226
20 Tachyons and Causality
and the observer in
registers according to (265) a function q
T
(x
, t
) for the
tachyon observed in o . For this function, we find, analogue to (261),
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
,
thus
q
T
(x
, t
) =
2a
π
arctan exp
π(x
+ u t
)
L o κ
+
a
2
.
(266)
We therefore find the tachyon (265) that is observed in o . The observer in
, who
when observed from o moves in the direction of positive x-values with the velocity
v, notices a tachyon moving in the direction of negative x-values with the velocity
−u = −c
2
o /v. In this respect, the state of the tachyon q
T
∞ (t) is the exact opposite of
the static kink solution q
I
(x), see (105), so that q
T
(x
, t
) corresponds to the moving
kink q
I
(x
, t
). For the observer moving with the velocity v with respect to o , the
kink resting in o moves with the velocity −v and shows a Lorentz contraction with
the factor γ. For tachyons, we have to replace v with u = c
2
o /2 and γ with κ = γ u/c o .
We will furthermore assume that the observer in
can also synchronise his clocks
using this tachyon. He would give, as if he were synchronising his clocks using
a sound signal, the following instructions: When the tachyon arrives at the clock
positioned at x
, then this clock must be started, according to the running time of the
tachyon, from the coordinate origin to x
with the hand setting t
according to
t
=
x
−u
=
−v x
c 2
o
.
(267)
Just as with Eq. (113a) x
= x/
1 − v 2 /c 2
o applies and therefore
t
=
−v x/c
2
o
1 − v 2 /c 2
o
.
(267a)
The internal observers in both reference systems would then have synchronised
their clocks using the same tachyon (here we have once again assumed that both
clocks had the hand setting 0 at their origin of coordinates). The result of this
synchronisation would be: If all clocks in o have the setting t = 0, then all clocks
in
have the setting t
according to (267a). This is however nothing other than
Lorentz transformation (151) for t = 0, or exactly the hand setting that we found
in Chap. 13, Eq.(142) (see also Fig.12.7), from the different pace of two clocks
moving with respect to one another. Tachyons would therefore be ideally suited
20 Tachyons and Causality
and the observer in
registers according to (265) a function q
T
(x
, t
) for the
tachyon observed in o . For this function, we find, analogue to (261),
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
,
thus
q
T
(x
, t
) =
2a
π
arctan exp
π(x
+ u t
)
L o κ
+
a
2
.
(266)
We therefore find the tachyon (265) that is observed in o . The observer in
, who
when observed from o moves in the direction of positive x-values with the velocity
v, notices a tachyon moving in the direction of negative x-values with the velocity
−u = −c
2
o /v. In this respect, the state of the tachyon q
T
∞ (t) is the exact opposite of
the static kink solution q
I
(x), see (105), so that q
T
(x
, t
) corresponds to the moving
kink q
I
(x
, t
). For the observer moving with the velocity v with respect to o , the
kink resting in o moves with the velocity −v and shows a Lorentz contraction with
the factor γ. For tachyons, we have to replace v with u = c
2
o /2 and γ with κ = γ u/c o .
We will furthermore assume that the observer in
can also synchronise his clocks
using this tachyon. He would give, as if he were synchronising his clocks using
a sound signal, the following instructions: When the tachyon arrives at the clock
positioned at x
, then this clock must be started, according to the running time of the
tachyon, from the coordinate origin to x
with the hand setting t
according to
t
=
x
−u
=
−v x
c 2
o
.
(267)
Just as with Eq. (113a) x
= x/
1 − v 2 /c 2
o applies and therefore
t
=
−v x/c
2
o
1 − v 2 /c 2
o
.
(267a)
The internal observers in both reference systems would then have synchronised
their clocks using the same tachyon (here we have once again assumed that both
clocks had the hand setting 0 at their origin of coordinates). The result of this
synchronisation would be: If all clocks in o have the setting t = 0, then all clocks
in
have the setting t
according to (267a). This is however nothing other than
Lorentz transformation (151) for t = 0, or exactly the hand setting that we found
in Chap. 13, Eq.(142) (see also Fig.12.7), from the different pace of two clocks
moving with respect to one another. Tachyons would therefore be ideally suited