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RELATIVITY
Chapter 1 The Space and Time of RelativityChapter 2 Relativistic
Mechanics
Two great theories underlie almost all of modern physics, both
of them discov-ered during the first 25 years of the twentieth
century. The first of these, rela-tivity, was pioneered mainly by
one person, Albert Einstein, and is the subjectof Part I of this
book (Chapters 1 and 2). The second, quantum theory, was thework of
many physicists, including Bohr, Einstein, Heisenberg,
Schrdinger,and others; it is the subject of Part II. In Parts III
and IV we describe theapplications of these great theories to
several areas of modern physics.
Part I contains just two chapters. In Chapter 1 we describe how
several ofthe ideas of relativity were already present in the
classical physics of Newtonand others. Then we describe how
Einsteins careful analysis of the relation-ship between different
reference frames, taking account of the observed in-variance of the
speed of light, changed our whole concept of space and time.
InChapter 2 we describe how the new ideas about space and time
required a rad-ical revision of Newtonian mechanics and a
redefinition of the basic ideas mass, momentum, energy, and force
on which mechanics is built. At the endof Chapter 2, we briefly
describe general relativity, which is the generalizationof
relativity to include gravity and accelerated reference frames.
PARTI
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1
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2C h a p t e r 1The Space and Time of Relativity
1.1 Relativity1.2 The Relativity of Orientation and Origin1.3
Moving Reference Frames1.4 Classical Relativity and the Speed of
Light1.5 The MichelsonMorley Experiment
1.6 The Postulates of Relativity1.7 Measurement of Time1.8 The
Relativity of Time; Time Dilation1.9 Evidence for Time Dilation1.10
Length Contraction1.11 The Lorentz Transformation1.12 Applications
of the Lorentz Transformation1.13 The Velocity-Addition Formula1.14
The Doppler Effect
Problems for Chapter 1Sections marked with a star can be omitted
without significant loss of continuity.
1.1 Relativity
Most physical measurements are made relative to a chosen
reference system. Ifwe measure the time of an event as seconds,
this must mean that t is 5seconds relative to a chosen origin of
time, If we state that the positionof a projectile is given by a
vector we must mean that the posi-tion vector has components
relative to a system of coordinates with adefinite orientation and
a definite origin, If we wish to know the kineticenergy K of a car
speeding along a road, it makes a big difference whether wemeasure
K relative to a reference frame fixed on the road or to one fixed
onthe car. (In the latter case of course.) A little reflection
should con-vince you that almost every measurement requires the
specification of a refer-ence system relative to which the
measurement is to be made. We refer to thisfact as the relativity
of measurements.
The theory of relativity is the study of the consequences of
this relativityof measurements. It is perhaps surprising that this
could be an important sub-ject of study. Nevertheless, Einstein
showed, starting with his first paper on rel-ativity in 1905, that
a careful analysis of how measurements depend oncoordinate systems
revolutionizes our whole understanding of space and time,and
requires a radical revision of classical, Newtonian mechanics.
K = 0,
r = 0.x, y, z
r = 1x, y, z2,t = 0.t = 5
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Section 1.2 The Relativity of Orientation and Origin 3
In this chapter we discuss briefly some features of relativity
as it appliesin the classical theories of Newtonian mechanics and
electromagnetism, andthen we describe the MichelsonMorley
experiment, which (with the supportof numerous other, less direct
experiments) shows that something is wrongwith the classical ideas
of space and time. We then state the two postulates ofEinsteins
relativity and show how they lead to a new picture of space and
timein which both lengths and time intervals have different values
when measuredin any two reference frames that are moving relative
to one another. InChapter 2 we show how the revised notions of
space and time require a revi-sion of classical mechanics. We will
find that the resulting relativistic mechan-ics is usually
indistinguishable from Newtonian mechanics when applied tobodies
moving with normal terrestrial speeds, but is entirely different
whenapplied to bodies with speeds that are a substantial fraction
of the speed oflight, c. In particular, we will find that no body
can be accelerated to a speedgreater than c, and that mass is a
form of energy, in accordance with thefamous relation
Einsteins theory of relativity is really two theories. The
first, called thespecial theory of relativity, is special in that
its primary focus is restricted tounaccelerated frames of reference
and excludes gravity. This is the theory thatwe will be studying in
Chapters 1 and 2 and applying to our later discussions ofradiation,
nuclear, and particle physics.
