An alternative derivation of the length contraction can be performed without the Lorentz transformation, using the conservation law of energy.

This derivation represents the final step in the sequence of relativistic demonstrations that started with the mass-energy equivalence principle derived from classical physics.

In fact, from the length contraction formula we easily derive the relativistic transformations for space and time (see demonstration).

With this last derivation we thus also reach the final purpose of the discussion presented in this website:

That of showing that the Lorentz transformation – and consequently the Theory of Special Relativity – can be interpreted as the result of a logical extension of Newtonian mechanics.

Description of the proof in reduced form

(For the detailed version of the proof we refer to the 11th chapter of the book “Newton and Relativity“).

For the derivation of length contraction, we imagine in an thought experiment the collision of an electron and a positron.

Collision of electron and positron detected by an observer at rest and by an observer in vertical motion

It is assumed that following the collision, a new particle is formed which is at the origin of a coordinate system at rest with respect to an observer O.

A second observer O’ moves at the same speed of the electron and positron, but in a vertical direction.

From the point of view of observer O:

According to the law of conservation of energy, the following energy balance relationship before and after the collision is valid:

\[ m_0c^2 = \frac{2m_{0e}c^2}{\sqrt{1-\frac{v^{2}}{c^{2}}}} \quad (11.2)\]

From the point of view of observer O’:

The particle formed after the collision moves downwards along the Y axis at the speed v (see animation).

Since the time t elapsed until the collision on the horizontal axis is the same for both observers, it follows:

\[ l^2+l’^2= v’^2t^2 \quad \quad (11.1)\]

Considering the principle of energy conservation from the point of view of the observer O’, we get the following relationship:

\[ \frac{m_0c^2}{\sqrt{1-\frac{v^2}{c^2}}} = \frac{2m_{0e}c^2}{\sqrt{1-\frac{v’^2}{c^2}}} \quad (11.3)\]

By replacing m0c2 with the term on the right of the relation (11.2), we get:

\[ 1-\frac{v^2}{c^2} = \sqrt{1-\frac{v’^2}{c^2}}\]

And taking into account the relation (11.1), after simple algebraic passages, we get the following relation, which expresses the relativistic contraction of lengths as a function of speed:

\[ l’=l\sqrt{1-\frac{v^2}{c^2}}\]
Alternative derivation of length contraction - Length Contraction as a Function of the Velocity

The detailed version of this alternative derivation of the length contraction and time dilation is given in the eleventh chapter of the book “Newton and Relativity“.


With the Relativistic Calculator you can calculate the length contraction as a function of speed.

Continue on the alternative path of relativistic proofs: Alternative derivation of the Lorentz transformations for space and time.

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