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V1=u2/'
Ụ =U,;
z2, (15.3)
rằ t— u#/c2
1_— u2/c2`
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namely, Maxwells equations remain in the same form when this transformation
is applied to theml Equations (15.3) are known as a Eoreniz transformation.
Hinstein, following a suggestion originally made by Poincaré, then proposed that
dÌl the phụs¿cal laes should be oŸ such a kind that they remain unchơngcd under
Loreniz transformation. In other words, we should change, not the laws of
electrodynamics, but the laws of mechanics. How shall we change Newton”s laws
so that £hew will remain unchanged by the Lorentz transformation? lf this goal is
set, we then have to rewrite Newton”s equations in such a way that the conditions
we have imposed are satisied. As it turned out, the only requirement is that the
mass ?m in Newton”s equations must be replaced by the form shown in Eq. (15.1).
'When this change is made, Newton's laws and the laws of electrodynamiecs will
harmomize. 'Phen if we use the Lorentz transformation in comparing Moe's
measurements with Joe”s, we shall never be able to detect whether either is
moving, because the form of all the equations wiïll be the same in both coordinate
systemsl
Tt is interesting to discuss what it means that we replace the old transformation
between the coordinates and time with a new one, because the old one (Galilean)
seems to be self-evident, and the new one (Lorentz2) looks peculiar. We wish
to know whether it is logically and experimentally possible that the new, and
not the old, transformation can be correct. 'To find that out, i% is not enough
to study the laws of mechanics but, as Einstein did, we too must analyze our
ideas of space and f#me in order to understand this transformation. We shall
have to discuss these ideas and their implications for mechanics at some length,
So we say in advance that the efort will be justifed, since the results agree with
experIment.
15-3 The Michelson-Morley experiment
As mentioned above, attempts were made to determine the absolute velocity
of the earth through the hypothetical “ether” that was supposed to pervade all
space. The most famous of these experiments is one performed by Michelson and
Morley in 1887. It was 18 years later before the negative results oŸ the experiment
were fñnally explained, by Einstein.
The Michelson-Morley experiment was performed with an apparatus like that
shown schematically in Fig. 15-2. This apparatus is essentially comprised of a
light source A, a partially silvered glass plate Ö, and two mirrors Œ and #, all
mounted on a rigid base. The mirrors are placed at equal distances Ù from ÿÖ.
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L4
Souce “|À 2 \⁄8» Là
xx”Š5————‹>— =
‹ .. lÍ
Waves 3 S Waves out
in phase S < LG of phase
: : Š € Đa
DF Dị!
Fig. 15-2. Schematic diagram of the Michelson-Morley experiment.
The plate splits an oncoming beam of light, and the two resulting beams
continue in mutually perpendicular directions to the mirrors, where they are
reflected back to . Ôn arriving back at , the two beams are recombined as
two superposed beams, D and ?'. Tf the time taken for the light to go from ?
to È and back is the same as the time from ?Ö to Œ and back, the emerging
beams D and # wïll be in phase and will reinforce each other, but If the two
times difer slightly, the beams will be slightly out of phase and interference will
result. If the apparatus is “at rest” in the ether, the times should be precisely
cqual, but iŸ it is moving toward the right with a velocity w, there should be a
diference in the times. Let us see why.
Pirst, let us calculate the time required for the light to go from to and
back. Let us say that the time for light to go from plate Ö to mirror 2 is É\,
and the time for the return is ¿¿. Now, while the light is on its way from
to the mirror, the apparatus moves a distance œ#, so the light must traverse a
distance Ù + œ#t, at the speed c. We can also express this distance as c1, so we
cị = Ù+uớa, Or tạ = L/(c— 0).
(This result is also obvious from the point of view that the velocity of light relative
to the apparatus is e— , so the time is the length Ƒ divided by c— 0.) In a like
manner, the time #s can be calculated. During this time the plate Ö advances a
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distance œ£¿, so the return distance of the light is Ù — uứ¿. Then we have
ca —= ÙL— tua, OT tạ = L/(c+ 0).
Then the total time is
t + ta = 2Lc/(c2 — u2).
For convenience in later comparison of times we write this as
fi+tạ= ——>—a- 15.4
¬—.. /cœ2 05-4)
Our second calculation will be of the time ¿¿ for the light to go trom to
the mirror Œ. As before, during tỉme ¿ the mirror Œ moves to the right a
distance œ‡a to the positlon C”; in the same time, the light travels a distance ca
along the hypotenuse oŸ a triangle, which is ĐC”. For this right triangle we have
(cts)? = L2 + (u£z)Ÿ
L2 = c”1 — u?tạ = (c?— u2)tã,
from which we get
tạ = L/VWc2— u2.
For the return trip from C” the distance is the same, as can be seen from the
symmetry of the fñgure; therefore the return time is also the same, and the total
time is 2f¿. With a little rearrangement of the form we can write
2L 2L/c
2fz = ————p — —2Hkc _. (15.5)
ve2—u2 v1-—u2/c2
WS are now able to compare the times taken by the two beams of light. In
expressions (15.4) and (15.5) the numerators are identical, and represent the
time that would be taken ïf the apparatus were at rest. In the denominators, the
term ”/c? will be small, unless is comparable in size to c. The denominators
represent the modifications in the times caused by the motion of the apparatus.
And behold, these modifcations are no the sœme—the time to go to C and
back is a little less than the time to # and back, even though the mirrors are
equidistant from 7, and all we have to do is to measure that diference with