The Quantization of Classical Fields Equations and the Cyclic ...

v + t dv = c (16)
dt
This differential equation (16) is similar to (11). It can be integrated by the same procedures as what we
do for (11) and obtain (17).
t −
2 2 1/
2
= t0
( 1 v c ) (17)
It is different from the time dilation principle. The discrepancy may because that in its derivation of
special relativity, the limitation of physical quantities was not considered. (17) may be considered as
quantization of time dilation principle equation.
3.8 Mass and Velocity Limits Coupling Formula and Einstein Equation
Mass and velocity may also form a coupling pair and similar limits coupling formula is composed as shown
in (18)
∆ p / ∆m
→ c (18)
Here p= mv. A simple differential equation will be obtained. Integrating it by similar procedures as above,
one gets (19)
2 2 1/
2
m = m0
( 1−
v c ) (19)
This equation is also different from what Einstein derived from relativity.
m0
m = (20)
2 2 1/
2
( 1−
v c )
The equation (19) does also show that velocity cannot be greater than the velocity of light, which has
shown in Einstein’s relativity theory. However, in (19) as velocity starts from rest and reaches to c, the
mass decreases from 0 m to zero, instead of increasing from m0 to infinity by Einstein equation in (20).
Actually, what happens in reality is that the mass disappears when its velocity reaches c. The mass is
2
transformed into radiation. At the end of the process, the total kinetic energy lost will be 1
2 m 0c
. As we
know, the kinetic energy has been transferred to radiation. Particle annihilation generally happens in pairs.
That gives
E = moc 2 (21)
It is another Einstein’s famous mass and energy exchange formula.
Electron annihilation and pair production of electron with positron by radiation have been observed in the
laboratories. Only the rest mass of electron was involved in these mass and energy conversion processes. No
indication of raising the mass of electron or energy beyond m was observed.
Tolman in his book of relativity theory (Tolman, 1934) was able to derive from Einstein equation the total
kinetic energy of a particle of rest mass m 0 moving with velocity v
m c
( 1−
v c
2
0c
2
E = 0
2 2
2
− m c
1 2 o (22)
)
which reduces as velocity small compared with that of light, to the familiar Newtonian expression
2
E = 1
2 m0v
Therefore Einstein equation works fine under ordinary condition as on astrophysics.
However, it was not possible to show that as the velocity reaches c, (22) reduced to (21). As pointed
out in the book, actually, (21) was not derived but postulated.