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Introduction

Lorentz transformations play a key role in relativistic physics, describing how the spatial and temporal coordinates of objects change as they move. They were originally introduced as a mathematical tool to explain the invariance of the speed of light, but later became the basis of the special theory of relativity. However, if we consider particles as wave structures, we can arrive at the same transformations through a simple geometrical interpretation.

1. How did Lorenz get his transformation?

The problem that Lorenz was solving

Until the end of the 19th century, it was assumed that light propagates in the ether, a hypothetical medium that fills all space.

But the Michelson-Morley experiment (1887) showed that the speed of light does not depend on the motion of the Earth. This was strange: if the Earth moves through the ether, why does not the speed of light change?

Lorentz proposed that objects moving through the ether contract in the direction of motion.

2. Length of the rod in a moving system

Lorentz imagined a rod of length L at rest.

• When it is stationary, its length is simply L.
• When it is travelling at velocity υ, the time at different points on the rod will not be the same due to relativistic time.

To find the new length L′, he used Lorentz transformations (named later in his honour).

3. Output of length reduction

Consider two observers:

• One in the resting system (where the rod length is L).
• The second in a moving system (the observing rod moving with velocity υ).

Step 1: Write down the coordinates of the rod ends

• In the rest frame, the ends of the rod are at x₁ and x₂. Then its length is:

L=x₂-x₁ ​

• In a moving system (a frame of reference moving with velocity υ) the coordinates are transformed by Lorentz formulas: x′=γ(x-υt)

where

Step 2: Calculate the length of the rod in a moving system

Let us write down the new coordinates of the rod ends:

Subtract:

since x₂​− x₁=L, we obtain:

where:

• L is the length of the object in the resting system,
• L’ is the length in the moving system,
• υ is the speed of movement,
• c is the speed of light.

1.2 Formal mathematical reasoning

Using his coordinate transformations, Lorentz considered two observers:

• One is in a resting system where the rod has length L.
• The second moves with the rod and measures its length as L’.

He derived the length conversion based on the assumption that the speed of light is constant in any frame of reference and more time is required for a signal travelling in a moving system.

2. Wave interpretation of length contraction

2.1 Particle as a wave

Instead of considering the particle as a point object, suppose it is a wave structure whose propagation in its own frame of reference is spherical at the speed of light c.

In this case, the particle has two velocities:

• The speed of wave propagation inside the particle (radial), which is c at rest.
• The velocity of the particle itself in spaceυ.

2.2 Geometric relationship of velocities

If a particle moves, its internal wave structure changes. Since the total velocity cannot exceed c, the radial component of the velocity υ_sub decreases:

Hence:

Since the size of a particle is determined by its internal wave, a reduction in this velocity means a reduction in the effective radius of the particle:

Since the length of an object is proportional to its radius, we get:

which corresponds exactly to the Lorentz formula!

3. Conclusions

• Lorentz transformations can be obtained not only through mathematical postulates, but also through the wave nature of particles.
• Length contraction is a natural consequence of limiting the speed of light to internal motion in a particle.
• The wave interpretation makes relativistic effects understandable in terms of the physical structure of particles, not just axioms.