The derivation of the Mass-Energy Equivalence E=mc² from classical physics is the first step on an alternative path leading to the Theory of Relativity, as described on this website.
The theory of relativity is not necessary for the derivation of the mass-energy equivalence E=mc².
In fact, as Einstein demonstrated (see: Max Born – Die Relativitätstheorie Einsteins – pages 244-247), one can derive E=mc2 using only the laws of classical physics.
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 the mass-energy equivalence E=mc² based on “radiation pressure”
The demonstration is based on the fact that by emitting radiation, a resting body receives a counter-impulse that causes it to move in the opposite direction.
In an isolated physical system, we consider the emission and absorption of radiation between two identical bodies of mass m:
Click on the rectangular area to visualise the process
After the internal energy exchange, the centre of mass of the physical system remains at rest.
Since, after emitting radiation, the emitting body is at a greater distance from the centre of mass than the absorbing body, it follows that its mass has decreased.
On the basis of a purely experimental analysis, it can therefore be deduced that:
A body that emits electromagnetic energy loses mass.
And, consequently :
A body that absorbs electromagnetic energy acquires mass.
From a strictly theoretical point of view, we have that:
The momentum balance between matter and radiation is sufficient to calculate the mass gain of an absorbing body and the mass loss of an emitting body.
By means of a thought experiment (see animation), the equation E=mc² is easily obtained from the electromagnetic momentum relationship p=E/c.
Description of the demonstration in abbreviated form
(For the detailed version of the proof see to the third chapter of the book "Newton and Relativity").
The thought experiment mentioned above allows us to consider three physical aspects. These give the possibility of setting up as many elementary relationships.
- The first relationship concerns the time:
In the time Δt that the radiation takes to reach the absorbing body on the right, the emitting body on the left travels the distance Δl at the speed v:
\[ \Delta t=\frac{l}{c}=\frac{\Delta l}{v} \]Where l is the initial distance between the two bodies and c is the speed of light.
- The second relationship relates to the position of the centre of gravity which remains unchanged despite the displacement of the emitting body:
Where Δm represents the mass lost by the emitting body and acquired by the absorbing body.
- The third and final relationship 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 is the energy of the electromagnetic radiation.
The solution of the system consisting of the three simple equations above gives the relationship:
\[ \Delta m=\frac{E}{c^2} \]of the equivalence principle between mass and energy.
Derivation of the mass-energy equivalence E=mc² from Doppler effect
Another simple derivation of the mass-energy equivalence E=mc² can be made using the Doppler effect of electromagnetic radiation.
The thought experiment used for this purpose is the annihilation of the electron-positron pair.
Description of the demonstration in abbreviated form
(For the detailed version of the proof see the fourth chapter of the book “Newton and Relativity“).
The decay of the particle created by the electron-positron collision can result in the emission of two photons in opposite directions (see animation).
For an observer moving at the speed of v (v << c) in the direction of one of the two photons, the following relationship applies, based on the conservation of momentum before and after the formation of the 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 resulting 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.
Continue on the alternative path of relativistic proofs: Alternative derivation of relativistic mass.
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