Time travel is a concept used very often in science fiction. Basically it deals with questions like: what if we could travel to any point in time in either the past or the future? How could this be achieved? What would be the implications and consequences of such an endeavor?
As I've explained in the post dealing with sublight interstellar travel, Einstein's theory of relativity shows us that travelling close to the speed of light is time travel into the future. Time passes faster for everyone else except you. The theory of relativity has also shown us that an intense gravitational field produces the same effect. The closer you are to a gravity source (a massive body) the slower time will pass for you. For example, let's say we have a black hole with an event horizon at a distance R from it's center.
The event horizon is the surface beyond which light can no longer escape the intense gravity field of the black hole. If you go beyond it, there's no turning back, but as long as you keep your distance from the event horizon the black hole is just another massive object (it doesn't pull you in any more than any other object of the same mass). If you were at a distance of 2R from the black hole and stayed there for 7 years, you would notice that about 10 years have passed on Earth. As you can see, gravitational time dilation is a tiny effect.
So, we've solved the problem of time travel into the future. The effect has been measured, and current technologies like GPS satellites have to take time dilation into account so as to keep synchronized. So far we don't have the means to send a person into the future a meaningful length of time, but as technology advances and we develop faster propulsion techniques that moment will come some time in the near future.
Now, we deal with the really big question concerning time travel: is travel into the past possible?
For starters, let's tackle this from a purely logical manner. What would happen if you could go back in time? We can immediately see that many paradoxes can arise. For example, if you were to go back in time and kill your own grandfather before he met your grandmother, then you would cease to exist. But if that happens how can you go back in time and kill your grandfather?! This is known as the grandfather paradox. Another type of paradox that arises comes from conservation of mass/energy and information, the ontological paradox. If you go back in time to a point in which you already exist there will be two of you. And if both of you travel to a previous point, there will be three of you, and so on. Essentially you could fill the Earth with these "time clones" and it would appear that all of these versions of you are coming out of nowhere, a clear violation of conservation of mass/energy. There is also the problem of information: let's say that one day you find on your desk the plans for building a time-machine. So you build it and decide to send the plans back in time to your earlier self so that he can find them. Now comes the question: who invented the time-machine? You received the schematics from the future, built the time machine and then sent them back in time for you to discover. So no one actually wrote them. So where did they come from? These are examples of violations of causality, the basic notion that a cause produces an effect.
Another problem that arises is: if time travel into the past is possible, how come we haven't seen any time travelers from the future? Well, there are many ways to answer this question. Stephen Hawking postulated the chronology protection conjecture, which basically states that the laws of physics are such that they prevent time travel (well, except for submicroscopic systems which can't affect the timeline). Essentially, that would mean that travel to the past is impossible. Another answer would be that a time machine only allows you to go back in time to a point in which the device already exists, and since no one has built a time machine yet we don't have any visitors from the future. However, even if this is true, there are still the grandfather and ontological paradoxes to be solved. Fortunately, there are two solutions available both of which also solve the problem of why we haven't been visited by time travelers.
To begin with, we have the Novikov self-consistency principle which states that if time travel is possible then all time travel events are consistent with our current history. So basically, you could go back in time but you couldn't change anything from our current history. For example if you went back in time to kill your grandfather, you could be killed by some accident before you found him. It would also mean that you yourself are part of history, since your presence in the past is part of our current history. This solution to the paradoxes of time travel is somewhat unsatisfying as it implies a form of predetermination since your actions in the past seem to be restricted by history.
The second solution, which is my personal favorite, is the existence of parallel universes. The idea is that when you go back in time you are no longer in your own universe, but in another that is the product of your travel through time. So, in this case if you go back in time to kill your grandfather, it won't be your grandfather, it will be the one from that universe. In fact you can do whatever you want because, if let's say you are from the year 2050 and go back to the year 1950, in that universe the present is 1950, and the future of 2050 hasn't happened yet and that future will be determined by your actions in that universe. The existence of parallel universes is postulated in one of the interpretations of quantum mechanics, the many-worlds interpretation, but it makes the exact same predictions as the Copenhagen interpretation, which is currently used by the scientific community, making it difficult to assert the existence of such universes. At least it means it's plausible.
Now, let's tackle this from physics' perspective: how would a time machine work and how could we build one?
The simplest time machine can be constructed using a wormhole.
As explained in the previous post, a wormhole is a hypothetical tunnel through space-time which acts like a shortcut essentially uniting two separated regions. For example you could have one end of the wormhole on Earth and the other close to the Sun. Going from Earth to the Sun the classical route, takes a long time (even light takes about 8 minutes for this trip), but stepping through the wormhole will take you there instantly. Now, from what we've learned about time dilation we can take one end (A) of the wormhole and put it on Earth in the year 2010, and accelerate the other end (B) to near light speed or place it in a powerful gravitational field. Due to time dilation, time will pass slower at end B so that it can still be 2010 at that mouth and 2050 at mouth A. Now if we were to bring mouth B to Earth (where it is the year 2050), we have a wormhole that takes you back to the year 2010. This is a type of time machine that can only send you back in time to a point in which it already exists. Since the wormhole didn't exist before 2010, you can't go to an earlier point in time. Unfortunately, as we know, creating a wormhole requires negative energy so this solution is impossible to implement for now. So what other options are there for time travel?
