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Unformatted text preview: 15 The Special Theory of Relativity 15—1 The principle of relativity For over 200 years the equations of motion enunciated by Newton were be
lieved to describe nature correctly, and the ﬁrst time that an error in these laws
was discovered, the way to correct it was also discovered. Both the error and its
correction were discovered by Einstein in 1905. Newton’s Second Law, which we have expressed by the equation F = d(mv)/dt, was stated with the tacit assumption that m is a constant, but we now know that
this is not true, and that the mass of a body increases with velocity. In Einstein’s
corrected formula m has the value mo \/1 — vz/cZ’ (15.1) where the “rest mass” m0 represents the mass of a body that is not moving and
c is the speed of light, which is about 3 X 105 km  sec"1 or about 186,000
mi  sec—1. For those who want to learn just enough about it so they can solve problems,
that is all there is to the theory of relativity—it just changes Newton’s laws by
introducing a correction factor to the mass. From the formula itself it is easy to
see that this mass increase is very small in ordinary circumstances. If the velocity
is even as great as that of a satellite, which goes around the earth at 5 mi/sec,
then v/c = 5/ 186,000: putting this value into the formula shows that the cor
recdon to the mass is only one part in two to three billion, which is nearly impossible
to observe. Actually, the correctness of the formula has been amply conﬁrmed by
the observation of many kinds of particles, moving at speeds ranging up to practi
cally the speed of light. However, because the effect is ordinarily so small, it
seems remarkable that it was discovered theoretically before it was discovered
experimentally. Empirically, at a sufﬁciently high velocity, the effect is very large,
but it was not discovered that way. Therefore it is interesting to see how a law
that involved so delicate a modiﬁcation (at the time when it was ﬁrst discovered)
was brought to light by a combination of experiments and physical reasoning.
Contributions to the discovery were made by a number of people, the ﬁnal result
of whose work was Einstein’s discovery. There are really two Einstein theories of relativity. This chapter is concerned
with the Special Theory of Relativity, which dates from 1905. In 1915 Einstein
published an additional theory, called the General Theory of Relativity. This
latter theory deals with the extension of the Special Theory to the case of the law
of gravitation; we shall not discuss the General Theory here. The principle of relativity was ﬁrst stated by Newton, in one of his corollaries
to the laws of motion: “The motions of bodies included in a given space are the
same among themselves, whether that space is at rest or moves uniformly forward
in a straight line.” This means, for example, that if a space ship is drifting along
at a uniform speed, all experiments performed in the space ship and all the phenom
ena in the space ship will appear the same as if the ship were not moving, pro
vided, of course, that one does not look outside. That is the meaning of the princi
ple of relativity. This is a simple enough idea, and the only question is whether it
is true that in all experiments performed inside a moving system the laws of physics 15—1 151 The principle of relativity
15—2 The Lorentz transformation 15—3 The MichelsonMorley
experiment 15—4 Transformation of time
15—5 The Lorentz contraction
15—6 Simultaneity 157 Fourvectors 15—8 Relativistic dynamics 15—9 Equivalence of mass and energy P Fig. 15—1. Two coordinate systems
1 in uniform relative motion along their
‘ xaxes. will appear the same as they would if the system were standing still. Let us ﬁrst
investigate whether Newton’s laws appear the same in the moving system. Suppose that Moe is moving in the xdirection with a uniform velocity u, and
he measures the position of a certain point, shown in Fig. 15—1. He designates the
“xdistance” of the point in his coordinate system as x’. Joe is at rest, and measures
the position of the same point, designating its xcoordinate in his system as x.
The relationship of the coordinates in the two systems is clear from the diagram.
After time t Moe’s origin has moved a distance ut, and if the two systems originally
coincided, x’ = x — ut,
I =
y, J” (15.2)
2 = z,
t, = 1. If we substitute this transformation of coordinates into Newton’s laws we ﬁnd
that these laws transform to the same laws in the primed system; that is, the laws
of Newton are of the same form in a moving system as in a stationary system, and
therefore it is impossible to tell, by making mechanical experiments, whether the
system is moving or not. The principle of relativity has been used in mechanics for a long time. It was
employed by various people, in particular Huygens, to obtain the rules for the
collision of billiard balls, in much the same way as we used it in Chapter 10 to
discuss the conservation of momentum. In the past century interest in it was
heightened as the result of investigations into the phenomena of electricity, mag
netism, and light. A long series of careful studies of these phenomena by many
people culminated in Maxwell’s equations of the electromagnetic ﬁeld, which
describe electricity, magnetism, and light in one uniform system. However, the
Maxwell equations did not seem to obey the principle of relativity. That is, if we
transform Maxwell’s equations by the substitution of equations 15.2, their form
does not remain the same; therefore, in a moving space ship the electrical and
optical phenomena should be different from those in a stationary ship. Thus
one could use these optical phenomena to determine the speed of the ship; in
particular, one could determine the absolute speed of the ship by making suitable
optical or electrical measurements. One of the consequences of Maxwell’s equa
tions is that if there is a disturbance in the ﬁeld such that light is generated, these
electromagnetic waves go out in all directions equally and at the same speed c, or
186,000 mi/sec. Another consequence of the equations is that 1f the source
of the disturbance is moving, the light emitted goes through space at the same
speed c. This is analogous to the case of sound, the speed of sound waves being
likewise independent of the motion of the source. This independence of the motion of the source, in the case of light, brings up
an interesdng problem: Suppose we are riding in a car that is going at a speed u, and light from the
rear is going past the car with speed c. Differentiating the ﬁrst equation in (152)
gives dx’/dt = dx/dt — u, which means that according to the Galilean transformation the apparent speed of
the passing light, as we measure it in the car, should not be c but should be c — u.
