Bookmark 9: Free motion of the Dirac particle with the CM spin along the OY axis and the CM velocity ≡=0.1. The transversal velocity of the CC is =- and the period, the time of a turn is T()=γ(), where is the time of a turn when the CM is at rest. The local clock is going at a slower rate when moving.
v
v
y
u
⟂
2
c
2
v
v
v
T
0
T
0
What we measure is that the local clock of the moving particle is going at a slower rate when moving. It takes more time to complete a cycle. This time dilation is independent of the relative orientation between the CM spin and the initial CM velocity and it is the same as the time dilation between the laboratory observer and the inertial observer at rest with the CM of the particle.
S
CM
v
17. Dynamic equations and Particle Initial Conditions
17. Dynamic equations and Particle Initial Conditions
In this first article we are considering the standalone particle in absence of external fields and forces, so the dynamic equations of the electron will be as follows.
The Center of Mass is not accelerated, it moves inertially at a constant speed, so:
The Center of Mass is not accelerated, it moves inertially at a constant speed, so:
2
q
2
t
v
t
(
39
)and the Center of Charge acceleration will be according to (29):
a
u
t
2
r
2
t
2
c
v
u
2
-
q
r
q
r
(
40
)Where we see that the fourth - order differential equations for the point can be decoupled into a system of second-order differential equations for the points and
r
q
r
To integrate the system of differential equations (39) and (40), we have to establish the appropriate boundary conditions for the positions and velocities of both points, the CC and the CM of each particle . These are the 12 boundary values for the variables , , and . These parameters are those parameters that define the relationship between the Center of Mass observer of the particle and any arbitrary inertial observer or laboratory observer who sees the CM of the particle moving at the speed .
r
0
u
0
q
0
v
0
v
Lets calculate now the initial or boundary conditions of the particle as a function of the parameters that describe the system (user defined parameters in the code of section 20): - The initial location of the Center of Mass of the particle is defined by the coordinates =(dx,dy,dz). - The velocity of the Center of Mass of the particle is defined by its module v, β the zenithal angle and λ the azimuthal angle. - The Center of Mass Spin is defined by its zenithal angle θ and azimuthal angle ϕ. - The Center of Charge initial location is defined through the initial phase ψ.
d
v