308
28 Particles and Tachyons
Then, per definition, the following applies to the velocity u of the tachyon (as for
every other particle) in the new coordinates
u =
d x
dt
= −u .
(432)
Furthermore, the momentum is, according to its physical attribution a spatial vector.
The components of spatial vectors change their signs during inversion, in other words
P
T = −P .
(433)
Due to u/sign u = u/sign u it follows from (425) that the tachyon parameter m
T
∗
must also change its sign so that (433) can be fulfilled,
m
T
∗ = −m
T
∗ .
(434)
We can make the following conclusion from this:
The tachyon parameter m T
∗ is a spatial vector.
This is the key needed to unlock and understand the properties of tachyons. The
quantity m
T
∗ has nothing at all in common with the restmass m o of a normal particle
of the type (a). The particle parameter m o , the restmass, is a spatial scalar, therefore
unchangeable with respect to any change of space coordinates. In comparison to
this, m
T
∗ multiplied by the constant quantity c o is a characteristic momentum value
that is assumed by the tachyon in the reference system where its energy is zero. The
occasional classification of the tachyon as a particle with an imaginary restmass is
therefore misleading.
The quantity m T
∗ is the momentum parameter of the tachyon.
Whilst the restmass m o of a ‘normal’ type (a) particle is always positive, the momentum parameter m
T
∗ of a tachyon can be both positive and negative. In our onedimensional Theory of Relativity, we have two types of tachyons, those with a positive and those with a negative momentum parameter. We will be able to observe these
in the next chapter directly using the solutions of the sine-Gordon equation. If we
change the direction of the x-axis defined as positive, then the signs of the tachyon’s
momentum parameters also change.
For all of these tachyon properties that have the transformation Eq. (427) in common with the ‘normal’ particles, but vary so distinctly in their kinematics, there is a
simple mathematical frame. In the formalism of Minkowski geometry all particles are
described using spacetime vectors. Our ‘normal’ particles of type (a) are described by
so-called time-like vectors with the consequence of a preferred frame in which only
the so-called time component P o = E o /c o = m o c o of this vector is different from
zero. In this way, the restmass m o is well-defined for all ‘normal’ particles of the type
(a). In comparison to this, tachyons are described using this geometry by so-called
28 Particles and Tachyons
Then, per definition, the following applies to the velocity u of the tachyon (as for
every other particle) in the new coordinates
u =
d x
dt
= −u .
(432)
Furthermore, the momentum is, according to its physical attribution a spatial vector.
The components of spatial vectors change their signs during inversion, in other words
P
T = −P .
(433)
Due to u/sign u = u/sign u it follows from (425) that the tachyon parameter m
T
∗
must also change its sign so that (433) can be fulfilled,
m
T
∗ = −m
T
∗ .
(434)
We can make the following conclusion from this:
The tachyon parameter m T
∗ is a spatial vector.
This is the key needed to unlock and understand the properties of tachyons. The
quantity m
T
∗ has nothing at all in common with the restmass m o of a normal particle
of the type (a). The particle parameter m o , the restmass, is a spatial scalar, therefore
unchangeable with respect to any change of space coordinates. In comparison to
this, m
T
∗ multiplied by the constant quantity c o is a characteristic momentum value
that is assumed by the tachyon in the reference system where its energy is zero. The
occasional classification of the tachyon as a particle with an imaginary restmass is
therefore misleading.
The quantity m T
∗ is the momentum parameter of the tachyon.
Whilst the restmass m o of a ‘normal’ type (a) particle is always positive, the momentum parameter m
T
∗ of a tachyon can be both positive and negative. In our onedimensional Theory of Relativity, we have two types of tachyons, those with a positive and those with a negative momentum parameter. We will be able to observe these
in the next chapter directly using the solutions of the sine-Gordon equation. If we
change the direction of the x-axis defined as positive, then the signs of the tachyon’s
momentum parameters also change.
For all of these tachyon properties that have the transformation Eq. (427) in common with the ‘normal’ particles, but vary so distinctly in their kinematics, there is a
simple mathematical frame. In the formalism of Minkowski geometry all particles are
described using spacetime vectors. Our ‘normal’ particles of type (a) are described by
so-called time-like vectors with the consequence of a preferred frame in which only
the so-called time component P o = E o /c o = m o c o of this vector is different from
zero. In this way, the restmass m o is well-defined for all ‘normal’ particles of the type
(a). In comparison to this, tachyons are described using this geometry by so-called