Chapter 30
On the Causality Problem:
Particle–Tachyon Collisions
Just merely changing the time order when observing two events E 1 and E 2 from
different reference systems doesn’t break causality. If the one event E 2 was caused
by the event E 1 , then E 2 must always occur after E 1 , regardless of the reference
system from which these events are observed. A reversal of the time order of these
events would severely confuse our causal view of the world. One can however ask
the question, when does a break of causality inside of the smallest spatial distances
not contradict our view of the world? Here, quantum theoretical deliberations play a
role. We refer the reader to the discussions of H. Treder [94].
The mere existence of tachyons as particles is not enough. The all decisive question
is, can we use tachyons to transmit signals, meaning:
Can we transmit energy using tachyons?
This is what it is all about. To begin with, we will examine, in full generality, elastic
collisions between a particle and a tachyon. We use the term ‘particle’ as defined in
Chap. 28, as a type (a) particle and the term ‘tachyon’ as a type (b) particle.
A particle with restmass m o shall collide with a tachyon with the momentum
parameter m
T
∗ . The velocities of the particle before and after the collision are w
and w
, the corresponding velocities of the tachyon are u and u
, respectively.
The momenta and energies of particles and tachyons are defined by the formulas
(297) and (442). Due to the presumed ‘elasticity’ of the collision process, the energy
and momentum conservation is valid, the Lorentz invariant parameters, the mass
parameter m o for the particle, and the momentum parameter m
T
∗ for the tachyon
are retained, no ‘other’ particles or tachyons are created. The following is valid for
the momenta and energies before and after the collision,
m o w
1−w 2 /c 2
o
+
m T
∗ u
sign u
u 2 /c 2
o −1
=
m o w
1−w 2 /c 2
o
+
m T
∗ u
sign u
u 2 /c 2
o −1
, Momentum
m o c 2
o
1−w 2 /c 2
o
+
m T
∗ c 2
o
sign u
u 2 /c 2
o −1
=
m o c 2
o
1−w 2 /c 2
o
+
m T
∗ c 2
o
sign u
u 2 /c 2
o −1
. Energy
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(457)
© 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_30
323
On the Causality Problem:
Particle–Tachyon Collisions
Just merely changing the time order when observing two events E 1 and E 2 from
different reference systems doesn’t break causality. If the one event E 2 was caused
by the event E 1 , then E 2 must always occur after E 1 , regardless of the reference
system from which these events are observed. A reversal of the time order of these
events would severely confuse our causal view of the world. One can however ask
the question, when does a break of causality inside of the smallest spatial distances
not contradict our view of the world? Here, quantum theoretical deliberations play a
role. We refer the reader to the discussions of H. Treder [94].
The mere existence of tachyons as particles is not enough. The all decisive question
is, can we use tachyons to transmit signals, meaning:
Can we transmit energy using tachyons?
This is what it is all about. To begin with, we will examine, in full generality, elastic
collisions between a particle and a tachyon. We use the term ‘particle’ as defined in
Chap. 28, as a type (a) particle and the term ‘tachyon’ as a type (b) particle.
A particle with restmass m o shall collide with a tachyon with the momentum
parameter m
T
∗ . The velocities of the particle before and after the collision are w
and w
, the corresponding velocities of the tachyon are u and u
, respectively.
The momenta and energies of particles and tachyons are defined by the formulas
(297) and (442). Due to the presumed ‘elasticity’ of the collision process, the energy
and momentum conservation is valid, the Lorentz invariant parameters, the mass
parameter m o for the particle, and the momentum parameter m
T
∗ for the tachyon
are retained, no ‘other’ particles or tachyons are created. The following is valid for
the momenta and energies before and after the collision,
m o w
1−w 2 /c 2
o
+
m T
∗ u
sign u
u 2 /c 2
o −1
=
m o w
1−w 2 /c 2
o
+
m T
∗ u
sign u
u 2 /c 2
o −1
, Momentum
m o c 2
o
1−w 2 /c 2
o
+
m T
∗ c 2
o
sign u
u 2 /c 2
o −1
=
m o c 2
o
1−w 2 /c 2
o
+
m T
∗ c 2
o
sign u
u 2 /c 2
o −1
. Energy
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(457)
© 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_30
323