The second of Einsteins theories is the general theory of
relativity,which is general in that it includes accelerated frames
of reference and grav-ity. Einstein found that the study of
accelerated reference frames led naturallyto a theory of
gravitation, and general relativity turns out to be the
relativistictheory of gravity. In practice, general relativity is
needed only in areas whereits predictions differ significantly from
those of Newtonian gravitational theo-ry. These include the study
of the intense gravity near black holes, of the large-scale
universe, and of the effect the earths gravity has on extremely
accuratetime measurements (one part in or so). General relativity
is an importantpart of modern physics; nevertheless, it is an
advanced topic and, unlike specialrelativity, is not required for
the other topics we treat in this book. Therefore,we have given
only a brief description of general relativity in an
optionalsection at the end of Chapter 2.
1.2 The Relativity of Orientation and Origin
In your studies of classical physics, you probably did not pay
much attention tothe relativity of measurements. Nevertheless, the
ideas were present, and,whether or not you were aware of it, you
probably exploited some aspects ofrelativity theory in solving
certain problems. Let us illustrate this claim withtwo
examples.
In problems involving blocks sliding on inclined planes, it is
well knownthat one can choose coordinates in various ways. One
could, for example, use acoordinate system S with origin O at the
bottom of the slope and with axes horizontal, vertical, and across
the slope, as shown in Fig. 1.1(a). An-other possibility would be a
reference frame with origin at the top of theslope and axes
parallel to the slope, perpendicular to the slope, and
across it, as in Fig. 1.1(b). The solution of any problem
relative to theframe S may look quite different from the solution
relative to and it oftenhappens that one choice of axes is much
more convenient than the other. (Forsome examples, see Problems 1.1
to 1.3.) On the other hand, the basic laws of
S,O z
O yO xOS
OzOyOx
1012
E = mc2.
(a)
(b)
O x
Frame Sy
Frame Sy
O
x
FIGURE 1.1(a) In studying a block on anincline, one could choose
axes horizontal and vertical and putO at the bottom of the
slope.(b) Another possibility, which isoften more convenient, is to
use anaxis parallel to the slope with
perpendicular to the slope, andto put at the top of the
slope.(The axes and point out ofthe page and are not shown.)
O zOzO
O yO x
OyOx
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4 Chapter 1 The Space and Time of Relativity
motion, Newtons laws, make no reference to the choice of origin
and orienta-tion of axes and are equally true in either coordinate
system. In the languageof relativity theory, we can say that
Newtons laws are invariant, or unchanged,as we shift our attention
from frame S to or vice versa. It is because thelaws of motion are
the same in either coordinate system that we are free to
usewhichever system is more convenient.
The invariance of the basic laws when we change the origin or
orienta-tion of axes is true in all of classical physics Newtonian
mechanics, electro-magnetism, and thermodynamics. It is also true
in Einsteins theory ofrelativity. It means that in any problem in
physics, one is free to choose the ori-gin of coordinates and the
orientation of axes in whatever way is most expedi-ent. This
freedom is very useful, and we often exploit it. However, it is
notespecially interesting in our study of relativity, and we will
not have muchoccasion to discuss it further.
1.3 Moving Reference Frames
As a more important example of relativity, we consider next a
question involv-ing two reference frames that are moving relative
to one another. Our discus-sion will raise some interesting
questions about classical physics, questions thatwere
satisfactorily answered only when Einstein showed that the
classicalideas about the relation between moving reference frames
needed revision.
Let us imagine a student standing still in a train that is
moving with con-stant velocity v along a horizontal track. If the
student drops a ball, where willthe ball hit the floor of the
train? One way to answer this question is to use areference frame S
fixed on the track, as shown in Fig. 1.2(a). In this
coordinatesystem the train and student move with constant velocity
v to the right. At themoment of release, the ball is traveling with
velocity v and it moves, under theinfluence of gravity, in the
parabola shown. It therefore lands to the right of itsstarting
point (as measured in the ground-based frame S). However, while
theball is falling, the train is moving, and a straightforward
calculation shows thatthe train moves exactly as far to the right
as does the ball.Thus the ball hits thefloor at the students feet,
vertically below his hand.
Simple as this solution is, one can reach the same conclusion
even moresimply by using a reference frame fixed to the train, as
in Fig. 1.2(b). In thiscoordinate system the train and student are
at rest (while the track moves tothe left with constant velocity ).
At the moment of release the ball is at rest(as measured in the
train-based frame ). It therefore falls straight down andnaturally
hits the floor vertically below the point of release.