For instance, if the car is going 100,000 mi/sec, and the light is going 186,000
mi/sec, then apparently the light going past the car should go 86,000 mi/sec.
In any case, by measuring the speed of the light going past the car (if the Galilean
transformation is correct for light), one could determine the speed of the car. A
number of experiments based on this general idea were performed to determine
the velocity of the earth, but they all failed—they gave no velocity at all. We
shall discuss one of these experiments in detail, to show exactly what was done
and what was the matter; something was the matter, of course, something was
wrong with the equations of physics. What could it be? 15—2 15—2 The Lorentz transformation When the failure of the equations of physics in the above case came to light,
the ﬁrst thought that occurred was that the trouble must lie in the new Maxwell
equations of electrodynamics, which were only 20 years old at the time. It seemed
almost obvious that these equations must be wrong, so the thing to do was to
change them in such a way that under the Galilean transformation the principle
of relativity would be satisﬁed. When this was tried, the new terms that had to
be put into the equations led to predictions of new electrical phenomena that did
not exist at all when tested experimentally, so this attempt had to be abandoned.
Then it gradually became apparent that Maxwell’s laws of electrodynamics were
correct, and the trouble must be sought elsewhere. In the meantime, H. A. Lorentz noticed a remarkable and curious thing when
he made the following substitutions in the Maxwell equations: x,= x—ut ,
y’=y,
z’:z, (15.3)
,lzt_ux/_c_2_,
m namely, Maxwell’s equations remain in the same form when this transformation
is applied to them! Equations (15.3) are known as a Lorentz transformation.
Einstein, followmg a suggestion originally made by Poincare, then proposed that
all the physical laws should be of such a kind that they remain unchanged under a t '
Lorentz transformation. In other words, we should change, not the laws of electro
dynamics, but the laws of mechanics. How shall we change Newton’s laws
so that they w111 remain unchanged by the Lorentz transformation? If this
goal is set, we then have to rewrite Newton’s equations in such a way that the
conditions we have imposed are satisﬁed. As it turned out, the only requiremént 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 clectrody
namics will harmonize. Then if we use the Lorentz transformation in comparing
Moe’s measurements with Joe’s, we shall never be able to detect whether either is ""“7
moving, because the form of all the equations will be the same in both coordinate
systems! It 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 (Galllean)
seems to be selfevident, and the new one (Lorentz) looks peculiar. We wish to
know whether it is logically and experimentally poss1ble that the new, and not the
old, transformation can be correct. To ﬁnd that out, it is not enough to study the Source
laws of mechanics but, as Einstein did, we too must analyze our ideas of space
and time 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 :ZKZZXL +U WW“
advance that the effort will be justiﬁed, since the results agree with experiment. OF 15—3 The MichelsonMorley experiment
Fig. 15—2. Schematic diagram of the
As mentioned above, attempts were made to determine the absolute velocity MichelsonMorley experiment. 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 of the experi
ment were ﬁnally explained, by Einstein. The MichelsonMorley 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 B, and two mirrors C and E, all
mounted on a rigid base. The mirrors are placed at equal distances L from B.
