Chapter 29
Tachyons of Plastic Deformation
We now come back to the question that we formulated at the beginning of the last
chapter: Is the field of the solution (261) of the sine-Gordon equation,
q
T
(x, t) =
2a
π
arctan exp
−π(x − u t)
L o κ
+
a
2
,
κ =
u
c o
1 −
v 2
c 2
o
= sign u
u 2
c 2
o
,
u =
c
2
o
v
,
|u| > c o ,
(435)
mechanically seen a particle? Can we really attribute the field (435) an energy E
T
and a momentum P
T in every reference system , so that these quantities are related
in various reference systems to each other via the Eq. (427)? Only then would we be
able to state that the solution (435) is a tachyon, a particle of type (b) as defined in
the last chapter.
First of all we formally construct, according to the rules (289), (290) the energymomentum tensor (276) for the field q
T
= q
T
(x, t). The calculations are done in
the same manner as for the solution q
I in Chap. 23. A difference between the two
consists in the exponents of the exponential function, where now a minus sign is
present, and where γ is replaced with the quantity κ = γ u/c o . Furthermore it must
be observed that a velocity u with |u| > c o replaces the velocity v with |v| < c o .
The additive constant a/2 in q
T disappears during differentiation. According to this
additive constant a/2 in q
T , in the cosine term of the Lagrangian L in (287) the
argument increases by π, which reverses the sign of the cosine term. We thus find
by applying the results of the calculations made in Chap. 23,
© The Editor(s) (if applicable) and The Author(s), under exclusive
license to Springer Nature Singapore Pte Ltd. 2020
H. Günther, Elementary Approach to Special Relativity,
https://doi.org/10.1007/978-981-15-3168-2_29
311
Tachyons of Plastic Deformation
We now come back to the question that we formulated at the beginning of the last
chapter: Is the field of the solution (261) of the sine-Gordon equation,
q
T
(x, t) =
2a
π
arctan exp
−π(x − u t)
L o κ
+
a
2
,
κ =
u
c o
1 −
v 2
c 2
o
= sign u
u 2
c 2
o
,
u =
c
2
o
v
,
|u| > c o ,
(435)
mechanically seen a particle? Can we really attribute the field (435) an energy E
T
and a momentum P
T in every reference system , so that these quantities are related
in various reference systems to each other via the Eq. (427)? Only then would we be
able to state that the solution (435) is a tachyon, a particle of type (b) as defined in
the last chapter.
First of all we formally construct, according to the rules (289), (290) the energymomentum tensor (276) for the field q
T
= q
T
(x, t). The calculations are done in
the same manner as for the solution q
I in Chap. 23. A difference between the two
consists in the exponents of the exponential function, where now a minus sign is
present, and where γ is replaced with the quantity κ = γ u/c o . Furthermore it must
be observed that a velocity u with |u| > c o replaces the velocity v with |v| < c o .
The additive constant a/2 in q
T disappears during differentiation. According to this
additive constant a/2 in q
T , in the cosine term of the Lagrangian L in (287) the
argument increases by π, which reverses the sign of the cosine term. We thus find
by applying the results of the calculations made in Chap. 23,
© The Editor(s) (if applicable) and The Author(s), under exclusive
license to Springer Nature Singapore Pte Ltd. 2020
H. Günther, Elementary Approach to Special Relativity,
https://doi.org/10.1007/978-981-15-3168-2_29
311