The justification of this second, simpler argument is actually
quite subtle.We have taken for granted that an observer on the
train (using the coordinates
) is entitled to use Newtons laws of motion and hence to predict
thata ball which is dropped from rest will fall straight down. But
is this correct?The question we must answer is this: If we accept
as an experimental fact thatNewtons laws of motion hold for an
observer on the ground (using coordi-nates ), does it follow that
Newtons laws also hold for an observer in thetrain (using )?
Equivalently, are Newtons laws invariant as we passfrom the
ground-based frame S to the train-based frame Within the frame-work
of classical physics, the answer to this question is yes, as we now
show.
Since Newtons laws refer to velocities and accelerations, let us
first con-sider the velocity of the ball. We let u denote the balls
velocity relative to theground-based frame S, and the balls
velocity relative to the train-based S.u
S ?x, y, z
x, y, z
x, y, z
S-v
S
S,
y
x
v
OFrame S fixed to ground
(a)
(b)
v
y
xO
Frame S fixed to train
FIGURE 1.2(a) As seen from the ground, thetrain and student move
to the right;the ball falls in a parabola and landsat the students
feet. (b) As seenfrom the train, the ball falls straightdown, again
landing at the studentsfeet.
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Section 1.3 Moving Reference Frames 5
Since the train moves with constant velocity v relative to the
ground, wenaturally expect that
(1.1)
We refer to this equation as the classical velocity-addition
formula. It re-flects our common-sense ideas about space and time,
and asserts that velocitiesobey ordinary vector addition. Although
it is one of the central assumptions ofclassical physics, equation
(1.1) is one of the first victims of Einsteins relativity.In
Einsteins relativity the velocities u and do not satisfy (1.1),
which is onlyan approximation (although a very good approximation)
that is valid when allspeeds are much less than the speed of light,
c. Nevertheless, we are for the mo-ment discussing classical
physics, and we therefore assume for now that theclassical
velocity-addition formula is correct.
Now let us examine Newtons three laws, starting with the first
(the lawof inertia): A body on which no external forces act moves
with constant veloc-ity. Let us assume that this law holds in the
ground-based frame S. This meansthat if our ball is isolated from
all outside forces, its velocity u is constant. Since
and the trains velocity v is constant, it follows at once that
isalso constant, and Newtons first law also holds in the
train-based frame Wewill find that this result is also valid in
Einsteins relativity; that is, in both clas-sical physics and
Einsteins relativity, Newtons first law is invariant as we
passbetween two frames whose relative velocity is constant.
Newtons second law is a little more complicated. If we assume
that itholds in the ground-based frame S, it tells us that
where F is the sum of the forces on the ball, m its mass, and a
its acceleration,all measured in the frame S. We now use this
assumption to show that
where are the corresponding quantities measured rela-tive to the
train-based frame We will do this by arguing that each of
is in fact equal to the corresponding quantity F, m, and a.The
proof that depends to some extent on how one has chosen to
define force. Perhaps the simplest procedure is to define forces
by their effecton a standard calibrated spring balance. Since
observers in the two frames Sand will certainly agree on the
reading of the balance, it follows that anyforce will have the same
value as measured in S and that is, *
Within the domain of classical physics, it is an experimental
fact that anytechnique for measuring mass (for example, an inertial
balance) will producethe same result in either reference frame;
that is,
Finally, we must look at the acceleration.The acceleration
measured in S is
where t is the time as measured by ground-based observers.
Similarly, the ac-celeration measured in is
(1.2)a =du
dt
S
a =dudt
m = m.
F = F.S;S
F = FF, m, a
S.F, m, aF = ma,
F = ma
S.uu = u - v
u
u = u + v
*Of course, the same result holds whatever our definition of
force, but with some defi-nitions the proof is a little more
roundabout. For example, many texts define force bythe equation
Superficially, at least, this means that Newtons second law istrue
by definition in both frames. Since and (as we will show shortly),
itfollows that F = F.
a = am = mF = ma.