The plate B splits an oncoming beam of light, and the two resulting beams con 15—3 tinue in mutually perpendicular directions to the mirrors, where they are reﬂected
back to B. On arriving back at B, the two beams are recombined as two superposed
beams, D and F. If the time taken for the light to go from B to E and back is the
same as the time from B to C and back, the emerging beams D and F will be in
phase and will reinforce each other, but if the two times differ 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 equal, but if it is moving toward
the right with a velocity u, there should be a difference in the times. Let us see why. First, let us calculate the time required for the light to go from B to E and
back. Let us say that the time for light to go from plate B to mirror E is t1, and
the time for the return is 12. Now, while the light is on its way from B to the mirror,
the apparatus moves a distance utl, so the light must traverse a distance L + ml,
at the speed c. We can also express this distance as ct], so we have ctl = L + ml, or 11 = L/(c — u). (This result is also obvious from the point of View that the velocity of light relative
to the apparatus is c — u, so the time is the length L divided by c — 11.) In a like
manner, the time 12 can be calculated. During this time the plate B advances a
distance utz, so the return distance of the light is L — utz. Then we have ct2 = L — ut2, or t2 = L/(c + u). Then the total time is
11 +12 = 2Lc/(c2 — u2). For convenience in later comparison of times we write this as 2L/c
11 + [2 = W. (15.4)
Our second calculation will be of the time Is for the light to go from B to the
mirror C. As before, during time t3 the mirror C moves to the right a distance mg
to the position C’; in the same time. the light travels a distance Ctg along the
hypotenuse of a triangle, which is BC’. For this right triangle we have (eta)2 = L2 + (ms)2
01‘ L2 = c2t§ — u2t§ = (c2 — u2)z§,
from which we get
t3 = L/x/c2 — 142. For the return trip from C’ the distance is the same, as can be seen from the
symmetry of the ﬁgure; therefore the return time is also the same, and the total
time is 2t3. With a little rearrangement of the form we can write 2:3 = _2L_ = 2L/c x/cz—u2 V1 —u2/c2. We 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 if the apparatus were at rest. In the denominators, the term
u2/c2 will be small, unless u is comparable in size to c. The denominators represent
the modiﬁcations in the times caused by the motion of the apparatus. And behold,
these modiﬁcations are not the same—the time to go to C and back is a little less
than the time to E and back, even though the mirrors are equidistant from B, and
all we have to do is to measure that diﬂerence with precision. Here a minor technical point arises—suppose the two lengths L are not
exactly equal? In fact, we surely cannot make them exactly equal. In that case
we simply turn the apparatus 90 degrees, so that BC is in the line of motion and BE is perpendicular to the motion. Any small ditterence in length then becomes
154 (15.5) unimportant, and what we look for is a shift in the interference fringes 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 drif ” should be at least that much at some time of the day or night and
at some time during the year. The apparatus was amply sensitive to observe such
an effect, but no time difference was found—the velocity of the earth through the
ether could not be detected. The result of the experiment was null. The result of the MichelsonMorley experiment was very puzzling and most
disturbing. The ﬁrst fruitful idea for ﬁnding a way out of 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 if
the length is L0 when a body is at rest, then when it moves with speed u parallel
to its length, the new length, which we call L” (Lparallel), is given by Lll = L0 \/1 — u2/c2. (15.6) When this modiﬁcation is applied to the MichelsonMorley interferometer appara
tus the distance from B to C does not change, but the distance from B to E is
shortened to Lvl — uZ/cz. Therefore Eq. (15.5) is not changed, but the L of
Eq. (15.4) must be changed in accordance with Eq. (15.6). When this is done we
obtain (2L/c) \/1 — u2/c2 2L/c
7 l~ 112/62 _ \/1__ “2/62 t1 ‘1' t2 _ (157) Comparing this result with Eq. (15.5), we see that II + 12 = 2t3. So if the ap—
paratus shrinks in the manner just described, we have a way of understanding why
the MichelsonMorley experiment gives no eﬁect 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
was in a “conspiracy” to thwart man by introducing some new phenomenon to
undo every phenomenon that he thought would permit a measurement of u. It was ultimately recognized, as Poincaré pointed out, that a complete conspiracy
is itself a law of nature! Poincare then proposed that there is such a law of nature,
that it is not possible to discover an ether wind by any experiment; that is, there
is no way to determine an absolute velocity. 154 Transformation of time In checking out whether the contraction idea is in harmony with the facts in
other experiments, it turns out that everything is correct prov1ded that the [lines
are also modiﬁed, in the manner expressed in the fourth equation of the set (15.3).
That is because the time t3, calculated for the trip from B to C 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 in the ship the time is simply 2L/c, but to the other observer it is
(2L/c)/\/1 — zﬂ/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 timemeasuring instruments (“clocks”) must appar
ently 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/\/ l — uZ/c‘z 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 is 15—5 Mirror algal y.
5’ system
, 1,, Photocoll Fluhtuho Fig. l5—3. (o) A “light clock" at rest
in the 5’ system. (b) The same clock,
moving through the S system. (c) Illustra
tion of the diagonal path taken by the
thtbeamina moﬁng‘hghtdock" rather difﬁcult, 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 two 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 ymeter stick as they
pass each other. By symmetry, the two marks must come at the same y and
y’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 moving. Now let us see what happens to the moving clock. Before the man took it
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
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 Spring '09
 LeeKinohara
 Physics, Theory Of Relativity

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