The Bizarre Case of the “Rod Light Clock”

 

The Bizarre Case of the “Rod Light Clock”

 

Pavle I. Premović

Laboratory for Geochemistry, Cosmochemistry&Astrochemistry,

University of Niš, pavleipremovic@yahii.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. From this postulate, time dilation, as well as the associated length contraction, inevitably results {1}. 

Simply speaking time dilation can be defined as the phenomenon that a clock at rest shows the time interval ΔT₀ which is shorter with respect to the time interval ΔT when it is moving with speed v. Physicists usually “use” as a clock a “traditional” light clock of SR (hereinafter “clock”) to illustrate this phenomenon and to derive the time dilation expression: 

ΔT = ΔT₀ /√(1-v²/c²)

where 1/√(1-v²/c²) is the Lorentz-Einstein factor or the time dilation factor γ.  

In this note, we will focus on length contraction which can be simply anticipated as a direct consequence of time dilation {1}. 

Again, simply speaking, the length l_c of a moving rod appears shorter than its length l when it rests. This effect is expressed by the length contraction relation:

l_c = l√(1-v²/c²).[1]

“Clock” consists of two plane-parallel mirrors M₁ and M₂ facing each other at a distance d apart as in Fig. 1a. The upper mirror M₁ has a light source at the center that emits a photon (or light signal) at 90 degrees in the direction of mirror M₂. The photon for a time interval ΔT₀ reaches M₂ traveling a distance d = cΔT₀.[1]

Let us assume that the “clock” is placed on the left end of a rod of a length l resting on the positive x-axis, Fig. 1a. Allow now this clock to move along the rod from its left end to the right end. Suppose now that at the start of measuring M₁ emits a photon towards M₂. A stationary observer records that the photon travels a larger distance D = cΔT for a longer (dilated) time ΔT (>ΔT₀) but an observer moving with the clock (hereinafter the moving observer) records that the photon moves a smaller distance d = cΔT₀. (Of course, ΔT₀ and ΔT are related by the time dilation expression ΔT = ΔT₀ /√(1-v²/c²). Simultaneously, “clock” travels l = vΔT.

We will now “use” the “rod light clock” (hereinafter “rod clock”) which is similar to the “clock” 

Figure 1: “Clock” (a) and ”rod clock” (b) experiments. (M₁^* and M₂^* are the subsequent positions of mirrors). For further details see the text.

except that M₁ and M₂ are “hooked” on the rod described above. If the “rod clock” is resting the photon emitted by M₁ reaches M₂ traveling a distance d = cΔT₀. If this clock is moving, we are dealing with its length contraction which is illustrated in Fig. 1b. The stationary observer measures that the photon travels a larger distance D_c = cΔT_c for a longer (dilated) time ΔT_c (>ΔT₀) but the moving observer records that the photon moves a smaller distance d = cΔT₀. Now, “rod clock” covers a contracted length l_c for a time interval ΔT_c: 

l_c = vΔT_c    ... (2). 

If ΔT_c and ΔT₀ are related by the time dilation relation ΔT_c = ΔT₀/√(1-v²/c²) then ΔT = ΔT_c or l = l_c. According to SR, this is impossible. 

This communication and several other our papers {2, 3, 4, 5}, as well as the works of numerous other researchers [1, and the references therein], show that the thought experiment with the “traditional” light clock appears to need at least some rethinking.

References

 

{1} J. A. Rybczyk, Millennium theory of relativity, mrelativity.net, 2018.

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

{3} P. I. Premović, Relativistic light clock experiments: time dilation or time contraction? The General Science Journal, December 2021.

{4} P. I. Premović, The moving light clock: which way the photon goes? The General Science Journal, December 2021.

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[1] I am slightly confused by the meaning of this expression which is mathematically correct. Indeed, it implies that when the speed of rod v tends to zero its “contracted” length l_c tends to the length l of the rod at rest.

[2] If we are dealing with a single photon as the light signal any process of its detection leads to its annihilation.