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6 Chapter 1 The Space and Time of Relativity
where is the time measured by observers on the train. Now, it is
a central as-sumption of classical physics that time is a single
universal quantity, the samefor all observers; that is, the times t
and are the same, or Therefore, wecan replace (1.2) by
Since
we can simply differentiate with respect to t and find that
(1.3)
or, since v is constant,We have now argued that and Substituting
into
the equation we immediately find that
That is, Newtons second law is also true for observers using the
train-basedcoordinate frame
The third law,
is easily treated. Since any given force has the same value as
measured in S orthe truth of Newtons third law in S immediately
implies its truth in
We have now established that if Newtons laws are valid in one
referenceframe, they are also valid in any second frame that moves
with constant veloc-ity relative to the first. This shows why we
could use the normal rules of pro-jectile motion in a coordinate
system fixed to the moving train. Moregenerally, in the context of
our newfound interest in relativity, it establishes animportant
property of Newtons laws: If space and time have the usual
proper-ties assumed in classical physics, Newtons laws are
invariant as we transferour attention from one coordinate frame to
a second one moving withconstant velocity relative to the
first.
Newtons laws would not still hold in a coordinate system that
wasaccelerating. Physically, this is easy to understand. If our
train were accelerat-ing forward, just to keep the ball at rest
(relative to the train) would require aforce; that is, the law of
inertia would not hold in the accelerating train. To seethe same
thing mathematically, note that if and v is changing, isnot
constant even if u is. Further, the acceleration as given by (1.3)
is notequal to a, since is not zero; so our proof of the second law
for the trainsframe also breaks down. In classical physics the
unaccelerated frames inwhich Newtons laws hold (including the law
of inertia) are often calledinertial frames. In fact, one
convenient definition (good in both classical and
Sdv>dt a
uu = u - v
S.S,
1action force2 = -1reaction force2
S.
F = ma
F = ma,a = a.m = m,F = F,
a = a.
a = a -dvdt
u = u - v
a =du
dt
t = t.t
t
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Section 1.4 Classical Relativity and the Speed of Light 7
*More precisely, In fact, the determination of c has become so
ac-curate that since 1984, the meter has been defined in terms of
c, as the fraction
of the distance traveled by light in 1 second. This means that,
by defini-tion, c is exactly.299,792,458 m>s1>299,792,458
c = 299,792,458 m>s.
Isaac Newton(16421727, English)
Newton was possibly the greatestscientific genius of all time.
In addi-tion to his laws of motion and histheory of gravity, his
contributionsincluded the invention of calculusand important
discoveries in op-tics. Although he believed in ab-solute space
(what we would callthe ether frame),Newton was wellaware that his
laws of motion holdin all unaccelerated frames of ref-erence. From
a modern perspec-tive, it is surprising that Newtondevoted much of
his time to find-ing ways to manufacture gold byalchemy and to
dating the creationof the world (3500 B.C. was hisanswer) using
biblical chronology.
v
Speed c seen from S
Speed c v seen from S Speed c v seen from S
SS
relativistic mechanics) of an inertial frame is just that it is
a frame where thelaw of inertia holds. The result we have just
proved can be rephrased to saythat an accelerated frame is
noninertial.
1.4 Classical Relativity and the Speed of Light
Although Newtons laws are invariant as we change from one
unacceleratedframe to another (if we accept the classical view of
space and time), the sameis not true of the laws of
electromagnetism.We can show this by separately ex-amining each law
Gausss law, Faradays law, and so on but the requiredcalculations
are complicated. A simpler procedure is to recall that the laws
ofelectromagnetism demand that in a vacuum, light signals and all
other electro-magnetic waves travel in any direction with
speed*
where and are the permittivity and permeability of the vacuum.
Thus ifthe electromagnetic laws hold in a frame S, light must
travel with the samespeed c in all directions, as seen in S.
Let us now consider a second frame traveling relative to S and
imag-ine a pulse of light moving in the same direction as as shown
on the left ofFig. 1.3.The pulse has speed c relative to
S.Therefore, by the classical velocity-addition formula (1.1), it
should have speed as seen from Similarly, apulse traveling in the
opposite direction would have speed as seen from
and a pulse traveling in any other oblique direction would have
a differentspeed, intermediate between and We see that in the frame
thespeed of light should vary between and according to its
directionof propagation. Since the laws of electromagnetism demand
that the speed oflight be exactly c, we conclude that these laws
unlike those of mechanics could not be valid in the frame
The situation just described was well understood by physicists
towardthe end of the nineteenth century. In particular, it was
accepted as obvious thatthere could be only one frame, called the
ether frame, in which light traveledat the same speed c in all
directions. The name ether frame derived from thebelief that light
waves must propagate through a medium, in much the sameway that
sound waves were known to propagate in the air. Since light
propa-gates through a vacuum, physicists recognized that this
medium, which no one
S.
c + vc - vSc + v.c - v
S,c + v
S.c - v
S,S
m0e0
c =11eo mo = 3.00 * 108 m>s
FIGURE 1.3Frame travels with velocity vrelative to S. If light
travels with thesame speed c in all directionsrelative to S, then
(according to theclassical velocity-addition formula) itshould have
different speeds as seenfrom S.
