From the beginning of the 20th century, scientists have realised that time is not absolute. However it was still considered to be a road on which you could go one way or the other. But normal roads have side streets, some of them loop-shaped. By always going forward on these roads and loop-shaped side road we can, sometime, come back through a town where we passed before. Using this analogy, can we imagine travelling into the future or the past?
Relativity shows that it is possible to travel in the future, at least in principle. If we had a rocket able to travel at a velocity close to the speed of light, then it is only necessary to travel in it and when we come back on the earth we would realise that more time has passed here that for us in the rocket. Such rockets are not available to day but who knows what will happen tomorrow?
Travelling in the past is not so obvious but in 1949, Kurt Gödel, a mathematician, discovered a new solution to Einstein's equations of relativity that showed that, again in principle, travelling in the past is possible. Gödel's space-time has the particularity to see the whole universe rotating. In this case a rotating universe means that distant matter rotate by reference to directions that little gyroscopes point in within the universe. A consequence of Gödel's space-time theory is that if you travel far away from the earth it would be possible to come back before you started. However Gödel's universe is different as our universe is not rotating and, moreover, his universe is not expanding. Later on other scientists have found other solutions to Einstein's equations that allow also travel in the past. But given the known microwave background and the presence of elements such as hydrogen and helium, all these universes do not have the kind of curvature that time travels require. Still, if the universe started with the wrong space-time curvature required for time travels, could we bend up some limited region of our universe space-time to allow them.
Time travels also involve travelling faster than light. It is certain that if one can travel at unlimited speed, one can travel backwards in time. The relativity theory says that time is not absolute, and that different observers watching a specific event will not necessarily agree at what time it happened. Going to the extreme it is also possible, according to relativity, that two observers will not agree on the order in which two different events A and B happened.
We have seen that going faster that the speed of light is in fact impossible due to the fact that when we are close to this limit the energy required to go faster grows steadily. To reach the speed of light an infinite amount of energy is required as the mass of the moving object become also infinite. There is at least a theoretical solution to this problem. It is, in principle, possible to bend space-time in such a way that there is a shortcut between A and b. This shortcut, a kind of wormhole between A and b, is a tube of space-time connecting two far away nearly flat regions of our universe. Travelling through this wormhole from A to B would be travelling in the future and travelling from B to A is travelling in the past.
In fact Einstein in collaboration with Nathan Rosen wrote a paper in 1935 that showed that the theory of general relativity allowed "bridges" (now wormholes) between two regions of space-time. However spaceship trying to go through these wormholes would meet singularities -in this case the tunnel would close at some point- that you destroy them. But it will perhaps be possible to keep wormholes open we need a region in space-time with negative curvature (like the surface of a saddle). Ordinary matter, however, has positive energy density that gives space-time a positive curvature (like a sphere). Classical laws of physics prohibited negative energy density impeding travelling backward in time. Quantum laws based on the uncertainty principle superseded classical laws and allow the energy density to be negative in some limited regions of space-time but the total energy density of the universe must be positive. If this is true then it is possible to "warp" some regions in space-time so that in their curvature is negative allowing, in principle, time travel in both directions.
According to Feynman's theory time travel in the past already occurs at
the level of single particles. According to this theory, a particle moving
forward in time is the same as an anti-particle moving backward in time.
Accordingly a particle/antiparticle pair created and then annihilated together
can be viewed as a single particle moving on a closed loop in space-time.
To explain this in a simplify way, let us assume that the pair of particle/antiparticle
are created at time A, then that they move forward in time until they are
annihilated at time B. According to Feynman's theory one can see this as
follow: at time A a single particle is created, it then moves forward to
time B and finally move back to A. What is going forward in time from A
to B is called a particle and what is moving backward in time from B to
a it is an antiparticle travelling forward in time. The Feynman theory seems
to show that travelling in time is possible but there are still many problems
to solve so no immediate solution will be found in the near future.