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 fourvector 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 v_{0}<c with respect to the system A. At the initial
moment the two systems coincide. The time coordinate in the position fourvector
of the tachion as measured in B is obtained by the Lorentz transformation:
where
Thus we can see that if v_{0} v > c^{2}, 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 fourvector of position (t,x) is being transformed
in the same way as the fourvector 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 socalled 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 (v_{0}<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.
Stanislaw Mrowczynski