S
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8 Chapter 1 The Space and Time of Relativity
had ever seen or felt, must have unusual properties. Borrowing
the ancientname for the substance of the heavens, they called it
the ether. The uniquereference frame in which light traveled at
speed c was assumed to be the framein which the ether was at rest.
As we will see, Einsteins relativity implies thatneither the ether
nor the ether frame actually exists.
Our picture of classical relativity can be quickly summarized.
In classi-cal physics we take for granted certain ideas about space
and time, all basedon our everyday experiences. For example, we
assume that relative velocitiesadd like vectors, in accordance with
the classical velocity-addition formula;also, that time is a
universal quantity, concerning which all observers agree.Accepting
these ideas we have seen that Newtons laws should be valid in
awhole family of reference frames, any one of which moves uniformly
relativeto any other. On the other hand, we have seen that there
could be no morethan one reference frame, called the ether frame,
relative to which the elec-tromagnetic laws hold and in which light
travels through the vacuum withspeed c in all directions.
It should perhaps be emphasized that although this view of
natureturned out to be wrong, it was nevertheless perfectly logical
and internallyconsistent. One might argue on philosophical or
aesthetic grounds (as Einsteindid) that the difference between
classical mechanics and classical electromag-netism is surprising
and even unpleasing, but theoretical arguments alonecould not
decide whether the classical view is correct. This question could
bedecided only by experiment. In particular, since classical
physics implied thatthere was a unique ether frame where light
travels at speed c in all directions,there had to be some
experiment that showed whether this was so.This was ex-actly the
experiment that Albert Michelson, later assisted by Edward
Morley,performed between the years 1880 and 1887, as we now
describe.
If one assumed the existence of a unique ether frame, it seemed
clearthat as the earth orbits around the sun, it must be moving
relative to the etherframe. In principle, this motion relative to
the ether frame should be easy todetect. One would simply have to
measure the speed (relative to the earth) oflight traveling in
various directions. If one found different speeds in
differentdirections, one would conclude that the earth is moving
relative to the etherframe, and a simple calculation would give the
speed of this motion. If, instead,one found the speed of light to
be exactly the same in all directions, one wouldhave to conclude
that at the time of the measurements the earth happened tobe at
rest relative to the ether frame. In this case one should probably
repeatthe experiment a few months later, by which time the earth
would be at a dif-ferent point on its orbit and its velocity
relative to the ether frame shouldsurely be nonzero.
In practice, this experiment is extremely difficult because of
the enor-mous speed of light.
If our speed relative to the ether is the observed speed of
light should varybetween and Although the value of is unknown, it
should onaverage be of the same order as the earths orbital
velocity around the sun,
(or possibly more if the sun is also moving relative to the
ether frame). Thusthe expected change in the observed speed of
light due to the earths motion is
v ' 3 * 104 m>s
vc + v.c - vv,
c = 3 * 108 m>s
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Section 1.4 Classical Relativity and the Speed of Light 9
about 1 part in This was too small a change to be detected by
direct mea-surement of the speed of light at that time.
To avoid the need for such direct measurements, Michelson
devised aninterferometer in which a beam of light was split into
two beams by a partiallyreflecting surface; the two beams traveled
along perpendicular paths and werethen reunited to form an
interference pattern; this pattern was sensitive to dif-ferences in
the speed of light in the two perpendicular directions and so
couldbe used to detect any such differences. By 1887, Michelson and
Morley hadbuilt an interferometer (described below) that should
have been able to detectdifferences in the speed of light much
smaller than the part in expected.Totheir surprise and chagrin,
they could detect absolutely no difference at all.
The MichelsonMorley and similar experiments have been
repeatedmany times, at different times of year and with
ever-increasing precision, butalways with the same final result.*
With hindsight, it is easy to draw the rightconclusion from their
experiment: Contrary to all expectations, light alwaystravels with
the same speed in all directions relative to an earth-based
refer-ence frame even though the earth has different velocities at
different times ofthe year. In other words, light travels at the
same speed c in all directions inmany different inertial frames,
and the notion of a unique ether frame with thisproperty must be
abandoned.
