28 Particles and Tachyons
301
ticle is therefore defined by means of its energy. We found our further considerations
on particles in SRT on the equation
E = m c
2
o .
(416)
We further assume that the interaction between particles during collisions is based
on the mechanical laws of energy and momentum conservation. We now begin with
the dependence of the mass m of a particle on its velocity u. In other words, we
look for the function f (u) in equation
m = ξ o f (u) .
(417)
For the velocity u we make no limitations, the constant ξ o is a parameter that, from
the start we do not see as a restmass.
We observe the total inelastic collision in the reference system
of two particles that have the same parameter ξ o and the same opposite momentum, so that
the total momentum disappears according to momentum conservation. For the
momentum conservation in
, which we would normally note as ξ o f (u
) u
+
ξ o f (−u
) (−u
) = 0 with an even function f (−u) = f (u), we make a more general ansatz, which is justificated only later on,
ξ o f (u
) u
+ δ · ξ o f (−u
) (−u
) = 0 .
Momentum conservation
in
(418)
Here, we introduced a factor δ for the parameter of the second particle that can
assume the values +1 or −1. We will make use of this later on. Furthermore, we
do not automatically have the function f (u) at our disposal. The ansatz (418) will
prove very useful when intend on registering particles of type (b) during the process
of collision.
According to our assumptions of a total inelastic collision, both particles should
remain static after the collision fused to one single particle in
with the mass
parameter M o . We can therefore write in
for the constancy of the total energy
before and after the collision,
ξ o f (u
) c
2
o + δ · ξ o f (−u
) c
2
o = M o f (0) c
2
o ,
Energy conservation
in
(419)
so that
ξ o
f (u
) + δ · f (−u
)
= M o .
(420)
In the reference system, o the mass M o has after the collision the velocity V . The
velocities of the colliding particles before the collision occurs can be calculated in
o using the composition of velocities (415). We insert u
in
for the velocity of
the first particle and for the second particle with the parameter δ · ξ o we insert −u
.
We therefore receive for the energy conservation in o
301
ticle is therefore defined by means of its energy. We found our further considerations
on particles in SRT on the equation
E = m c
2
o .
(416)
We further assume that the interaction between particles during collisions is based
on the mechanical laws of energy and momentum conservation. We now begin with
the dependence of the mass m of a particle on its velocity u. In other words, we
look for the function f (u) in equation
m = ξ o f (u) .
(417)
For the velocity u we make no limitations, the constant ξ o is a parameter that, from
the start we do not see as a restmass.
We observe the total inelastic collision in the reference system
of two particles that have the same parameter ξ o and the same opposite momentum, so that
the total momentum disappears according to momentum conservation. For the
momentum conservation in
, which we would normally note as ξ o f (u
) u
+
ξ o f (−u
) (−u
) = 0 with an even function f (−u) = f (u), we make a more general ansatz, which is justificated only later on,
ξ o f (u
) u
+ δ · ξ o f (−u
) (−u
) = 0 .
Momentum conservation
in
(418)
Here, we introduced a factor δ for the parameter of the second particle that can
assume the values +1 or −1. We will make use of this later on. Furthermore, we
do not automatically have the function f (u) at our disposal. The ansatz (418) will
prove very useful when intend on registering particles of type (b) during the process
of collision.
According to our assumptions of a total inelastic collision, both particles should
remain static after the collision fused to one single particle in
with the mass
parameter M o . We can therefore write in
for the constancy of the total energy
before and after the collision,
ξ o f (u
) c
2
o + δ · ξ o f (−u
) c
2
o = M o f (0) c
2
o ,
Energy conservation
in
(419)
so that
ξ o
f (u
) + δ · f (−u
)
= M o .
(420)
In the reference system, o the mass M o has after the collision the velocity V . The
velocities of the colliding particles before the collision occurs can be calculated in
o using the composition of velocities (415). We insert u
in
for the velocity of
the first particle and for the second particle with the parameter δ · ξ o we insert −u
.
We therefore receive for the energy conservation in o