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

(This is an abridged version of the alternative derivation of length contraction. For the detailed version, click here).

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 from the Pythagorean theorem:

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

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“.

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

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