UPDATE October 6, 2023: This blog post is erroneous. The velocity v_y of a particle slows down if the particle is pulled to the x direction with a force F. This is because the y momentum does not change.
In our box example we assumed that v_y stays the same. That lead us to claim that the inertia is different for a box where particles are bouncing up and down.
----
-------------------
| |
| | ----> F
| |
-------------------
M
We pull the box with a force F. Does the acceleration only depend on M? We assume that the inside of the box stays approximately constant during the pull. The mass M does not slide to the left end of the box.
^ v_y
|
|
• --> v_x
m
Assume that we have the box filled with very fast moving particles that bounce up and down. We want to give them a velocity vector to the x direction. The rest mass of a particle is m. The velocity is v, and it is divided into components v_x and v_y.
Let us denote
γ = 1 / sqrt(1 - v² / c²).
The x momentum of the particle m, in the laboratory frame, is
p_x = γ m v_x
= m v_x
* 1 / sqrt(1 - (v_x² + v_y²) / c²).
However, we have to take into account that the y momentum of the particle cannot grow at all if we only accelerate it with a force which pulls it to the x direction. The velocity v_y must slow down if we accelerate the box to the x direction.
It is well known that the inertial mass of a moving particle is not its mass-energy. See the reply by "Paul T." in the link above.
Paradox of two boxes with the same mass-energy but different inertia
rope
__ __
|__| ------------- |__|
box 1 box 2
Let us have two boxes which are initially filled with idle, non-moving, matter. The boxes pull the ends of a rope and eventually meet at the center of mass.
But if the box 1 decides to convert its contents into moving matter before pulling the rope, then it will have more inertia, and the boxes meet to the left from the center of mass.
This breaks conservation of the center of mass. We firmly believe that the conservation law is true.
Theorem. If special relativity is true and conservation of the center of mass is true, then all mass-energy must be kinetic energy. There cannot exist idle, non-moving mass-energy at all.
If all mass particles are essentially boxes full of photons, then special relativity may be reduced to newtonian physics plus the Doppler effect. It might be that the aether hypothesis of the 19th century produces special relativity. In the aether hypothesis, particles with a rest mass were difficult to handle.
For example, the equivalence of mass and energy in special relativity comes from the fact that also the rest mass of a particle really is kinetic energy.
The hypothesis that all mass-energy is kinetic energy is mentioned in literature, but we have not seen any proof that it is implied by special relativity.
Conclusions
Is there a calculation error above? It is astonishing if people have not realized that special relativity plus a simple conservation law entirely ban the existence of a rest mass.
We must check if boxes of photons behave according to the laws of special relativity. It would be a two-tier system: at the low level, light-speed particles bouncing around. At a higher level, boxes which contain these particles.
The zitterbewegung of the electron suggests that it is a light-speed particle in a tight loop.
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