9.1 Tachyons (Without Special Provisions)
Description
This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.
9.1 Tachyons (Without Special Provisions)
Tachyons are hypothetical/theoretical particles which would travel FTL.
The concept of the tachyon attempts to get around the infinite energy
requirements which the light speed barrier problem poses on a particle as it
approaches the speed of light. This was accomplished by demanding that the
particle have certain characteristics which we will discuss here.
First, consider the energy and momentum. Recall that we can write the
energy (E) and the momentum (p) of a particle of mass m as expressed in
Equation 1:8 and Equation 1:6 which are duplicated here:
(Eq 9:1--Copy of Eq 1:8)
E = gamma * m * c^2
(Eq 9:2--Copy of Eq 1:6)
p = gamma * m * v
Where gamma is defined in Equation 1:5 as gamma = 1/(1 - v^2/c^2)^0.5. From
this we find that |p*c|/|E| = |v|/|c|, which is greater than 1 if v is
greater than c. We can thus write
(Eq 9:3)
E^2 < p^2*c^2 (for an FTL particle).
But since we can also express the energy squared as defined in Equation 1:7:
(Eq 9:4--Copy of Eq 1:7)
E^2 = p^2 * c^2 + m^2 * c^4
we find that the only way to get E^2 < p^2*c^2 is if the mass squared is
negative (because then m^2*c^2 reduces the sum in Equation 9:4). The mass
would then be the square root of a negative number, and such an obviously
unreal number is called an imaginary number (imaginary numbers may seem odd,
but they have important uses in mathematics). In general we express such
imaginary numbers as a product of a real number multiplied by something that
symbolizes the imaginary square-root of negative one: i = sqrt(-1). So, the
mass of a tachyon is imaginary. Further, from the equation for gamma, we
find that it too is imaginary if v is greater than c, but it is also
negative because we have the i in the denominator of gamma, and 1/i = -i.
(We can show this as follows: start with 1/i = 1/sqrt(-1) and multiply and
divide the right-hand side by sqrt(-1) (which doesn't change the value): i =
sqrt(-1)/(sqrt(-1)*sqrt(-1)). The top of that equation is just i, and the
bottom is sqrt(-1)^2 = -1. Thus 1/i = i/(-1) = -i.) That would mean that
from Equation 9:1, the energy would still be a real, positive number
(because to get E we multiply the i in the imaginary m by the -i in gamma to
get -i^2 = -(sqrt(-1)^2) = -(-1) = +1). The same would be true for the
momentum, p = gamma*m*v.
I would like to note that I have read elsewhere that the energy would
be negative for a tachyon, but this doesn't seem to be the case.
The final interesting property of tachyons I will mention comes from
noting that as their velocity increases, the value of their gamma will
become a smaller, negative, imaginary number (because when v/c > 1,
1/sqrt(1-v^2/c^2) is a negative, imaginary number that decreases as v gets
larger). That means that the value of a tachyons energy will decrease as the
speed of the tachyon increases--or in other words, as the tachyon loses
energy, it gains speed. One result of this is that if a charged tachyon were
to exist, then because it would travel faster than light, it would give off
a radiation known as Cherenkov radiation. This would take energy away from
the tachyon and cause it to go faster and faster, continually giving off
more and more energy. Neutral tachyons, however, wouldn't do this.
In any case, we can consider the possibility that tachyons exist and
always travel faster than light. They then never have to cross the light
speed barrier, and they do not have infinite energy (but their mass is
imaginary and their energy decreases as their velocity increases). However,
they still cause trouble because of the second problem--if you can use them
for FTL communication, they can be used to create unsolvable paradoxes using
the same arguments as we used in our "FTL bullet" example.
To explore the question of using tachyons for FTL communication, one
can apply quantum mechanics to the energy equation of the tachyon. What one
finds is that either (1) the tachyons cannot be localized, or (2) the actual
effects of a tachyon cannot themselves move faster than light. In either of
these cases, the tachyon cannot be used to produce an FTL signal.
A third idea would also allow the tachyon to exist without the
possibility of using the tachyon to send FTL signals. The basic idea is that
there would be no way to distinguish between the situation through which you
could receive a tachyon and the situation though which you could transmit a
tachyon. To show what I mean, consider Diagram 8-1 yet again. From the O
frame of reference, a tachyon could be sent "from" * and "to" the origin.
However, as long as you cannot distinguish between the transmitter and the
receiver, then the Op observer could reinterpret this as a tachyon being
sent "from" the origin "to" *. Neither, then, will believe that the tachyon
went backwards in time. Obviously, there is no way for a message to be sent
(because then you could identify the sender and decide which way the tachyon
"really" went), and it wouldn't be quite right to call this FTL travel.
However, it would allow tachyons to exist (though uselessly) without causing
any problems.
And so, we find that with tachyons, one of the following must be true:
1. Tachyons do not exist,
2. Tachyons exist but cannot be used to send FTL signals, or
3. Tachyons exist and can be used to send FTL signals, but some special
provision will keep anyone from using them to produce an unsolvable
paradox.
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