We have already seen that time is perceived differently by observers in relative motion to each other. We are now going to determine an equation that shows this mathematically. Much of the work that follows uses the relationship that distance = velocity × time or, when we are talking about the passage of light, distance = ct. This relationship comes from the definition of velocity.
The following thought experiment uses a ‘light clock’. As shown in figure 5.8, a light pulse is released by a lamp, travels the length of the clock and is then reflected back to a sensor next to the lamp. When the sensor receives the pulse of light, it goes ‘click’.
For this thought experiment we shall return to the train scenario favoured by Einstein. Imagine a traveller, seated in a speeding train. The light clock is arranged vertically, with the lamp at the ceiling and the
(a) As seen by train traveller
(b) As seen by stationary observer
Back door open Front door open
v
v Original position of lamp
Light waves haven’t reached front door yet.
Front door still closed Back door open
Figure 5.7 A thought experiment to illustrate the relativity of simultaneity
Mirror Lamp
Receiver L
‘Click’
Figure 5.8 A light clock
CHAPTER 5 SPACE AND TIME
79
mirror on the floor. An observer is watching from the embankment out-side the train. Our question now is this: when a light pulse is released, how long does it take to travel down to the mirror and return back to the ceiling, as seen by both the train traveller and the observer?
Let us first examine the situation as seen by the train traveller in the rest frame; that is, the frame within which the event occurs. If L is the height of the train carriage, for the total journey we can say that:
distance = 2L= ct0
where
t0= time taken as seen by traveller L= height of the carriage
so that
t0 = .
Examine now the situation as seen by the observer on the embank-ment. Figure 5.9 (page 80) compares the way the situation is viewed by each person. From outside the train the observer sees the light travelling along a much longer journey, and its length can be determined using Pythagoras’ theorem:
Total journey = ctv =
= .
Squaring this expression gives: c2tv
2= 4L2 + v2tv 2. Rearranging this leads to: tv
2= .
Taking the square root of both sides: tv =
but, from above, t0 = .
Substituting this into the expression gives: tv =
(the time dilation equation) where
t0 = time taken in the rest frame of reference
= proper time
tv = time taken as seen from the frame of reference in relative motion to the rest frame
v = velocity of the train c = speed of light.
Note that t0 is the time taken for the clock to go ‘click’ as observed by the train traveller, while tv is the time taken as observed by the person on the embankment. Looking at the last expression above, we can see that A rest frame is the frame of
reference within which a measured event occurs or a measured object lies at rest.
2L ---c
2 L2 vtv
---2
2 + 4L2+v2ttvv
2
4L2 c2–v2
( )
---2L c 1 v2
c2 ---–
---2L ---c t0
1 v2 c2 ---–
---SPACE
80
the term is always less than one so that tv is always greater than t0. This means that the clock takes longer to go ‘click’ as observed by the person on the embankment or, put another way, the outside observer hears the light clock clicking slower than does the train traveller. Time is passing more slowly on the train as observed by the person outside the train!
Figure 5.9 The length of the path of the light is perceived differently by the train traveller and the observer on the embankment. The observer sees the light travel further but with the same speed, hence time slowed down on the train.
This effect is called time dilation and can be generally stated as follows:
the time taken for an event to occur within its own rest frame is called the proper time t0. Measurements of this time, tv, made from any other inertial reference frame in relative motion to the first, are always greater.
The degree of time dilation varies with velocity as shown in figure 5.10.
It can be most simply stated as: moving clocks run slow.
This rather startling conclusion has been experimentally verified by comparing atomic clocks that have been flown over long journeys with clocks that have remained stationary for the same period. These experi-ments are possible only because of the extreme accuracy of atomic clocks built over the last few decades, even though Einstein predicted this effect about 100 years ago.
Further supporting evidence has been found in the abundance of mesons striking the ground after having been created in the upper atmosphere by incoming cosmic rays. What is surprising is that the mesons have a velocity of about 0.996c and, at that speed, should take approximately 16 µs to travel through the atmosphere. However, when
1 v2 c2 ---eBookplus –
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doc-0037 (a) As seen by the train traveller
(b) As seen by an observer on an embankment outside the train L
vtv L
vtv
Time dilation is the slowing down of events as observed from a reference frame in relative motion.
1 00
12 34 56 78
109 Time dilation
Velocity v_c tv
t0
—
Figure 5.10 The degree of time dilation varies with velocity.
CHAPTER 5 SPACE AND TIME
81
measured in a laboratory, mesons have an average lifetime of approxi-mately 2.2 µs. This anomaly can be explained by the fact that 2.2 µs represents their proper lifetime, as measured in their rest frame, whereas 16 µs is a dilated lifetime due to their relativistic speed.