29 Tachyons of Plastic Deformation
315
There is a further relationship less well known in the field theory: The energymomentum tensor T once again has the structure (279),
T =
u
2
ρ
uρ
−u c
2
o ρ −c
2
o ρ
, div T = 0 .
(279)
This time however, the condition |u| > c o applies to the velocity u. The velocity
u in (279) cannot therefore stand for the velocity of a reference system. Our tensor
(439a) has just this structure. Such a tensor can be associated with a particle of type
(b), a tachyon with the velocity u according to the following rule,
m
T
= sign u
+∞
−∞
ρ
T dx ,
(440)
and
P
T
= m
T u ,
E
T
= m
T c
2
o .
(441)
According to these equations we can calculate the momentum and the energy of a
tachyon (435). This would be
+∞
−∞
ρ
T dx =
1
c 2
o
1
κ 2
4 σa
2
L 2
o
+∞
−∞
exp
−2
π(x − ut)
Ł o κ
1 + exp
−2
π(x − ut)
L o κ
2
=
1
c 2
o
1
κ 2
4 σa
2
L 2
o
L o κ
2π
sign u
+∞
−∞
e
−x
1 + e −x
2 dx
=
1
c 2
o
1
κ 2
2a
2
σ
πL o
1
1 + e −x
+∞
−∞
=
sign u
κ
1
c 2
o
2a
2 E
πL o
.
Due to the fact that we introduce the new integration variable 2
π(x−ut)
L o κ
into the second
line the direction of integration changes, depending on the sign of u in κ. We kept
the direction of integration from −∞ to +∞ and introduced the factor sign u .