The derivation of E=mc² from classical physics represents the first step on an alternative path leading to the Theory of Relativity, as described on this website.

(This is an abridged version of the derivation of the equivalence between mass and energy. For the detailed version, click here).

The theory of relativity is not required to derive the equivalence between mass and energy.

Indeed, as Einstein demonstrated (see: Max BornDie Relativitätstheorie Einsteins – pages 244-247), E=mc2 can be derived using only classical physics laws.

In the third and fourth chapters of the book “Newton and Relativity”, it is shown that the equivalence of mass and energy is a consequence of the conservation of momentum in the interaction between matter and electromagnetic radiation.

Derivation of E=mc² based on “radiation pressure”

By emitting radiation, a body at rest receives a counter-impulse that causes it to move in the opposite direction:

Click on the rectangular area to visualize the process

The momentum balance between matter and radiation is sufficient to prove an increase or decrease in the mass of an absorbing or emitting body.

With the help of a thought experiment (see animation), the equation E=mc² can be obtained in a simple way, considering that the center of gravity of an inertial frame of reference remains at rest after an internal energy exchange.

Derivation of E=mc² based on the Doppler effect

Another simple demonstration of the mass-energy equivalence principle can be carried out based on the Doppler effect of electromagnetic radiation.

The thought experiment used is the annihilation of the electron-positron pair.

From the decay of the particle formed by the electron-positron collision, the emission of two photons in opposite directions can follow (see animation).

Derivation of E=mc² from classical physics based on the Doppler effect - Decay of unstable particle with emission of two photons in opposite directions

For an observer who moves at the speed of v (v << c) in the direction of one of the two photons, the following relationship is valid based on the conservation of momentum before and after the formation of photons:

\[m_0v=\frac{hf}{c}(1+\frac{v}{c})-\frac{hf}{c}(1-\frac{v}{c})\quad\quad(1) \]

Where: 

m0  is the mass of the unstable particle formed after the annihilation

c      is the speed of light in a vacuum

h      is Planck’s Constant

f       is the electromagnetic frequency associated with the photons

Considering that 2hf is the total energy E of the system, we obtain from the relation (1): E = m0c2.

The proofs of E=mc² derived from classical physics represent the connecting link between Newtonian and relativistic mechanics, as can be seen from the derivation of the relativistic mass formula.

Continue on the alternative path of relativistic proofs: Alternative derivation of relativistic mass.

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