The proof of the constancy of the speed of light can be obtained starting from classical physics, with a sequence of derivations.
The following text is taken from the thirteenth chapter of the book “Newton and Relativity“.
Two different types of constancy
The speed of light occurs as a constant c in many physical formulas. One of these formulas is that of the momentum p = E/c of light radiation.
When the light source is at rest, the speed of light is constant for every frequency, from radio waves to gamma radiation.
Constancy for all frequencies does not violate the laws of classical physics.
In this article, however, 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 among all inertial frames, as demonstrated by the Michelson and Morley experiment.
Since this constancy occurs for every relative velocity between sender and receiver, it no longer conforms to Galilei’s transformation and therefore conflicts with 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). This involves the exclusion of the existence of an absolute time.
It follows that in each inertial system, a speed-dependent proper time flows.
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.
Hence, the need to give the natural phenomenon of the constancy of the speed of light the attribute of a fundamental postulate of the laws of physics was recognized.
On the other hand, it was believed that this postulate was not compatible with Newtonian laws.
This conviction was then followed by the renunciation of the attempt to explain the phenomenon of the constancy of the speed of light with classical mechanics.
Rather, scientists became convinced of the need to come up with a new physical theory.
Therefore, the birth of the theory of relativity is closely linked to the supposed 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 treatise (see here), we possess 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 carrying out the proof, we want to briefly summarize backwards the sequence of proofs that led to the velocity addition formula:
- The expression \[ v_{12} = \frac{v_1 + v_2}{1 + \frac{v_1v_2}{c^2}} \quad (10.6)\] of the velocity addition 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 resting energy) 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 with the use of the relation \[ m = \frac{m_0}{\sqrt{1-\frac{v^{2}}{c^{2}}}} \quad\quad\quad(5.4)\] which expresses the dependence of inertia on speed.
- In the fifth chapter, we have, on the other hand, demonstrated that the relation (5.4) is, in turn, a direct consequence of Newton‘s Second Law of Motion and of the mass-energy equivalence principle.
- The mass-energy equivalence principle was obtained in the third and fourth chapters with the sole use of classical physics.
The conclusion of this argumentation is that the proof of the relativistic addition formula of velocities can be carried out starting from classical physics and without using the postulate of the constancy of the speed of light.
A theoretical proof
Thus, the velocity addition formula theoretically proves the principle of the constancy of the speed of light among all inertial frames of reference.
To this end, we consider a light source moving with respect to an observer. The latter, wanting to calculate the relative speed of light vl, 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:
\[ v_{l} = \frac{c + v_s}{1 + \frac{cv_s}{c^2}} \]Based on this relationship, vl equals c for any velocity of the light source.
This means that the speed of light is the same for every inertial frame, regardless of its velocity.
Considering the complete procedure that was used to arrive at this proof, we can state that:
The constancy of the speed of light can be demonstrated in a theoretical way, that is, even without the use of experiments, but in confirmation of these.
The detailed version of the proof of the constancy of the speed of light is described in the thirteenth chapter of the book “Newton and Relativity”.
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Continue on the alternative path of relativistic proofs: Alternative derivation of length contraction.
Very interesting!