320
29 Tachyons of Plastic Deformation
lim
u→∞
lim
r →∞
+r
−r
· · · ρ
T
dx .
(456)
Using this procedure we get the correct solution (444) and thus the correct tachyons
with all their peculiar properties.
One final objection could be made against the physical reality of the tachyon
solution (435). This solution does not have the same physical importance as the kink
solution (111). Whilst the kink realises a deviation from the stabile equilibrium, the
tachyon solution (435) is an instabile solution—a dislocation on a potential barrier, as
we saw in Chap. 20 during the discussion concerned with Fig.20.1. It will therefore be
difficult to verify such conditions. This does not however make their physical reality
uncertain. The instabile position of a mechanical pendulum above its suspension is
not something one sees everyday. No human would thus come to the conclusion that
such a position does not exist—the artists in a circus earn their money by showing
such positions.
The existence of tachyons can therefore not be questioned. It is of no relevance
that we concern ourselves with the tachyons of internal observers of an infinitely
extended crystal. The important factor is that tachyons, tachyons with both negative
and positive momentum parameters actually exist inside of the world of our internal
observers, for whom the Special Theory of Relativity is valid. Let us now take another
closer look at these tachyons.
The tachyon q
T
∞ (t), the special solution (265) of the sine-Gordon equation,
describes the exact simultaneous gliding of a straight dislocation along the x-axis
from one potential barrier of the lattice to a neighbouring. During the process of a
plastic deformation the motion of the dislocation elements, arbitrarily placed at large
distances from each other on the x-axis, occurs in exactly the same timed sequence.
A motion of the complete dislocation line, embedded in the crystalline lattice, which
is taken into account using the sine-Gordon equation, can only occur if its line tension
can be sustained for arbitrary distances whilst upholding the correlations caused by
the lattice. Using the expression ‘tachyon’ we can state for this procedure that: The
stress state of a moving straight dislocation line is realised by a tachyon, which moves
in the x-direction with infinite velocity with a finite momentum, but with vanishing
energy. We can also interpret the tachyon belonging to q
T
(x, t) according to (435).
One easily realises that its initial state at t = −∞ and its final state at t = +∞
coincide with the corresponding states of the tachyon belonging to q
T
∞ (t). Hence,
a dislocation line moved from one potential barrier to the neighbouring. Even this
motion occurs strictly correlated, but not in one and the same timed procedure for all
dislocation segments. The dislocation pieces with large x-coordinates limp behind
if u is positive. The inevitable creation of inhomogeneity in the stress state can be
interpreted using the momentum and the energy of a tachyon, which moves with
|u| > c o along the x-axis.
For every solution q
T
(x, t) there is always a preferred frame of the internal
observer, in which this tachyon has the special form q
T
∞ (t). For the internal observer
in this frame, the gliding of the dislocation line occurs simultaneously. Mathemat-
29 Tachyons of Plastic Deformation
lim
u→∞
lim
r →∞
+r
−r
· · · ρ
T
dx .
(456)
Using this procedure we get the correct solution (444) and thus the correct tachyons
with all their peculiar properties.
One final objection could be made against the physical reality of the tachyon
solution (435). This solution does not have the same physical importance as the kink
solution (111). Whilst the kink realises a deviation from the stabile equilibrium, the
tachyon solution (435) is an instabile solution—a dislocation on a potential barrier, as
we saw in Chap. 20 during the discussion concerned with Fig.20.1. It will therefore be
difficult to verify such conditions. This does not however make their physical reality
uncertain. The instabile position of a mechanical pendulum above its suspension is
not something one sees everyday. No human would thus come to the conclusion that
such a position does not exist—the artists in a circus earn their money by showing
such positions.
The existence of tachyons can therefore not be questioned. It is of no relevance
that we concern ourselves with the tachyons of internal observers of an infinitely
extended crystal. The important factor is that tachyons, tachyons with both negative
and positive momentum parameters actually exist inside of the world of our internal
observers, for whom the Special Theory of Relativity is valid. Let us now take another
closer look at these tachyons.
The tachyon q
T
∞ (t), the special solution (265) of the sine-Gordon equation,
describes the exact simultaneous gliding of a straight dislocation along the x-axis
from one potential barrier of the lattice to a neighbouring. During the process of a
plastic deformation the motion of the dislocation elements, arbitrarily placed at large
distances from each other on the x-axis, occurs in exactly the same timed sequence.
A motion of the complete dislocation line, embedded in the crystalline lattice, which
is taken into account using the sine-Gordon equation, can only occur if its line tension
can be sustained for arbitrary distances whilst upholding the correlations caused by
the lattice. Using the expression ‘tachyon’ we can state for this procedure that: The
stress state of a moving straight dislocation line is realised by a tachyon, which moves
in the x-direction with infinite velocity with a finite momentum, but with vanishing
energy. We can also interpret the tachyon belonging to q
T
(x, t) according to (435).
One easily realises that its initial state at t = −∞ and its final state at t = +∞
coincide with the corresponding states of the tachyon belonging to q
T
∞ (t). Hence,
a dislocation line moved from one potential barrier to the neighbouring. Even this
motion occurs strictly correlated, but not in one and the same timed procedure for all
dislocation segments. The dislocation pieces with large x-coordinates limp behind
if u is positive. The inevitable creation of inhomogeneity in the stress state can be
interpreted using the momentum and the energy of a tachyon, which moves with
|u| > c o along the x-axis.
For every solution q
T
(x, t) there is always a preferred frame of the internal
observer, in which this tachyon has the special form q
T
∞ (t). For the internal observer
in this frame, the gliding of the dislocation line occurs simultaneously. Mathemat-