20 Tachyons and Causality
229
However, we need an equation allowing tachyon solutions as well as such a particle
tachyon collision as a solution. Only in that case, we could speak of the physical
reality of such processes. In our case, this is the sine-Gordon equation, but in this
equation we can find no such processes. We will delve deeper into this problem in
Chaps. 28–30, with the result:
The tachyons of the sine-Gordon equation do not realise interactions, they realise a correlation.
We will now show that the property |u| > c o of the tachyons discussed by us is
an immediate consequence of the relativity of simultaneity of the internal observers
of the crystal, if such a solution (265) q
T
∞ applies in the crystal. Nevertheless, these
tachyons exist inside of the crystal just as they do for the outside observer.
In the reference system o , the tachyon q
T
∞ (x, t) represents a state that depends
according to (268) only on time t and not on x, so that everywhere in o the same
original change in state occurs simultaneously. One can show, see Chap. 28, that this
is a state of stress t that changes synchronically with hand settings of the clocks in
o . If all clocks in o have the hand setting t, then the clocks of an observer in
,
who moves with the velocity v with respect to o , show the hand setting t
at the
position x
in
according to our Eq. (151a), thus
t =
t
+ v x
/c
2
o
1 − v 2 /c 2
o
.
The observer in
registers a state q
T
(x
, t
) due to his clock settings according to
q
T
(x
, t
) =
2a
π
arctan exp
⎡
⎢
⎢
⎣
πc o
L o
(t
+
v x
c 2
o
)
1 − v 2 /c 2
o
⎤
⎥
⎥
⎦ +
a
2
.
We can however rewrite this using (257), (258) and (259) with the result
: q
T
(x
, t
) =
2a
π
arctan exp
π (x
+ u t
)
L o κ
+
a
2
.
(268a)
If we explicitly choose u = −2c o , then we get the plot in Fig. 20.1. Therefore, whilst
the observer in o registers the constant function according to Fig. 20.2, the observer
in
registers the function, based on the effect of the relativity of simultaneity, as
plotted in Fig. 20.1. An outside observer could state that the observations made from
are artificial and thus ignore them, donating only the observations made in o
any meaning. Can we conclude that the state registered by the observer in
, a
state produced only using a peculiar time measuring method, resulting in an artificial
operand, that this operand has no physical meaning? Certainly not! The observer in
also does have a solution q
T
∞ (t
),
229
However, we need an equation allowing tachyon solutions as well as such a particle
tachyon collision as a solution. Only in that case, we could speak of the physical
reality of such processes. In our case, this is the sine-Gordon equation, but in this
equation we can find no such processes. We will delve deeper into this problem in
Chaps. 28–30, with the result:
The tachyons of the sine-Gordon equation do not realise interactions, they realise a correlation.
We will now show that the property |u| > c o of the tachyons discussed by us is
an immediate consequence of the relativity of simultaneity of the internal observers
of the crystal, if such a solution (265) q
T
∞ applies in the crystal. Nevertheless, these
tachyons exist inside of the crystal just as they do for the outside observer.
In the reference system o , the tachyon q
T
∞ (x, t) represents a state that depends
according to (268) only on time t and not on x, so that everywhere in o the same
original change in state occurs simultaneously. One can show, see Chap. 28, that this
is a state of stress t that changes synchronically with hand settings of the clocks in
o . If all clocks in o have the hand setting t, then the clocks of an observer in
,
who moves with the velocity v with respect to o , show the hand setting t
at the
position x
in
according to our Eq. (151a), thus
t =
t
+ v x
/c
2
o
1 − v 2 /c 2
o
.
The observer in
registers a state q
T
(x
, t
) due to his clock settings according to
q
T
(x
, t
) =
2a
π
arctan exp
⎡
⎢
⎢
⎣
πc o
L o
(t
+
v x
c 2
o
)
1 − v 2 /c 2
o
⎤
⎥
⎥
⎦ +
a
2
.
We can however rewrite this using (257), (258) and (259) with the result
: q
T
(x
, t
) =
2a
π
arctan exp
π (x
+ u t
)
L o κ
+
a
2
.
(268a)
If we explicitly choose u = −2c o , then we get the plot in Fig. 20.1. Therefore, whilst
the observer in o registers the constant function according to Fig. 20.2, the observer
in
registers the function, based on the effect of the relativity of simultaneity, as
plotted in Fig. 20.1. An outside observer could state that the observations made from
are artificial and thus ignore them, donating only the observations made in o
any meaning. Can we conclude that the state registered by the observer in
, a
state produced only using a peculiar time measuring method, resulting in an artificial
operand, that this operand has no physical meaning? Certainly not! The observer in
also does have a solution q
T
∞ (t
),