Our August 25, 2023 post may offer the explanation. A particle with a rest mass is really a box where massless subparticles bounce around.
The inertia of a box of wave packets
Let us assume that the subparticles actually are wave packets which travel at the speed of light c in the laboratory frame.
^
| | F pull
|
| rope
|
--------------------
|~~~~~~~~~~|
|~~~~~~~~~~|
|~~~~~~~~~~|
--------------------
box filled with
wave packets
^ y
|
-------> x
If we pull the box with a force F, the wave packets will bounce against the walls of the box and become blueshifted. In quantum mechanics, a blueshift means more energy in a photon.
In newtonian mechanics, a wave packet which is reflected from a wall, exerts a pressure on the wall. If we fight against this pressure by pushing the wall, we put more energy into the wave packet. This may be the origin of the quantum mechanical formula
E = h f,
where the energy E of a quantum is linear in the frequency f. There h is the Planck constant. We cannot explain the value of the Planck constant, but we may be able to explain the linearity in f.
We have to calculate how does the energy of the wave packets increase when we accelerate the box. Do we obtain the relativistic formula
m / sqrt(1 - v² / c²) ?
One may object that the laboratory frame is not special, and our calculation assumes an "aether" where waves propagate. But if we can reproduce all the equations of special relativity in this aether, then the aether hypothesis makes sense. The hypothesis does not fix the "true" speed of the aether. One is still free to use any inertial frame.
Let us assume that we have the box in the diagram above, and wave packets are bouncing back and forth in the x direction.
Our method of accelerating is such that the x momentum p_x of a wave packet must stay constant. To get the box to move up, we have to add y momentum p_y.
Let us assume that the box was initially static in the laboratory frame, and we were able to make it to move at a velocity v_y upward.
We have for the wave packet
v_x² + v_y² = c².
The momentum to the x direction is
p_x = E / c² * v_x
where E is the energy of the wave packet. Then
E = p_x c² * 1 / sqrt(c² - v_y²)
= p_x c * 1 / sqrt(1 - v_y² / c²).
We got the right formula!
In the derivation we need:
1. A hypothesis that space is filled with aether, relative to which light moves at a speed c.
2. The laboratory is at rest relative to aether.
3. Every particle is a box filled with light.
4. Newtonian mechanics.
We can probably drop condition 2 by defining simultaneousness with light signals, just like it is done in special relativity. If we make a clock by letting a light signal bounce inside a box, we get the slowdown of time in special relativity from newtonian mechanics:
t' = t sqrt(1 - v² / c²).
The inertia of the box is
E / c².
The inertia of the box truly grows with the velocity v, in the sense of newtonian mechanics.
An application: charges which attract each other
q
• ----> v
r = distance (q, Q)
● ---> v
Q
Let us have opposite charges Q and q moving at a velocity v relative to the laboratory frame.
They approach each other slower than static charges because the inertia is larger. But now we face a dilemma. The acceleration is reduced by the time dilation, by a factor
1 - v² / c²,
while the inertia only grew by a factor
1 / sqrt(1 - v² / c²).
The discrepancy should be explained by the electric and magnetic fields of the moving charge Q?
Let us calculate the magnetic repulsion. Let the distance of q and Q be r. The magnetic field of the moving charge Q is
B = μ₀ / (4 π) * Q v / r²
= 1 / (c² * 4 π ε₀) * Q v / r².
The magnetic force on q is
F_B = q v² / c² * F_E,
where F_E is the electric force. The magnetic force is repulsive.
The electric field of Q is squeezed by length contraction, and is stronger by a factor
1 / sqrt(1 - v² / c²).
Now it matches. The slow acceleration is explained by
1. the increased inertia of q, and
2. the repulsive force of the magnetic field of Q, and
3. the increased electric attraction by Q.
Conclusions
An "aether" model may be able to explain all the equations of special relativity, if we treat particles as boxes where light-speed, massless subparticles bounce around.
The practical value of such an aether model is not large, since it is easier to use the equations of special relativity. However, the aether model might give us some hints about what is the "right" fundamental model of gravity.
We will not analyze the aether model further now, but will continue to study gravity once again.
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