Relativistic Light Clock Experiments: Time Dilation or Time Contraction?

 

Relativistic Light Clock Experiments: Time Dilation or Time Contraction?

 

Pavle I. Premović, Laboratory for Geochemistry, Cosmochemistry&Astrochemistry, 

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

Abstract

Time dilation of Special relativity can be easily derived from the thought experiments using the traditional light clock. However, time contraction is also possible to derive but using the “novel” light clock described in this communication. The question is which of these two clocks is right? 

Keywords: Light clock, thought experiment, Special relativity, time dilation, time contraction.

Introduction

One of the concepts of Special relativity (SR) is time dilation which depends upon the second postulate of SR that the speed of light c (= 2.99792×10⁸ m sec^-1) is the same in all inertial frames of reference [1]. According to this theory if ΔT₀ is the (proper) time interval of an event that occurs at the same position in an inertial frame, then the (improper) time interval ΔT of the same event has a longer duration as measured by an observer in an inertial frame that is in uniform motion relative to the first frame. Of course, this initial choice of which of frames is stationary and which is moving is arbitrary and it could be vice versa. It appears that time dilation is successfully tested by - muon experiment [2] and the experiment of synchronizing two atomic clocks [3].

Many physics textbooks demonstrate time dilation using a device known as a light clock. In this note, we will consider a set of thought experiments for time intervals ΔT₀ and ΔT employing the two different light clocks. Of course, you may find some of the following derivations in many elementary physics texts.

The traditional light clock consists of two-plane parallel mirrors M₁ and M₂ facing each other at a distance d apart as in Fig. 1a. The lower mirror M₁ has a light source at the center that emits a photon (or light signal/pulse) at 90 degrees in the direction of mirror M₂. For the sake of simplicity, we will consider in this note only the time interval needed for the photon to travel from mirror M₁ to mirror M₂. For the light clock at rest this is the (proper) time interval ΔT₀ = d/c.

Fig. 1. The traditional light clock: (a) no relative motion and (b) the clock moving at speed υ.

Now allow the same clock to be moving with a relative speed υ horizontally in the direction of the positive x-axis, Fig. 1b. The photon will now travel a larger distance D, and thus will take a longer time: the (improper) time interval ΔT = D/c. Elementary SR shows that ΔT₀ and ΔT are related by the following formula: ΔT = ΔT₀/√(1-υ²/c²) where 1/√(1-υ²/c²) is the Lorentz factor or the time dilation factor. Thus, the stationary observer measures time dilation for the moving classical light clock.

Let us now perform the thought experiments using a somewhat different (“novel”) light clock. This clock is similar to the traditional light clock except that the two plane-parallel mirrors M₁ and M₂ not facing each other, and they are at a distance D away, as shown in Fig. 2a. The proper time interval required for the photon to reach then M₂ is now ΔT₀ = D/c.

Fig. 2. The “novel” light clock: (a) no relative motion and (b) the clock moving at speed υ.

In the next thought experiment, we assume that the “novel” light clock moves, as the previously classical light clock, in the direction of the positive x-axis with the same speed v. The stationary observer observes that the photon travels from the mirror M₁ to the mirror M₂ following the path shown in Fig. 2b. She/he now measures the improper time interval ΔT = d/c. ΔT₀ and ΔT are now related with the following expression: ΔT₀ = ΔT√(1-υ²/c²). In other words, the stationary observer measures time contraction with the “novel” light clock. 

Thus, the two light clocks give different results. The classical light clock shows time dilation but the “novel” light clock time contraction. The question is now which of these two clocks is relativistically right?

 

References

[1] E. F. Taylor and J. A.Wheeler, Spacetime Physics: Introduction to Special Relativity, 2nd ed.  Freeman & Company, 1992.

[2] J. Bailey, K. Borer K, Combley, et al., Measurements of relativistic time dilation for positive and negative muons in a circular orbit, Nature 268, 301–305 (1977).

[3] J. Hafele R. Keating, Around the world atomic clocks: observed relativistic time gains, Science,  177, 167–168 (1972).