately.
Another obstacle is that these solutions might exist but the laws of physics could prevent any kind of time evolution that would lead to a Universe similar to these exotic solutions. If there exists this kind of insurance, one should explain what are exactly the features of spacetime that the evolution prohibits and how this ban is enforced.
There exist many proposals and so far, none of them is convincing enough to be generally accepted. It is very likely that quantum gravity or string theory has some "upper bound" on the most exotic or pathological phenomenon that resembles time machines but the question is what the "upper bound" is and how Nature manages to enforce it.
There exists another general argument in favor of wormholes and perhaps even time machines: the topology of space can change, after all, as can be demonstrated by completely well-defined, smooth, and continuous calculations in string theory. But it doesn't necessarily mean that it is becoming too easy to construct closed time-like curves.
Incidentally, one of the most universal constraints that applies to all cases of time travel that are at least slightly scientific is that we won't be allowed to travel anywhere to the past - just to the moment when the first time machine was created or later moments. You cannot rewrite the history: if there existed no time machine before 1492, you won't ever be able to see those times.
Arguments against: chronology, causality, energy conditions
Some of the wormholes and time machines require the energy density to be negative or they need similar unusual circumstances. But you should realize that the energy conditions (various generalizations of the statement that the energy density can't be negative) are as misunderstood as the criteria to eliminate time machines. Energy conditions may be formulated in various ways and they sound plausible and have reasonable consequences. However, there exist pretty good examples showing that many of them are probably not universally valid. The subject is not completely understood.
The least controversial energy condition is the so-called null energy condition. I am ready to believe that a construction that relies on a violation of the null energy condition is unphysical. In fact, I personally believe that even constructions where a violation of the dominant energy condition or strong energy condition is essential for the whole construction are unphysical, too. Note that the previous statement doesn't quite imply that the dominant and strong energy conditions must be universally valid. I just tend to believe that they are effectively valid for all qualitative considerations involving wormholes and time machines.
Chronology should eventually be protected. It means that it should always be possible to insist on a chronology i.e. to cut spacetime into slices that can be ordered and where no event on a "future" slice can be a cause of an event on a "past" slice. The principle that an event can only be influenced by the events in the past and not by those in the future is known as causality. It has been believed to be an important principle for many centuries. Special relativity made this principle more constraining because causes of an event "A" not only have to occur in the past: they have to belong to the past light cone of "A". Signals cannot propagate superluminally.
While special relativity has made the requirements of causality more constraining, general relativity has apparently made it easier to violate even the older, weaker principles of causality.
Because spacetime can get seriously warped and the notion of chronology may get curved as well, it is not clear how the evolution protects the chronology. Many authors have claimed that general relativity itself could break chronology but the effects of quantum gravity or string theory save the day.
The last decade in chronology protection
Stephen Hawking was among the first people who claimed that the laws of physics protect chronology but he couldn't say much about the internal regulations of his "Chronology Protection Agency". The beginning of the 21st century has seen dozens of stringy papers that have shed some light on the question.
For example, Boyda, Ganguli, Hořava, Varadarajan argued that string theory only protects chronology by removing a region behind a horizon from the realm describable by a holographic screen. Note that the non-gravitational, holographic dual theory on the boundary manifestly preserves chronology and you may try to "import" this feature to the gravitational bulk, too. Lisa Dyson claimed that stringy corrections modify the geometry in a similar way as "enhancons" that save us from a certain kind of naked singularities.
Dan Israel has argued that the existence of closed time-like curves is equivalent to an instability in which long strings may get a condensate; see also Huang. Caldarelli, Klemm, and Silva have claimed that the closed time-like curves are avoided because of the Pauli exclusion principle, at least in the LLM setup; see also a talk by Caldarelli.
In 2005, Costa, Herdeiro, Penedones, Sousa have offered an inherently stringy perspective on the puzzle. Strings wound around (almost) null curves in spacetime may get condensed and protect the spacetime against chronology violation. If necessary, a Hadedorn phase transition changes the spacetime completely. It is not clear to me whether they think that chronology is protected even in 11-dimensional M-theory that has no strings.
Most of these papers are intriguing and solid to some extent but they always study the question in the context of particular backgrounds that might or might not be physical. So even though most physicists think that it is hard if not impossible to create time machines that we know from science-fiction movies, the question how Nature precisely avoids the provocative solutions to Einstein's equations remains somewhat open.
But despite the science-fiction character of these debates, the questions and arguments are not only provoking but also fully scientific; only haters of physics suggest otherwise. Sometime in the future when all the relevant equations and principles are understood properly, the question will be settled much like the questions that have been settled by the Standard Model and other established theories, regardless of how easily empirically verifiable these insights are.
And that's the memo.