In this gedanken experiment, illustrated in Figure 2.2, we let the plane of a mirror M9 be perpendicular to the Y9-axis at some distance Dy9 above the origin of coordinates. An observer in S9 sends a light pulse up the Y9-axis, where it is reflected upon its incident path by the mirror M9 and even-tually absorbed at its point of origin O9. In accordance with Figure 2.2, any S9 observer will measure the distance from O9 to M9 to be given by
(2.3)
where Dt9 is defined as the time it takes the light pulse to travel from O9 to M9 and back to O9.
Observers in the S frame of reference do not see the motion and path of the light pulse in S9 as being vertical, since M9 (at rest relative to S9) is moving at a uniform speed u relative to their reference frame. Instead, they observe the motion and path of the light pulse in S9 to be something like the isosceles triangle depicted in Figure 2.2, where Dt ; tB2 tAis the time interval, according to S observers, for the light pulse to go from A to M9 to B. It should be obvious that, whereas S9 observers need only one clock to measure the time interval Dt9, S observers need two clocks for their measurement of the corresponding time interval Dt. In system S,
9 5 9 ,
5 9
y c c
2 2
D D 1
t Dt
c m Ch. 2 Basic Concepts of Einsteinian Relativity 41
Z
A B
Z9
Y9 Y
Z9 Z9
09 09
12
M9 M9 M9
09 uDt Dy9 cDt9
Dy
X, X 9
Y9(tA) Y9(tB)
u u
u
12 cDt
12
Figure 2.2
The path of a vertical pulse of light being re-flected from a mirror in S9, as viewed by ob-servers in S9 and S.
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we need one clock at A and another at B, and since the two clocks are spa-tially separated, it is essential that they be synchronized. Later, we will con-sider a method by which clocks can be synchronized, but for now, we simply assume the synchronization of clocks at A and B has been effected.
Clearly, Dt corresponds to the difference between the reading of the two clocks at A and B in system S.
Viewing the left triangle of Figure 2.2 and invoking the Pythagorean theorem, we have
(Dy)21 ( uDt)25 ( cDt)2, which is easily solved for Dy in the form
(2.4) where we have set
(2.5)
Here, Dy is the vertical displacement of the mirror M9, as measured by observers in system S. Now, using Equation 2.1 from the previous gedanken experiment (Dy 5 Dy9) along with Equations 2.3 and 2.4, we have
(2.6) By defining
(2.7)
and solving Equation 2.6 for Dt, we obtain
Dt 5 gD t9. (2.8)
The result expressed by Equation 2.8 gives a comparison of time in-tervals measured in two different inertial reference frames. The meaning and implications of this result may need some elaboration. First, consider that in the limit as u approaches c in Equation 2.7 (remember b 5 u/c), g approaches `. However, for u ,, c, g approaches 1, and Equation 2.8 re-duces to the Galilean transformation (Equation 1.29) in accordance with
,
5 2
y c 1
2
1 b2
D Dt
.
;c b u
.
2 5
c 1 c
2
1 2
2
b 1
Dt D 9t
2
; 1
1
2
g
b
1 2
1 2
2.3 Time Interval Comparisons 42
Time Dilation
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the correspondence principle. Thus, the range of values for g can be ex-pressed as
1 # g # `, (2.9)
which implies that Dt . D t9 in Equation 2.8 for u not small. That is, the time interval Dt measured between two events (emission and absorption of the light pulse) occurring at spatially different positions in system S is greater than the proper time D t9, which is the time interval measured in sys-tem S9 between two events occurring at the same position. Because of the factor g in Equation 2.8, Dt is greater than the proper time and is referred to as a dilated time. Hence, the name time dilation that is normally associ-ated with Equation 2.8.
It is important to keep in mind Einstein’s first postulate, because if the experiment were performed in system S, observers in S9 would con-clude that Dt9 5 gDt. The significance of the time dilation result is that it emphasizes how time interval measurements differ between inertial ob-servers in relative motion, because of differing physical measurements. A time interval between two events is always shortest in a system where the events occur at the same position and dilated by the factor g in all other in-ertial systems. In solving problems associated with the time dilation equa-tion, it is most convenient and less confusing to identify the proper time as occurring in the primed system, with the dilated time then given by Equation 2.8 for the unprimed inertial system. As the concept of time di-lation is most incredible, a few examples will be presented in an attempt to clarify the subtleties of the phenomenon.
If you believe the secret of eternal youth is in keeping on the move, you are not far from wrong. In fact, time dilation predicts that if you have a twin, your biological clocks will be different, if one twin is traveling uni-formly at a relativistic speed (e.g., u . 0.4c) relative to the other. For ex-ample, consider that at age twenty you take off in a rocket ship traveling at 0.866 the speed of light. You leave behind a younger brother of age ten and travel through space for twenty years (according to your clock). When you return home, at age forty, you will find your kid brother to be fifty years old! He will have aged by Dt = 40 years, while you have aged by only Dt9 = 20 yrs. That is, time dilation gives
As compared to your brother, your biological clock and aging process was slowed by the time dilation effect. It should be emphasized that time dila-tion is a real effect that applies not just to clocks but to time itself—time flows at different rates to different inertial observers.
/ . / 40 .
5 9 5 yrs 2 5 yrs
t g t 20 1 866c c2
D D ^ h ^ h
Ch. 2 Basic Concepts of Einsteinian Relativity 43
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Other time dilation effects have been observed in laboratories for the lifetimes of radioactive particles. As a particular example, consider the decay of unstable elementary particles called muons (or mu-mesons). A muon is observed in a laboratory to decay into an electron in an average time of 2.20 3 1026sec, after it comes into being. Normally, muons are created in the upper atmosphere by cosmic ray particles and travel with a uniform mean speed of 2.994 3 108m/s 5 0.9987c toward the earth’s sur-face. In their lifetime, the muons should be able to travel a distance of
according to the laws of classical kinematics. Since they are created at al-titudes exceeding 6000 m, they should rarely reach the earth’s surface. But they do reach the earth’s surface and in profusion—approximately 207 muons per square meter per second are detected at sea level.
The muon paradox is immediately resolved, if we take time dilation into account. According to observers in the earth’s reference frame, the muon’s proper mean lifetime is Dt9 5 2.20 3 1026s. The mean time that we observe the muons in motion is a time dilated to the value
Thus, according to Einstein’s postulates, the muons should (and do) travel a mean distance of
in our reference frame.