228
20 Tachyons and Causality
o :
E 1 : t 1 = 0, 0 < x 1 ,
E 2 : t 2 = 0, x 1 < x 2 .
t 1 = t 2
(270)
These events are observed from
as follows. Firstly, the distance x from the origin
of coordinates in o is the Lorentz contracted length x
, thus x = x
1 − v 2 /c 2
o .
For the events E 1 and E 2 to following can be derived from (269), in other words
applying (151),
o :
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
E 1 : t
1 =
−v x 1
c 2
o
1 − v 2 /c 2
o
, 0 < x 1 ,
E 2 : t
2 =
−v x 2
c 2
o
1 − v 2 /c 2
o
, x 1 < x 2 .
t
2 < t
1
(270a)
The observer in also finds using x = ¯
x
1 − v 2 /c 2
o , and if we only replace v with
−v,
:
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
E 1 : ¯
t 1 =
+v x 1
c 2
o
1 − v 2 /c 2
o
, 0 < x 1 ,
E 2 : ¯
t 2 =
+v x 2
c 2
o
1 − v 2 /c 2
o
, x 1 < x 2 .
¯
t 1 < ¯
t 2
(270b)
According to (270b), a detective in would observe the event E 2 occurring after
event E 1 . For a detective in the event E 2 (maybe a murder, for example the death
of Mr Stus) could in fact be the result of event E 1 (Dr Fast’s fatal shot caused
by the tachyon). A detective in the reference system
who relies on the principle
of causality would contradict this energetically. He states according to (270a) that
event E 1 takes place after event E 2 that Mr Stus was already dead (E 2 ) before (E 1 ),
the fatal shot, even occurred. Those relying on the principle of causality will see the
observations of a Sherlock Holmes in
as undeniable proof of Dr Fast’s innocence.
Only the strict formalist (who hopefully does not act as a juror) would object,
stating that this time the principle of causality did not hold, according to the motto—
because I watered the plants today they started blooming yesterday. We do not want to
continue philosophising about this here. However, because of the physics of tachyons,
or at least as much as we can mathematically discuss it in the frame of our crystalline
solid, the hypothesis of an acausal interaction cannot be supported. The transmission
of a signal is always combined with the transference of energy, be it as minute as
possible. Now, we have already stated that with tachyons an energy density does
in fact move through space (in our case through the crystal). This alone does not
suffice. This energy has to be able to be released. In Chap. 30, we will show using
an elementary calculation that an ‘elastic collision’ between a particle and a tachyon
results in nothing. This, at least, is one way that a tachyon does not release energy
to a particle, and therefore, also does not transmit a signal.
Inelastic collisions between particles and tachyons, where the transfer of energy
occurs, are however compatible with the energy–momentum conservation laws.
20 Tachyons and Causality
o :
E 1 : t 1 = 0, 0 < x 1 ,
E 2 : t 2 = 0, x 1 < x 2 .
t 1 = t 2
(270)
These events are observed from
as follows. Firstly, the distance x from the origin
of coordinates in o is the Lorentz contracted length x
, thus x = x
1 − v 2 /c 2
o .
For the events E 1 and E 2 to following can be derived from (269), in other words
applying (151),
o :
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
E 1 : t
1 =
−v x 1
c 2
o
1 − v 2 /c 2
o
, 0 < x 1 ,
E 2 : t
2 =
−v x 2
c 2
o
1 − v 2 /c 2
o
, x 1 < x 2 .
t
2 < t
1
(270a)
The observer in also finds using x = ¯
x
1 − v 2 /c 2
o , and if we only replace v with
−v,
:
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
E 1 : ¯
t 1 =
+v x 1
c 2
o
1 − v 2 /c 2
o
, 0 < x 1 ,
E 2 : ¯
t 2 =
+v x 2
c 2
o
1 − v 2 /c 2
o
, x 1 < x 2 .
¯
t 1 < ¯
t 2
(270b)
According to (270b), a detective in would observe the event E 2 occurring after
event E 1 . For a detective in the event E 2 (maybe a murder, for example the death
of Mr Stus) could in fact be the result of event E 1 (Dr Fast’s fatal shot caused
by the tachyon). A detective in the reference system
who relies on the principle
of causality would contradict this energetically. He states according to (270a) that
event E 1 takes place after event E 2 that Mr Stus was already dead (E 2 ) before (E 1 ),
the fatal shot, even occurred. Those relying on the principle of causality will see the
observations of a Sherlock Holmes in
as undeniable proof of Dr Fast’s innocence.
Only the strict formalist (who hopefully does not act as a juror) would object,
stating that this time the principle of causality did not hold, according to the motto—
because I watered the plants today they started blooming yesterday. We do not want to
continue philosophising about this here. However, because of the physics of tachyons,
or at least as much as we can mathematically discuss it in the frame of our crystalline
solid, the hypothesis of an acausal interaction cannot be supported. The transmission
of a signal is always combined with the transference of energy, be it as minute as
possible. Now, we have already stated that with tachyons an energy density does
in fact move through space (in our case through the crystal). This alone does not
suffice. This energy has to be able to be released. In Chap. 30, we will show using
an elementary calculation that an ‘elastic collision’ between a particle and a tachyon
results in nothing. This, at least, is one way that a tachyon does not release energy
to a particle, and therefore, also does not transmit a signal.
Inelastic collisions between particles and tachyons, where the transfer of energy
occurs, are however compatible with the energy–momentum conservation laws.