Walk to wrong ideas (E/=mc2)

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Albert Einstein in his 1905 paper "Ist die Trägheit eines Körpers von
seinem Energieinhalt abhängig?" ("Does the Inertia of a Body Depend
Upon Its Energy Content?"- Annalen der Physik- September 27, said:

"If a body gives off the energy L in the form of radiation, its
mass
diminishes by L / c^2, "radiation" means electromagnetic
radiation
or light, and mass means the ordinary newtonian mass of a slow
moving object."

Let's see how Einstein rationalized a 'correct' derivation.
Einstein's thought experiment:

"Considered a body at rest with mass m. If the body is
examined in a frame moving with nonrelativistic velocity
v, it is no longer at rest and in the moving frame it has
momentum mv.
Suppose now that the body emits two pulses of light to the
left and to the right, each carrying an equal amount of energy
E/2. Since the two pulses are equal, the object remains at rest
after the emission since the two beams are equal in strength
and carry opposite momentum."

At this point, I ask:
Can a body emit two opposite pulses simultaneously, each one
with energy E/2? Say that body is an electron. A spontaneous
emission means you can't predict when a photon will be emitted.
Only within an ensemble of atoms/molecules you can perform
statistics to compute a probability of emission. Thus, Einstein's
thought experiment is fatally flawed. The probabibility that an
electron emitted two simultaneous opposite photons of equal
energy is actually zero for an electron alone. But, let's give
that thought experiment the benefit of the doubt and assume the
simultaneous emission takes place in a macroscopic body
(ensemble).

Einstein continued his razionale:
"But if we consider the same process in a frame moving
with velocity v to the left, the pulse moving to the left will
be redshifted while the pulse moving to the right will be
blueshifted. The blue light carries more momentum than
the red light, so that the momentum of the light in the moving
frame is not balanced. The light is carrying some net momentum
to the right.
But the object hasn't changed its velocity before or after
the emission. Yet in this frame it lost some right-momentum to
the light. The only way it could have lost momentum is by losing
mass."

Now, I ask: Is losing mass actually the only way it could have lost
momentum? At this step of Einstein's rationale, I can see how the
previous flawed assumption is deeply influencing on this one.
the claim "the object hasn't changed its velocity before or after the
emission" can't be admitted. Two simultaneous opposite pulses of
equal energy E/2 is a phenomenon with no physical meaning.
There must exist at least an infinitesimal random time interval
between
both pulses. So, the body actually change its momentum when the
first pulse is emitted, and then it will recover part of the lost
momentum
when the opposite pulse is emitted. The question is, will it recover
the
whole lost momentum? Since the time interval between those two
opposite pulses of equal energy is a random infinitesimal time, dt,
then the body will not be at rest after the former emission. So, we
can
see losing mass is not the only way a body could have lost momentum.
After the first of those two non-simultaneous emissions has taken
place
we will be able to see the body slightly moving either in one
direction or
in its opposite one, but we will be unable to predit which one the
body will
be following. Therefore, in a frame moving with velocity v to the left
before
emissions, we will see the body, after the first emission, is moving
to the
right with a velocity v' = - v + dv, where dv is an infinitesimal
velocity whose
sign +/- can't be predicted. After the latter emission, the body will
become
at rest, but in a different location. Only when that thought
experiment has
been performed many times in a more realistic framework, we could
compute
statistics on spontaneous emissions, so we would then be able to
predict certain
probability for the sign +/- of dv. Once that statistics has been
performed,
we will have a more complete picture or the motion of that inertial
macroscopic body (ensemble of atoms/molecules).


Conclusion: Kinetic energy of a body with mass m moving at
speed v can be statistically computed, taking into account
spontaneous emissions, as

E_k = mc^2 (Exp(-v/c) - 1 + v/c)
and since momentum is p = mv, we get
E_k = mc^2 (Exp(-v/c) - 1) + pc

Therefore, total energy is
E = mc^2 + E_k ,
and energy-momentum relation is then
E = mc^2 Exp(-v/c) + pc

This latter equation and the relativistic relation

E^2 = m^2c^4 + (pc)^2,

result experimentally undistinguisable if the momentum
p in the latter is the relativistic momentum, and in the former
it is the Newtonian one.

Proof:
Let's put both equations as

E_n = mc^2 Exp(-v/c) + p_nc
E_r^2 = m^2c^4 + (p_rc)^2,

such that p_r = p_n/sqrt(1 - v^2/c^2).

Then
E_r^2 = m^2c^4 + (p_nc)^2 / (1 - v^2/c^2),
E_r^2 = m^2c^4 + m^2v^2c^2 / (1 - v^2/c^2),
E_r = mc^2 sqrt(1 + (v^2/c^2)/(1 - v^2/c^2)),
and
E_n = mc^2 Exp(-v/c) + mvc,
E_n = mc^2 ( Exp(-v/c) + v/c)

The expansion series for E_r is

E_r = mc^2( 1 + v^2/2c^2 - v^3/6c^3 + ... ),

and the expansion series for E_n is

E_n = mc^2( 1 + v^2/2c^2 + 3v^4/8c^4 + ... ).

Both match to second order approximation of v/c.
Thus, they are experimentally undistinguishable, but
we clearly can see that E_n > E_r.



.

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