First of all, we need to find a proper definition for it. As we know from relativity, it is postulated that nothing that carries mass/energy or information can travel faster than the speed of light. Therefore we can construct the following space-time diagram (Minkowski diagram):
As we can see, points on the light cone are where x^2 = (ct)^2, which is equivalent to (x/t)^2 = c^2. Distance over time is velocity, so these are points that can be reached by travelling at the speed of light, hence the name light cone. Inside the cone are points where x^2 < (ct)^2 or (x/t)^2 < c^2 so these are points that can be reached by sublight travel and outside the cone we have the points which can only be reached at superluminal velocity. Keep in mind that these aren't points in space, they are points in space-time, meaning that each point has an associated space and time coordinate and so travelling from one point to another is restricted by the lightspeed limit. For example, the origin can be (Earth, now); this is a space-time point, it defines the position on Earth and time as the present. A point which would be in the future light cone would be (Earth, now + 5 minutes), and a point in the past (Earth, now - 5 minutes). If we were to travel to the Sun, a point on our world line could be (Sun, now + 5 days), and the world line would be the curve between that point and (Earth, now). This would be inside the future light cone, as travelling from Earth to the Sun in 5 days can be achieved at sublight speed. What if I wanted to go to the point (Sun, now + 5 seconds)? That's not possible, since light takes 8 minutes to travel the distance so we can't go from Earth to the Sun in under that time, therefore the point (Sun, now + 5 seconds) is outside the light cone. Due to the lightspeed restriction, observers are "trapped" inside the light cone, moving from past to future. The trajectory they follow is called a world line. For reasons beyond the scope of this post, points inside the cone are called "timelike" and points outside the cone are "spacelike". Now, if we could somehow make this world line loop back on itself it would essentially mean that the observer would be returning to a previous point in space-time and would have thus time traveled. Therefore time travel is defined by the existence of closed world lines, called closed timelike curves (CTC). They are called timelike, because they would be inside the light cone and so would not be in conflict with breaking the lightspeed barrier. An example of points which could be on a CTC are: (Earth, now), (Sun, now + 5 days), (Earth, now). Notice how we went from the present to 5 days in the future (which would have become the new present) and then 5 days back into the past.
How can we make CTCs?
The trick is to do something similar to FTL travel. If you recall, in that post, when I talked about the Alcubierre drive I explained how you couldn't pass the speed of light locally but you could globally. The situation is very similar. If the light cone only lets us go from past to future, then lets make a series of local light cones in which we're going from past to future but each light cone will be slightly tilted with respect to the previous so that in the global light cone we will have formed a closed curve like in the figure:
Pretty ingenious, but can it be done? Well, the good news is there is nothing in the laws of physics which prevents this from happening. The tilting of light cones can be achieved in curved space-time like the one we live in, and many theoretical models have been found. The bad news is none of those models can be physically constructed in our Universe. The very first such model was discovered by Kurt Gödel, one of the greatest mathematicians of the 20th century and a very close friend of Einstein. Gödel presented this model to Einstein as a gift for his birthday. However, his model only worked in a rotating universe, which isn't the case for the one we inhabit. After that, scientists began to search for alternatives in our own universe. One of them was the Tipler cylinder, a large rotating cylinder which would allow the existence of CTCs in its interior. Unfortunately, it was proven that this could only happen if the cylinder had infinite length, something which is physically impossible. All other models have similar problems, though in some cases it may only be a question of lack of technology, for example one model suggests that a rotating superconductor (electrical conductor with 0 resistance) would form CTCs provided the number of Cooper pairs (special groups of electrons) is several orders of magnitude greater than it is in current superconductors. This could be merely a question of developing better superconductors, but on the other hand it could be physically impossible.
Can we define time travel in some other way?
Yes, however these alternate definitions don't even have theoretical solutions. One way to define it is simply by "true" faster than light travel, that is, travelling locally faster than light. The theory of relativity predicts that this is equivalent to travelling backwards in time as seen by outside observers. There is even a funny limerick based on this definition:
There once was a girl named Blight,
Who could travel faster than light.
She took off one day,
In a relative way,
And came back the previous day.
The only true FTL travel method we know of is represented by tachyons, the theoretical particles which could only travel at superluminal speeds, mentioned in the previous post. But even if they do exist and we could detect them, it has been proven that they cannot be used to send information into the past and thus violate causality, since tachyons of the future are indistinguishable from tachyons of the present.
Another definition comes from quantum mechanics. I mentioned antimatter in a previous post, the opposite of matter which annihilates it. An interesting property of antimatter is that in quantum mechanics it is viewed as matter travelling backwards through time at the same rate that normal matter travels forward through time. So if you would reverse the time axis on matter you would get antimatter. It is unknown whether this could be used to engineer a time machine, it's debatable if this even qualifies as time travel. Since time passes at the same rate if you would travel through time using this method it could mean that to go back in time 5 years, you'd have to wait 5 years.
There is also an interesting insight from the Einstein-Cartan theory. This theory completes general relativity, by including spin into the theory. Spin is seen as dislocations in the fabric of space-time and one of the predictions made is that particles with spin, travelling on certain trajectories can translate into the past or the future by a tiny amount, but yet again there is no theoretical time machine based on this method.
Time travel and FTL travel are very similar, though it would seem that time travel has a better chance of being achieved. This is due to the fact that it only seems to be restricted by paradoxes not actual physical laws, while FTL travel faces a major obstacle: the lightspeed limit. Also, both concepts find ways to circumvent their restrictions but the solutions to time travel are somewhat less exotic, in my opinion, and many solutions to the problem of FTL travel would also result in time travel while the inverse is not always true. It may turn out that both are possible or neither. History has shown us that many challenges viewed as impossible by scientists of the time, were eventually proven to be possible and are currently all around us. Examples include: airplanes, supersonic flight, X-rays, space travel, atomic energy and others. Given enough time, man's understanding of physics advances, the number of possibilities increases, to quote the Russian writer Anton Chekhov from one of his plays: "The human race progresses, perfecting its powers. Everything that is unattainable now will some day be near at hand and comprehensible..."
If time travel is possible then it would be the ultimate irony, for the only obstacle we face in achieving it, is time itself.