29 Tachyons of Plastic Deformation
317
with
m
T
∗ = − f m a , f =
2a
πL o
.
(447)
In the critical case u −→ ±∞ we now get
lim
u→±∞
P
T
=
P
T
∞ =
m
T
∗ c o = −
2a
2
σ
πL o c o
,
lim
u→±∞
E
T
=
E
T
∞ = 0 .
(448)
For the tachyon solution (445) we get the following special solution for the critical
case u −→ ∞
q
T
∞ (t) = lim
u→∞
q
T
(x, t) =
2a
π
arctan exp
−
πc o
L o
t
+
a
2
.
(449)
The negative sign of the momentum parameter for the tachyon q
T
(x, t) belonging
to (445) simply follows from the validity of
∂ q
T
(x, t)
∂t
= + u
∂ q
T
(x, t)
∂x
.
(450)
However, for the tachyon solution (435) the following applies,
∂q
T
(x, t)
∂t
= − u
∂q
T
(x, t)
∂x
.
(450a)
This results in a difference of sign for the components p and −s of the energymomentum tensor according to (289) and (290), and we get the following equation
(451) instead of (439a),
T
T
=
u
2
ρ
T
−u ρ
T
u c
2
o ρ
T
−c
2
o ρ
T
with |u| > c o ,
(451)
as well as a function (436), ρ
T
(x + ut) for the solution q
T
(x, t). With (279), (440)
and (441) we get a tachyon with the negative momentum parameter
m
T
∗ .
The function q
T
(x, t) according to (445) differs from the function q
T
(x, t)
according to (435) alone in the direction of motion. We get the same snapshot as
shown in Fig. 20.1, but the direction of motion is reversed, see Fig. 29.1. The same
situation occurs for the tachyon q
T
∞ (t) belonging to (449), whose graphic representation corresponds to that in Fig. 20.2 with the exception of the direction of motion,
see Fig. 29.2.
Of the tachyon we can say: it moves into the past if its velocity u and its momentum
P
T have opposite signs. If and whether such a situation is present depends on the
reference system. In every reference system the momentum P
T of a tachyon has
one and the same sign defined by its momentum parameter m
T
∗ . The sign of the
tachyon’s velocity u can change during the transition into another reference system,
317
with
m
T
∗ = − f m a , f =
2a
πL o
.
(447)
In the critical case u −→ ±∞ we now get
lim
u→±∞
P
T
=
P
T
∞ =
m
T
∗ c o = −
2a
2
σ
πL o c o
,
lim
u→±∞
E
T
=
E
T
∞ = 0 .
(448)
For the tachyon solution (445) we get the following special solution for the critical
case u −→ ∞
q
T
∞ (t) = lim
u→∞
q
T
(x, t) =
2a
π
arctan exp
−
πc o
L o
t
+
a
2
.
(449)
The negative sign of the momentum parameter for the tachyon q
T
(x, t) belonging
to (445) simply follows from the validity of
∂ q
T
(x, t)
∂t
= + u
∂ q
T
(x, t)
∂x
.
(450)
However, for the tachyon solution (435) the following applies,
∂q
T
(x, t)
∂t
= − u
∂q
T
(x, t)
∂x
.
(450a)
This results in a difference of sign for the components p and −s of the energymomentum tensor according to (289) and (290), and we get the following equation
(451) instead of (439a),
T
T
=
u
2
ρ
T
−u ρ
T
u c
2
o ρ
T
−c
2
o ρ
T
with |u| > c o ,
(451)
as well as a function (436), ρ
T
(x + ut) for the solution q
T
(x, t). With (279), (440)
and (441) we get a tachyon with the negative momentum parameter
m
T
∗ .
The function q
T
(x, t) according to (445) differs from the function q
T
(x, t)
according to (435) alone in the direction of motion. We get the same snapshot as
shown in Fig. 20.1, but the direction of motion is reversed, see Fig. 29.1. The same
situation occurs for the tachyon q
T
∞ (t) belonging to (449), whose graphic representation corresponds to that in Fig. 20.2 with the exception of the direction of motion,
see Fig. 29.2.
Of the tachyon we can say: it moves into the past if its velocity u and its momentum
P
T have opposite signs. If and whether such a situation is present depends on the
reference system. In every reference system the momentum P
T of a tachyon has
one and the same sign defined by its momentum parameter m
T
∗ . The sign of the
tachyon’s velocity u can change during the transition into another reference system,