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.

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

Indeed, as Einstein demonstrated (see: Max BornDie Relativitätstheorie Einsteins – pages 244-247), one can derive E=mc2 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.

Using a thought experiment (see animation), the equation E=mc² is easily obtained from the relationship of the electromagnetic momentum p=E/c.

The proof is based on the fact that the center of mass of a inertial system remains at rest after an internal energy exchange.

##### Brief description of the demonstration in an abridged version

(For the detailed version of the proof we refer to the third chapter of the book "Newton and Relativity").

The thought experiment mentioned above allows us to take into consideration three physical aspects. These give the possibility to set as many elementary relationships.

• The first relationship concerns the times:

In the time Δt that the radiation takes to reach the absorbing body on the right, the emitting body on the left travels the stretch Δl at the speed v:

$\Delta t=\frac{l}{c}=\frac{\Delta l}{v}$

Where l represents the initial distance between the two bodies and c the speed of light.

• The second relation refers to the position of the center of gravity which - despite the displacement of the emitting body - remains unchanged:
$(m-\Delta m)(\Delta l+\frac{l}{2})=(m+\Delta m)\frac{l}{2}$

Where Δm represents the mass lost by the emitting body and acquired by the absorbing body.

• The third and final relation is provided by the principle of conservation of momentum.

According to this principle, the momentum of the emitted radiation E/c is equal to the momentum of the emitting body:

$(m-\Delta m)v=\frac{E}{c}$

Where E represents the energy of electromagnetic radiation.

The resolution of the system made up of the three simple equations above gives the relation:

$\Delta m=\frac{E}{c^2}$

of the principle of equivalence between mass and energy.

#### Derivation of E=mc² based on the Doppler effect

Another simple demonstration of the mass-energy equivalence principle can be done using the Doppler effect of electromagnetic radiation.

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

##### Brief description of the demonstration in an abridged version

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

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

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 particle that arises from the electron-positron collision

c      is the speed of light in a vacuum

h      is Planck’s Constant

f       is the electromagnetic frequency of 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 the derivation of the relativistic mass formula shows.

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

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