224
20 Tachyons and Causality
The function q
T has no position of rest, whereas the kink q
I does. The velocity
u with which this figure moves in the direction of increasing x-values is, according
to (257), principally always larger than the critical velocity c o ,
u > c o for v < c o .
(264)
The interval on the x-axis in which the function q
T changes from q = 3a/2 to
q = a/2 is determined by the quantity L o κ. With exception of a small limit c o <
u ≈ 1, 4 c o this interval is always larger than the analogue interval of an arbitrary
kink solution where this changes from one minimum of the lattice potential to the
neighbouring; see Figs. 9.1, 10.2 and 18.7. Notice that only very high energy tachyons
are attributed velocities near the limit c o , as we will explicitly see in Chap. 29,
(442). Generally, the function q
T changes, in comparison to the kink q
I cf. (111),
along an extended region, this region extending infinitely with increasing velocity u.
Eilenberger [11] was the first to point at the tachyon character of this solution (261),
q
T
(x, t). There are numerous such tachyon states inside of a crystal; see Günther
[37]. Our derivation of the sine-Gordon equation, developed along the same lines as
our wave equation, shows that a ‘disturbance’ running through a medium does in
fact represent an energy transfer. In Chaps. 22 and 23, we will discuss the definitions
of an energy density e, of stress t, as well as momentum density p and a density
of energy current s for the solutions of the sine-Gordon equation; see also Günther
[37]. We will also discuss the particle aspect of a sine-Gordon field in Chaps. 22 and
23. In Chaps. 27–29, we will discuss the characteristics of a tachyon, and based upon
this we will then analyse the questions of a violation of causality for those problems
that we can actually calculate mathematically.
We will show that a tachyon moving with a permanent increasing velocity u
results in a decreasing energy density moving through ‘space’, thus through a crystal,
until a ‘vanishing energy density with infinite velocity’ moves through the crystal so
that the only thing remaining is a permanent state of stress.
Does the internal observer with his tachyons come into conflict with his Special
Theory of Relativity?
Let us consider the extreme case u −→ ∞. The solution q
T
(x, t) transforms
into a solution q
T
∞ (t). With (261) and
κ =
u
c o
1 −
c 2
o
u 2
we find for this function
q
T
∞ (t) = lim
u→∞
q
T
(x, t) =
2a
π
arctan exp
πc o
L o
t
+
a
2
.
(265)
20 Tachyons and Causality
The function q
T has no position of rest, whereas the kink q
I does. The velocity
u with which this figure moves in the direction of increasing x-values is, according
to (257), principally always larger than the critical velocity c o ,
u > c o for v < c o .
(264)
The interval on the x-axis in which the function q
T changes from q = 3a/2 to
q = a/2 is determined by the quantity L o κ. With exception of a small limit c o <
u ≈ 1, 4 c o this interval is always larger than the analogue interval of an arbitrary
kink solution where this changes from one minimum of the lattice potential to the
neighbouring; see Figs. 9.1, 10.2 and 18.7. Notice that only very high energy tachyons
are attributed velocities near the limit c o , as we will explicitly see in Chap. 29,
(442). Generally, the function q
T changes, in comparison to the kink q
I cf. (111),
along an extended region, this region extending infinitely with increasing velocity u.
Eilenberger [11] was the first to point at the tachyon character of this solution (261),
q
T
(x, t). There are numerous such tachyon states inside of a crystal; see Günther
[37]. Our derivation of the sine-Gordon equation, developed along the same lines as
our wave equation, shows that a ‘disturbance’ running through a medium does in
fact represent an energy transfer. In Chaps. 22 and 23, we will discuss the definitions
of an energy density e, of stress t, as well as momentum density p and a density
of energy current s for the solutions of the sine-Gordon equation; see also Günther
[37]. We will also discuss the particle aspect of a sine-Gordon field in Chaps. 22 and
23. In Chaps. 27–29, we will discuss the characteristics of a tachyon, and based upon
this we will then analyse the questions of a violation of causality for those problems
that we can actually calculate mathematically.
We will show that a tachyon moving with a permanent increasing velocity u
results in a decreasing energy density moving through ‘space’, thus through a crystal,
until a ‘vanishing energy density with infinite velocity’ moves through the crystal so
that the only thing remaining is a permanent state of stress.
Does the internal observer with his tachyons come into conflict with his Special
Theory of Relativity?
Let us consider the extreme case u −→ ∞. The solution q
T
(x, t) transforms
into a solution q
T
∞ (t). With (261) and
κ =
u
c o
1 −
c 2
o
u 2
we find for this function
q
T
∞ (t) = lim
u→∞
q
T
(x, t) =
2a
π
arctan exp
πc o
L o
t
+
a
2
.
(265)