The “Color” of Light in the Light Clock Experiment

 

The “Color” of Light in the Light Clock Experiment

 

Pavle I. Premović

Laboratory for Geochemistry, Cosmochemistry&Astrochemistry, 

University of Niš, pavleipremovic@yahoo.com, Niš, Serbia

A fundamental tenet of Special relativity (SR) is that the light speed c (= 2.99792×10⁸ m s^-1) is constant in all inertial frames.[1] From this postulate, time dilation, as well as the associated length contraction, inevitably result {2} Simply speaking time dilation can be defined as the phenomenon that a clock at rest shows the time interval ΔT₀ which is shorter than the time interval ΔT when it is moving with a speed υ. Physicists usually “use” as a clock a “traditional” light clock of SR ((hereinafter “the light clock”) to illustrate this phenomenon and to derive the time dilation expression

ΔT = ΔT₀ /√1 – υ²/c²)

where 1/√1 – υ²/c²) is the Lorentz-Einstein factor or the time dilation factor γ.

Elementary physics tells us that when a ray of light strikes a plane mirror, the light ray reflects off the mirror. According to the law of reflection, the angle of incidence equals the angle of reflection. In our previous paper {2}, we have shown that when the light clock stops then the light angle of incidence is greater than zero but its angle of reflection is zero degrees. This observation disagrees with the law of reflection.

We will here consider the light clock experiment and the associated blueshift of the reflected light, i. e.  the “colors” of the incidence and reflected light.

The traditional light clock usually consists of two plane-parallel mirrors M₁ and M₂ that face each other and are separated by a distance d, Fig. 1a. A light beam originating from mirror M₁ reaches mirror M₂. We will employ a monochromatic light beam. This light beam traces out a path of length d.

Figure 1: a) the light clock at rest and b) the light clock moves at a speed υ and then abruptly stops. See the text.

After reaching mirror M₂ the light beam reflects to mirror M₁ following the law of reflection {2}.

Now allow the same light clock to be moving with a certain relative speed υ horizontally in the direction of the positive x-axis. The light beam of mirror M₁ reaches mirror M₂ traveling the larger distance D. After reaching mirror M₂ the light beam would reflect to mirror M₁ according to the law of reflection {2}.

Allow now that the light clock makes a full stop when the light beam reaches mirror M₂. In this case, a stationary observer who is watching the light clock could design the following diagram, Fig. 1b. Of course, the number of periods would be identical for the incidence and reflected lights. We know that the frequency equals the speed of light divided by the wavelength (or ν = c/λ) then the number of their wavelengths would be also the same. In expression,

d = Nλ_d and D = Nλ_D

_awhere λ_D is the wavelength of light emitted by M₁ and λ_d is the wavelength of light reflected by M₂. N can be approximated as a very large natural number if it is larger or equal to say 10⁴ {3}.

Since λ_D > λ_d the reflected light has a different "color" with a shorter wavelength than the incidence light or the reflected light is blueshifted. In other words, the stationary observer finds that the energy of the reflected light is higher than the energy of the incidence light. The question is now is where does this extra energy come from?

The intrinsic energy of the photon emitted by this clock in rest (the reflected light) and in motion (the incidence light) would be hν₀ and hν, where h (= 6.63×10^-34 J sec) is Planck’s constant and ν₀ and ν₀ are the corresponding frequencies. The only possible answer to the above question would be that the difference between the two energies hν₀ – hν is equal to the relativistic kinetic energy of this clock or

E_K = hν₀ – hν    … (1)

where hν₀ and hν are the intrinsic energy of the photon emitted by the light clock in rest and in motion, respectively. This is in accordance with the law of conservation energy.  Eqn. (1) shows that if the speed of the light clock υ tends to zero (or υ → 0) then hν → hν₀ or E_K (= hν₀ – hν) → 0. If this speed approaches the speed of light c then hν → 0 or E_K (= hν₀ – hν) → hν₀.[2]

If the rest mass of the light clock is m₀ then this theory says that its relativistic mass m would increase by the Lorentz-Einstein factor or

m = m₀/√(1 – υ²/c²) … (2).[3]

SR now gives the following equation for the relativistic kinetic energy of this clock

E_K = mc² – m₀c² … (3).

Combining eqns. (2) and (3) we find that the relativistic kinetic energy of the light clock (or in general of a massive particle) is given by

E_K = {[1/√(1- υ²/c²)] ^_ 1}m₀c²    … (4).

If the speed of the light clock υ tends to zero (or υ → 0) then E_K (= hν₀ – hν) → 0. However, when the speed of the source υ approaches the speed of light c then according to the above equation E_K (= hν₀ – hν) → ∞. This contradicts the above conclusion for υ → c drawn from eqn. (1) based on the Principle of energy conservation. Moreover, the value of hν₀ (or ν₀) is initially fixed in the above thought experiment there is no way to make hν₀ – hν → ∞.

The question which arises now is how to explain this discrepancy?

One possibility is that the law of conservation energy does not hold in the above thought experiment. However, this law is a fundamental concept of physics.

If the law of energy conservation holds then eqn. (4) for the relativistic kinetic energy of the light clock described – i. e. of a massive particle in general - is not correct. It follows that eqns. (2) and (3) of SR are not correct either. This is a serious statement because this theory formulated by Einstein in 1905 is one of the cornerstones of modern physics.[4]

The only possible explanation is that the traditional thought experiment with the light clock cannot be used to demonstrate time dilation and hence length contraction. As we pointed out in previous communication {5}, it appears that the thought experiment with the “traditional” light clock appears to need at least some rethinking.

References

{1} R. G. Ziefle, Einstein’s special relativity violates the constancy of the velocity c of light under one-way conditions and thus contradicts the behavior of electromagnetic radiation. Physics Essays 34, 275-279 (2021).

{2} P. I. Premović, The light clock experiments and the law of reflection. The General Science Journal, December 2021.

{3} P. I. Premović, The Big Bang Universe and the Principle of energy conservation. The General Science Journal, December 2021.

{4} R. G. Ziefle, Refutation of Einstein's relativity on the basis of the incorrect derivation of the inertial mass increase violating the principle of energy conservation. A paradigm shift in physics. Physics Essays 33, 466-478 (2020).

{5} P. I. Premović, The “bizarre” case of the “rod light clock”. The General Science Journal, December 2021.

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[1] It is important to note here that Ziefle [1, and references therein] demonstrated that SR is not compatible with the constancy of light that is measured on Earth.

[2] Mathematically speaking, hν = hν₀√(1 – υ²/c²). In other words, the intrinsic energy of a photon depends on the speed of its source.

[3] Of note, Ziefle [4, and references therein] shows that the explanation of the inertial mass increase by SR violates the Principle of energy conservation. 

[4] Ziefle [1] argued SR is an unrealistic theory.