An alternative proof of the relativistic addition of velocities can be performed using the conservation principles of energy and momentum, without the Lorentz transformation.

After the demonstrations of the principle of equivalence between energy and mass E = mc², of relativistic mass formula and of relativistic energy, the following derivation of relativistic addition of velocities represents the fourth link in the chain of proofs which, starting from classical physics, leads to Special Theory of Relativity on a simple and intuitive alternative path.

Description of the proof in reduced form

(For the detailed version of the derivation we refer to the tenth chapter of the book “Newton and Relativity”).

For the derivation, we consider a central collision of two particles, which leads to the formation of a new particle.

Central collision of two particles with the formation of a new particle

The two observers, O and O1, in relative motion to each other, examine the experiment independently of each other (see the animation) by applying the principles of conservation of energy and momentum.

Observer O measures the velocity of the particles

Observer O, being at rest with the formed particle, can measure the velocities of the colliding particles.

He calculates the energy of the system as a function of these speeds with the expression:

\[ \frac{m_{01}c^2}{\sqrt{1-\frac{v_1^{2}}{c^{2}}}}+\frac{m_{02}c^2}{\sqrt{1-\frac{v_2^{2}}{c^{2}}}}=m_0c^2 \quad \]

by equating the sum of the energies of the colliding particles with the energy of the particle formed after the collision.

Observer O1 measures the relative speed between the particles

Observer O1, being at rest with one of the colliding particles, can measure the speed of the other particle. This corresponds to the relative velocity between the two particles.

O1 calculates the momentum of the system before and after the collision as a function of the relative speed v12 and the speed v1 of the particle formed following the collision:

\[ \frac{m_{02}v_{12}}{\sqrt{1-\frac{v_{12}^{2}}{c^{2}}}}=\frac{m_{0}v_{1}}{\sqrt{1-\frac{v_1^{2}}{c^{2}}}} \]

The relative velocity is the relativistic addition of velocities

The relative velocity v12 between the particles, measured by the observer O1, represents the relativistic addition of their velocities.

To complete the proof, we set an equation using the results of the two observers.

In this way, we get the final expression of the relative velocity as a function of the speeds of the individual particles:

\[ v_{12} = \frac{v_1 + v_2}{1 + \frac{v_1v_2}{c^2}} \]

This expression represents the relativistic addition of velocities.

The detailed version of this alternative proof of the relativistic addition of velocities is reported in the tenth chapter of the book “Newton and Relativity“.

Another simpler proof applies to the relativistic addition of equal velocities.

Alternative proof of the addition of velocities - Simple sum and relativistic addition of equal velocities

Continue on the alternative path of relativistic proofs: Proof of the constancy of the speed of light.

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