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Introduction

Modern physics considers elementary particles as objects possessing both corpuscular and wave properties. In relativistic mechanics the energy of a particle is defined by the expression:

This formula shows that the energy of the particle grows with increasing momentum. However, if the particle is a closed wave structure of an electromagnetic wave and a standing wave in space created by the propagation of the electromagnetic wave, then its energy must be conserved within the system. This leads to an important question: how does a change in velocity affect the internal structure of the wave?

De Broglie wavelength and energy redistribution

According to de Broglie’s hypothesis, a moving particle possesses an associated wave with length:

where is Planck’s constant, a is momentum. An increase in velocity leads to an increase in momentum, and hence to a reduction in wavelength. This means that when a particle accelerates, its wave structure shrinks, changing the distribution of energy within the system itself.

Particle as a closed object

If a particle is a wave structure localised in space, its energy should not change, but only redistribute. Then:

• For an external observer, the energy of the particle grows due to the growth of momentum.
• Inside the particle system, the energy remains unchanged, changing only its configuration.

If the momentum increases with increasing velocity, the second term must decrease so that the total energy balance remains unchanged. This means that the de Broglie wavelength contraction is not just a consequence of motion, but a mechanism of energy redistribution inside the particle.

Derivation of the energy equation

If we assume that the energy of a particle at change of its speed of motion is conserved, then in this case there should be its redistribution between wave components in space and along the sphere. We again obtain the equation of the circle, similar to the equation in the derivation of the Lorentz transformation:

Let’s rewrite the second summand:

Then:

Let’s take out m₀²υ²c² in the first summand:

Now let’s solve the fraction:

Then:

Thus, we end up with the same result as the standard relativistic expression, but we emphasise the splitting of the energy into two contributions:

• One depends on speed and resembles kinetic energy,
• The second reduces the internal energy of the particle as it accelerates.

If we consider the equation for energy in the form:

Then it becomes clear why the photon formally has no rest mass. It would be more correct to say that the summand responsible for the effective mass:

will tend to zero.

Consequences

This conclusion confirms that the energy of the particle does not change at acceleration, but only redistributes:

• The first term expresses the momentum-dependent kinetic energy.
• The second term decreases the internal energy of the particle as the velocity increases.

Thus, the motion of a particle can be considered as a process of deformation of its wave structure, which naturally leads to relativistic effects without changing the total energy of the system.