The theoretical proof of the constancy of the speed of light can be obtained by a sequence of derivations.
The following text is taken from the thirteenth chapter of the book “Newton and Relativity“.
Two different kinds of constancy
The speed of light appears as a constant c in many physical formulae. One of these formulae is that of the momentum p = E/c of light radiation.
When the light source is at rest, the speed of light is constant at every frequency, from radio waves to gamma rays.
Constancy at all frequencies does not violate the laws of classical physics.
However, in this article we are not referring to this concept of constancy, but to the fact that the speed of light remains constant even with a moving light source and between all inertial frames, as demonstrated by the Michelson and Morley experiment.
Since this constancy holds for any relative velocity between sender and receiver, it no longer conforms to Galilei’s transformation and thus contradicts a fundamental principle of classical mechanics.
The direct consequence of the constancy of the speed of light for all frames of reference is the relativity of simultaneity (see animation).
It follows that in every inertial system flows a velocity dependent proper time. This excludes the existence of an absolute time.
A postulate in conflict with classical mechanics
When Michelson and Morley made known the results of their experiments in 1887, scientists all over the world were faced with a great surprise: the experimental observations were in conflict with the principles of mechanics.
The experiment with Michelson’s interferometer had indeed shown that the speed of light is always constant in a vacuum, regardless of the state of rest or motion of the light source.
It was therefore recognised that the natural phenomenon of the constancy of the speed of light had to be given the attribute of a fundamental postulate of the laws of physics.
On the other hand, it was believed that this postulate was incompatible with Newton’s laws.
As a result of this conviction, the attempt to explain the speed of light constancy using classical mechanics was abandoned.
Instead, scientists became convinced of the need to develop a new physical theory.
The birth of the theory of relativity is therefore closely linked to the alleged incompatibility of Newtonian mechanics with the natural phenomenon of the constancy of the speed of light.
From classical physics to the constancy of the speed of light
However, on the basis of the results achieved in this paper (see here), we have the premises to prove the constancy of the speed of light for any relative speed between source and observer in a purely theoretical way.
In fact, we can prove the constancy of the speed of light with the relativistic velocity addition formula.
Before we carry out the demonstration, let us summarise backwards the proofs that led to the velocity addition formula:
- The velocity addition expression \[ v_{12} = \frac{v_1 + v_2}{1 + \frac{v_1v_2}{c^2}} \quad (10.6)\] was obtained by applying the principles of conservation of energy and momentum to the central collision of two particles. For the energy balance, the total energies of the particles (i.e., the sum of their kinetic and rest energies) were used.
- The formula for the total energy of a particle \[ E = \frac{m_0c^2}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\quad\quad (6.5) \] was obtained in the sixth chapter using the relation \[ m = \frac{m_0}{\sqrt{1-\frac{v^{2}}{c^{2}}}} \quad\quad\quad(5.4)\] expressing the dependence of inertia on speed.
- In the fifth chapter, however, we showed that relation (5.4) is a direct consequence of Newton‘s Second Law of Motion and of the mass-energy equivalence.
- The mass-energy equivalence principle was obtained in the third and fourth chapters using only classical physics.
The conclusion of this argumentation is that the proof of the relativistic addition formula of velocities can be done starting from classical physics and without using the postulate of the constancy of the speed of light.
A theoretical proof
The velocity addition formula thus provides a theoretical proof of the constancy of the speed of light in all inertial frames of reference.
To this end, let us consider a source of light moving with respect to an observer. If the observer wishes to calculate the relative speed of light vl, he can use the expression (10.6):
\[ v_{12} = \frac{v_1 + v_2}{1 + \frac{v_1v_2}{c^2}} \quad (10.6)\]Substituting the speed vs of the light source in place of v1 and in place of v2 the speed c which is measured for the light emitted by a source at rest, we obtain:
If we replace v1 by the speed vs of the light source and v2 by the speed c of light emitted by a source at rest, we obtain:
\[ v_{l} = \frac{c + v_s}{1 + \frac{cv_s}{c^2}} \]Based on this relationship, vl is equal to c for any velocity of the light source.
This means that the speed of light is the same for any inertial frame, regardless of its velocity.
If we look at the whole process by which this proof was obtained, we can say that:
The constancy of the speed of light can be proved theoretically. That is, even without the use of experiments, but by confirming them.
The detailed version of the proof is described in the thirteenth chapter of the book “Newton and Relativity”.
Continue on the alternative path of relativistic proofs: Alternative derivation of length contraction.
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Very interesting!