Relativistic Time Dilation or Contraction?

 

                                      

                                      Relativistic Time Dilation or Contraction? 

Pavle I. Premović

Laboratory for Geochemistry, Cosmochemistry&Astrochemistry, 

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

Abstract

We consider the time intervals of the half cycles of the classical light clock aligned along the direction of its motion. When the photon travels in this direction the light clock shows time dilation which is different from the Lorentz-Einstein type. If the photon moves opposite the clock shows time contraction. 

Keywords: Light clock, photon, time dilatation, time contraction, Lorentz-Einstein. 

Introduction

One of the tenets of Special relativity is its concept of time dilation which depends upon the second postulate of Special relativity that the speed of light c is the same in all inertial frames of reference [1]. According to Special relativity, if ΔT₀ is the time interval of an event that occurs at the same position in an inertial frame of reference, then the time interval of this event has a longer duration ΔT as measured by an observer in an inertial frame of reference that is in uniform motion relative to the first frame. Thus, both of the time intervals ΔT₀ (so-called proper time) and ΔT (so-called improper time) refer to the time of the same event occurring in the two different inertial frames of reference moving with a relative speed v. According to the Lorentz-Einstein transformations the two-time intervals ΔT and ΔT₀ are related by the formula: ΔT = ΔT₀/√(1-v²/c²) where c (≈ 3×10⁸ m sec^-1) is the speed of light and 1/√(1-v²/c²) is the Lorentz factor or time dilation factor. Apparently, time dilation has been demonstrated in many experiments, including muon experiments [2 - 4].

 

In this note, we especially focus our attention on the time intervals of the half cycles of the classical light clock colinear to the direction of its motion.

 

Discussion and Conclusions

The 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 at 90 degrees in the direction of mirror M₂. For the sake of simplicity, we will first consider the time for the photon to travel from mirror M₁ to mirror M₂. Of course, you may find all the following derivations in many elementary physics texts.

Fig. 1. Measurements and analyses made in different frames. The light clock is positioned perpendicular to the x-axis. (a): No relative motion. (b): light clock moving at speed v.

 For the light clock at rest, the proper time interval is then ΔT₀ = d/c. Now allow the same light clock to be moving with a certain relative speed v horizontally in the direction of the positive x-axis. An observer in the rest frame who is watching this clock could then design a diagram associated with a simple Pythagorean triangle, Fig. 1b. Clearly, a photon will now travel the larger distance L and thus it will take a longer (improper) time ΔT = L/c. Of course, ΔT₀ and ΔT are related by the relativistic time dilation expression ΔT = ΔT₀ /√(1-v²/c²).

Let us now perform the experiments using the same light clock as in Fig 1a but now aligned along the positive x-axis, as shown in Fig. 2a. Obviously, if this clock is resting the time interval is ΔT₀ = d/c.

Fig. 2. Measurements and analyses for the light clock positioned along the positive x-axis.

(a): No relative motion; (b) and (c): the light-clock moving at speed v.

Allow this clock to move in the direction of the positive x-axis with the speed v, Fig. 2b. To reach now the mirror M₂ the photon will travel the distance

 

cΔT₊ = D + vΔT₊

 

in the time interval

 

ΔT₊ = D/(c – v)

 

where D is the length of the light clock measured by a stationary observer.

 

To acquire a relationship between ΔT₀ and ΔT₊ one may invoke the relativistic
 length contraction D = d√(1-v²/c²) and then write

 

ΔT₊ = d√[(1 - v²/c²)/(c – v)].

 

Factoring out c in the dominator

 

ΔT₊ = d/c(√[(1 - v²/c²)/(1 – v/c)]

or

 

ΔT₊ = ΔT₀√[(1 - v²/c²)/(1 – v/c)].

 

After simplification, we get

 

ΔT₊ = ΔT₀√[(1 + υ/c)/(1 – υ/c)].

 

ΔT₊ represents time dilation as 1 + v/c greater than 1 – v/c. Of course, this dilation is different than the Lorentz-Einstein type of time dilation. 

For the back-reflected photon, the traveling time interval to the mirror M₁ is 

                                                                       cΔT_-  =  D - vΔT_-.

 

After a simple rearrangement

 

ΔT_- = D/(c + v).

Applying the same mathematics as above we get 

ΔT_- = ΔT₀√[(1 – υ/c)/(1 + υ/c)].

 

Clearly, ΔT_- describes time contraction. However, we must be aware that any photon detections at mirror M₁ or M₂ would involve its annihilation.

Thus, the same light clock shows time contraction ΔT₊ if its photon moves in the direction of motion and time dilation ΔT_- if it moves opposite to that direction. The question is now: which of these two-time intervals is “relativistically” right?

Finally, in the classical light clock experiment we usually consider the total time needed for a photon to complete the cycle

 

ΔT = ΔT₊ + ΔT_- = 2cD/(c² -  v²).

 

Factoring out c² in the denominator

 

ΔT = 2D/c(1 - v²/c²).

 

After simplification including length contraction, we get the relativistic time dilation 

ΔT = 2ΔT₀ /√(1-v²/c²).

We see, however, no reason why the complete time interval of the classical light clock aligned along the direction of motion is more “relativistically" speaking appropriate than the other two-time intervals.

 

References

[1]   N. D. Mermin. Space and Time in Special Relativity. McGraw-Hill, 1968.

[2] H. Rossi, Cosmic Rays. Chapter 8, McGraw-Hill, 1964.

[3] D. Frisch and J. Smith, Measurement of the relativistic time dilation using muons. Am. J. Phys. 31, 342–355 (1963).

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