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alike with wheels and gears, or perhaps based on radioactive decay, or something
else. Then we adjust these clocks so they both run in precise synchronism with
our frst clocks. When light goes up and back in the frst clocks and announces
1ts arrival with a click, the new models also complete some sort of cycle, which
they simultaneously announce by some doubly coincident flash, or bong, or other
signal. One of these clocks is taken into the space ship, along with the fñrst kind.
Perhaps #h2s clock will not run slower, but will continue to keep the same time
as its stationary counterpart, and thus disagree with the other moving clock. Ah
no, 1ƒ that should happen, the man ïn the ship could use this mismatch between
his two clocks to determine the speed of his ship, which we have been supposing
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Mirror
Sr system D
xí Ề |
Photocell
¬ reflected ========e
.. | 3UT _""
S system +
x x4 N5 D
Pulse _—> bàce-g Pulse
emitted () received
wc2— ư]
Fig. 15-3. (a) A “tight clock” at rest in the S” system. (b) The same
clock, moving through the S system. (c) Illustration of the diagonal
path taken by the light beam In a moving “light clock.”
--- Trang 295 ---
is Impossible. We need not knou anthing about the machinerg of the new clock
that might cause the efect——we simply know that whatever the reason, it will
appear to run slow, just like the first one.
Now ïf alÏ moving clocks run sÌower, iÝ no way oŸ measuring time gives anything
but a slower rate, we shall Just have to say, In a certain sense, that #ữne ?‡sclƒ
appears to be slower in a space ship. All the phenomena there—the man”s
pulse rate, his thought processes, the time he takes to light a cigar, how long 1
takes to grow up and get old—all these things must be slowed down in the same
proportion, because he cannot tell he is moving. The biologists and medical men
sometimes say it is not quite certain that the time it takes for a cancer to develop
will be longer in a space ship, but from the viewpoint of a modern physicist
1t is nearly certain; otherwise one could use the rate oŸ cancer development to
determine the speed of the ship!
A very interesting example of the slowing of time with motion is furnished by
mu-mesons (muons), which are particles that disintegrate spontaneously after an
avcrage lifetime of 2.2 x 10”8 sec. They come to the earth in cosmic rays, and
can also be produced artifcially in the laboratory. 5ome of them disintegrate
in midaïr, but the remainder disintegrate only after they encounter a piece of
material and stop. It is clear that in is short lifetime a muon cannot travel,
even at the speed of light, mụch more than 600 meters. But although the muons
are created at the top of the atmosphere, some 10 kilometers up, yet they are
actually found in a laboratory down here, in cosmic rays. How can that be?
The answer is that diferent muons move at various speeds, some of which are
very close to the speed of light. While from their own point of view they live
only about 2 /sec, from our point of view they live considerably longer—enough
longer that they may reach the earth. 'Phe factor by which the tỉme is increased
has already been given as 1/4/1 — ^2/c2. The average life has been measured
quite accurately for muons of diferent velocities, and the values agree closely
with the formula.
W©e do not know why the meson disintegrates or what its machinery 1s, but
we do know its behavior satisfes the principle of relativity. That is the utility of
the principle of relativity——it permits us to make predictions, even about things
that otherwise we do not know mụch about. Eor example, before we have any
idea at all about what makes the meson disintegrate, we can still predict that
when it is moving at nine-tenths of the speed of light, the apparent length of time
that it lasts is (2.2 x 10”8)/4/1 — 92/102 sec; and our prediction works—that is
the good thing about ït.
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15-5 The Lorentz contraction
Now let us return to the Lorentz transformation (15.3) and try 6o get a better
understanding of the relationship between the (z,,z,f) and the (z',,z,t)
coordinate systems, which we shall call the S and 5” systems, or Joe and Moe
systems, respectively. We have already noted that the first equation is based on
the Lorentz suggestion of contraction along the z-direction; how can we prove
that a contraction takes place? In the Michelson-Morley experiment, we now
appreciate that the fransuerse arm BC cannot change length, by the principle
of relativity; yet the null result of the experiment demands that the £#mes must
be equal. So, in order for the experiment to give a null result, the longitudinal
am BE must appear shorter, by the square root 4/1 — u2/c2. What does thìs
contraction mean, in terms of measurements made by Joe and Moe? Suppose
that Moe, moving with the Š” system in the z-direction, is measuring the #“-
coordinate of some point with a meter stick. He lays the stick down #“ tỉmes, so
he thinks the distance is ø“ meters. From the viewpoint of Joe in the Š system,
however, Moe is using a foreshortened ruler, so the “real” distance measured is
#“V1— u2/c2 meters. Then if the 5“ system has travelled a distance uý away
from the Š system, the Š observer would say that the same point, measured in
his coordinates, is at a distance œ = #/4/1— u2/c2 + uÈ, or
; % — UuÈ
= — ma. n.')
V1= u2/e
which is the first equation of the Lorentz transformation.
15-6 Simultaneity
In an analogous way, because of the difference in time scales, the denominator
expression is introduced into the fourth equation of the Lorentz transformation.
The most interesting term in that equation is the #/c in the numerator, because
that is quite new and unexpected. Now what does that mean? If we look at the
situation carefully we see that events that occur at two separated places at the
same time, as seen by Moe in ®”, do nø‡ happen at the same tỉme as viewed by
Joe in 6. lf one event occurs at point zø+ at time #o and the other event at #s
and £o (the same time), we ñnd that the two corresponding times /¡ and £2 difer
by an amount
tứ — u(Œ1 — +2) /c?
¿2 V1—u2/c2 `
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