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Drecision.
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Here a minor technical point arisessuppose the two lengths Ù are not exactÌy
cqual? In fact, we surely cannot make them exactly equal. In that case we simply
turn the apparatus 90 degrees, so that BƠ is in the line of motion and
is perpendicular to the motion. Any small diference in length then becomes
unimportant, and what we look for is a shØf in the interference Íringes when we
rotate the apparatus.
In carrying out the experiment, Michelson and Morley oriented the apparatus
so that the line BE was nearly parallel to the earth's motion in its orbit (at
certain times of the day and night). This orbital speed is about 18 miles per
second, and any “ether drift” should be at least that much at some time of the day
or night and at some time during the year. 'Phe apparatus was amply sensitive
to observe such an efect, but no time difference was found—the velocity of the
carth through the ether could not be detected. The result of the experiment was
'The result of the Michelson-Morley experiment was very puzzling and most
disturbing. The frst truitful idea for fñnding a way out oŸ the impasse came from
Lorentz. He suggested that material bodies contract when they are moving, and
that this foreshortening is only in the direction of the motion, and also, that 1Í
the length is họ when a body is at rest, then when it moves with speed œ parallel
to its length, the new length, which we call Lịị (L-parallel), is given by
Lị = LoV1— u2/c2. (15.6)
'When this modification is applied to the Michelson-Morley interferometer appa-
ratus the distance from #Ö to Œ does not change, but the distance trom #Ö to #
is shortened to 4/1 — u2/c2. Therefore Eq. (15.5) is not changed, but the of
Edq. (15.4) must be changed in accordance with Eq. (15.6). When this is done we
obtain
take (2L/c)w1—u2/c2 —— 2Lƒc 15.7
"ốm. d5.)
Comparing this result with Eq. (15.5), we see that £+-+ta = 24. So ifthe apparatus
shrinks in the manner just described, we have a way of understanding why the
Michelson-Morley experiment gives no efect at all. Although the contraction
hypothesis successfully accounted for the negative result of the experiment, it was
open to the objection that it was invented for the express purpose of explaining
away the difficulty, and was too artificial. However, in many other experiments
to discover an ether wind, similar difficulties arose, until it appeared that nature
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was in a “conspiracy” to thwart man by introducing some new phenomenon ©o
undo every phenomenon that he thought would permit a measurement of ứ.
It was ultimately recognized, as Poincaré pointed out, that ø comgplete con-
spfrac is iselƒ a lau oƒ naturel Poincaré then proposed that there ¡s such a law
of nature, that it is not possible to discover an ether wind by øng experiment;
that is, there is no way to determine an absolute velocity.
15-4 Transformation of tỉme
In checking out whether the contraction idea is in harmony with the facts
in other experimentfs, i% turns out that everything is correct provided that the
tmes are also modifiled, in the manner expressed in the fourth equation of the
set (15.3). That is because the time £z, calculated for the trip trom Ö to Œ and
back, is not the same when calculated by a man performing the experiment in a
moving space ship as when calculated by a stationary observer who is watching
the space ship. To the man ín the ship the time is simply 2/e, but to the other
observer it is (21/c)/1— u2/c2 (Eq. 15.5). In other words, when the outsider
sees the man in the space ship lighting a cigar, all the actions appear to be
slower than normal, while to the man inside, everything moves at a normal rate.
So not only must the lengths shorten, but also the time-measuring instruments
(“clocks”) must apparently slow down. That is, when the clock in the space ship
records 1 second elapsed, as seen by the man in the ship, it shows 1/4/1 — u2/c2
second to the man outside.
This slowing of the clocks in a moving system is a very peculiar phenomenon,
and is worth an explanation. In order to understand this, we have to watch the
machinery of the clock and see what happens when it is moving. Since that 1s
rather dificult, we shall take a very simple kind of clock. The one we choose is
rather a silly kind of clock, but it will work in principle: it is a rod (meter stick)
with a mirror at each end, and when we start a light signal between the mirrors,
the light keeps going up and down, making a click every time it comes down,
like a standard ticking clock. We build ©wo such clocks, with exactly the same
lengths, and synchronize them by starting them together; then they agree always
thereafter, because they are the same in length, and light always travels with
speed c. We give one of these clocks to the man to take along in his space ship,
and he mounts the rod perpendicular to the direction of motion of the ship; then
the length of the rod will not change. How do we know that perpendicular lengths
do not change? The men can agree to make marks on each other's -meter stick
--- Trang 293 ---
as they pass each other. By symmetry, the 6wo marks must come at the same 1
and #-coordinates, since otherwise, when they get together to compare results,
one mark will be above or below the other, and so we could tell who was really
1noving.
Now let us see what happens to the moving clock. Before the man took 1§
aboard, he agreed that it was a nice, standard clock, and when he goes along in
the space ship he will not see anything peculiar. If he did, he would know he
was moving—If anything at all changed because of the motion, he could tell he
was moving. But the prineiple of relativity says this is impossible in a uniformly
moving system, so nothing has changed. Ôn the other hand, when the external
observer looks at the clock going by, he sees that the light, in going from mirror
to mirror, is “really” taking a zigzag path, since the rod is moving sidewise all
the while. We have already analyzed such a zigzag motion in connection with
the Michelson-Morley experiment. lfin a given time the rod moves forward
a distance proportional to in Eig. 15-3, the distance the light travels in the
same tỉme is proportional to e, and the vertical distance is therefore proportional
to W2 — u2.
That is, it takes a longer tứmne for light to go tom end to end in the moving
clock than in the stationary clock. Therefore the apparent time bebween clicks is
longer for the moving clock, in the same proportion as shown in the hypotenuse
of the triangle (that is the source of the square root expressions in our equations).
trom the figure it is also apparent that the greater œ is, the more slowly the
moving clock appears to run. Not only does this particular kind of clock run
more slowly, but if the theory of relativity is correct, any other clock, operating
on any principle whatsoever, would also appear to run slower, and in the same
proportion—we can say this without further analysis. Why is this so?
To answer the above question, suppose we had two other clocks made exactÌy