Faster than light?
Relativity theory has a solid base in the independence of the velocity of light with respect to reference systems, i.e, in the invariance of this velocity. Now, one may ask: are there objects, called tachions, which move with a velocity greater than that of light? In other words, is the velocity of light only invariant, or maybe maximal too?
The energy of a particle of mass m increases together with its velocity v according to the following formula:
where c is the velocity of light. Therefore a particle cannot be accelerated to a velocity greater than c, for this would require crossing the barrier of infinite energy at v=c. Some may thereby deduce that the existence of tachions is not possible. This, however, reminds the argument of an old Hindu sage who claimed that no one can live north of the Himalaya, for a man cannot cross any mountain so big. The sage did not take into account that the peoples inhabiting Central Asia did not have to cross the Himalaya. The case is similar with the tachions, which always move faster than light - if they exist. Their energies and momenta would be determined by the following formulas:
Nevertheless, a very serious argument against the existence of tachions was brought to light. It turns out that a tachion moving forwards in time in one reference system may be moving backwards (in time) in another.
Think of a tachion emitted in a system A from a point x=0 at the moment t=0. The tachion moves with a velocity v>c until it is absorbed. Thus its four-vector of position is of the form (t,x)=(t,vt).
Now, let's look at the tachion from another system B which which moves with velocity v0<c with respect to the system A. At the initial moment the two systems coincide. The time coordinate in the position four-vector of the tachion as measured in B is obtained by the Lorentz transformation:
Thus we can see that if v0 v > c2, then t'<0, i.e. the tachion moves backwards in time in the system B. This can lead to paradoxical causal connections. Here is one of them.
A keen constructor devises a gun to shoot tachions. He launches a missile, which runs towards the past and kills the constructor's grandfather before the constructor's father is born. By killing his grandfather the constructor annihilates his father, so he cannot exist himself either. Now, who is the murderer? All this mess is due to a violation of the fundamental principle of causality which claims that the cause should always precede the effect.
However, a solution to such paradoxes has been found. It is based on the fact that tachions which move backwards in time carry negative energy. This is because their four-vector of position (t,x) is being transformed in the same way as the four-vector of momentum (E,p). Hence the following postulate, called the reinterpretation rule, was suggested: a tachion moving backwards in time with negative energy is an antitachion moving forwards in time. This rule makes all the objects always move forwards in time, so no causal paradoxes are possible any more.
Nevertheless, the price to be paid for this solution is quite high. The distinction between cause and effect is only possible within a given reference system. What is a cause in one system may turn to be the effect in another. What is the emission of a tachion for one observer may become the absorption of an antitachion for another. Hence the principle of causality looses its absolute meaning. This is not much of a disaster, though. After all, relativity theory deprived physics of absolute time and absolute space, so the relativization of the notions of cause and effect seems to fit very well along the line.
Unfortunately, the reinterpretation rule gives birth to another serious problem, which is the so-called paradox of the free will. Assume that an observer A which moves with respect to another observer B sends a tachion to B. As we have seen, it is possible to choose the velocity of the tachion (v>c) and the relative velocity of the observers (v0<c) in such a way that B will think that it was him who sent an antitachion to A and not A who sent a tachion to him. This implies that B's free will is limited: he sends an antitachion to A only because A had sent a tachion to him.