This conclusion is so surprising that it was not taken seriously
for nearly20 years. Rather, several ingenious alternative theories
were advanced that ex-plained the MichelsonMorley result but
managed to preserve the notion of aunique ether frame. For example,
in the ether-drag theory, it was suggestedthat the ether, the
medium through which light was supposed to propagate,was dragged
along by the earth as it moved through space (in much the sameway
that the earth does drag its atmosphere with it). If this were the
case, anearthbound observer would automatically be at rest relative
to the ether, andMichelson and Morley would naturally have found
that light had the samespeed in all directions at all times of the
year. Unfortunately, this neat expla-nation of the MichelsonMorley
result requires that light from the stars wouldbe bent as it
entered the earths envelope of ether. Instead, astronomical
ob-servations show that light from any star continues to move in a
straight line asit arrives at the earth.
The ether-drag theory, like all other alternative explanations
of theMichelsonMorley result, has been abandoned because it fails
to fit all thefacts. Today, nearly all physicists agree that
Michelson and Morleys failure todetect our motion relative to the
ether frame was because there is no etherframe. The first person to
accept this surprising conclusion and to develop itsconsequences
into a complete theory was Einstein, as we describe, starting
inSection 1.6.
104
104.
*From time to time experimenters have reported observing a
nonzero difference, butcloser examination has shown that these are
probably due to spurious effects such asexpansion and contraction
of the interferometer arms resulting from temperature vari-ations.
For a careful modern analysis of Michelson and Morleys results and
many fur-ther references, see M. Handschy, American Journal of
Physics, vol. 50, p. 987 (1982).Because of the earths motion around
the sun, the apparent direction of any one starundergoes a slight
annual variationan effect called stellar aberration. This effect
isconsistent with the claim that light travels in a straight line
from the star to the earthssurface, but contradicts the ether-drag
theory.
Albert Michelson(18521931, American)
Michelson devoted much of his ca-reer to increasingly accurate
mea-surements of the speed of light,and in 1907 he won the
NobelPrize in physics for his contribu-tions to optics. His failure
to de-tect the earths motion relative tothe supposed ether is
probably themost famous unsuccessful ex-periment in the history of
science.
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10 Chapter 1 The Space and Time of Relativity
1.5 The MichelsonMorley Experiment
More than a hundred years later, the MichelsonMorley experiment
remains the sim-plest and cleanest evidence that light travels at
the same speed in all directions in all in-ertial frames what
became the second postulate of relativity. Naturally, we think
youshould know a little of how this historic experiment worked.
Nevertheless, if you arepressed for time, you can omit this section
without loss of continuity.
Figure 1.4 is a simplified diagram of Michelsons interferometer.
Light fromthe source hits the half-silvered mirror M and splits,
part traveling to the mir-ror and part to The two beams are
reflected at and and returnto M, which sends part of each beam on
to the observer. In this way the ob-server receives two signals,
which can interfere constructively or destructively,depending on
their phase difference.
To calculate this phase difference, suppose for a moment that
the twoarms of the interferometer, from M to and M to have exactly
the samelength l, as shown. In this case any phase difference must
be due to the differ-ent speeds of the two beams as they travel
along the two arms. For simplicity,let us assume that arm 1 is
exactly parallel to the earths velocity v. In this casethe light
travels from M to with speed (relative to the interferome-ter) and
back from to M with speed Thus the total time for theround trip on
path 1 is
(1.4)
It is convenient to rewrite this in terms of the ratio
which we have seen is expected to be very small, In terms of
(1.4)becomes
(1.5)t1 =2lc
1
1 - b2L
2lc
11 + b22
b,b ' 10-4.
b =vc
t1 =l
c + v+
lc - v =
2lc
c2 - v2
c - v.M1c + vM1
M2 ,M1
M2M1M2 .M1 ,
c u
v
(a)
(b)
v
l
l2
1M1
M2
M Observer
Lightsource
FIGURE 1.4(a) Schematic diagram of theMichelson interferometer.
M is ahalf-silvered mirror, and aremirrors. The vector v indicates
theearths velocity relative to thesupposed ether frame. (b)
Thevector-addition diagram that givesthe lights velocity u,
relative to theearth, as it travels from M to The velocity c
relative to the etheris the vector sum of v and u.
M2 .
M2M1
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Section 1.5 The MichelsonMorley Experiment 11
In the last step we have used the binomial approximation
(discussed inAppendix B and in Problems 1.121.14),
(1.6)
which holds for any number n and any x much smaller than 1. (In
the presentcase and )
The speed of light traveling from M to is given by the
velocity-addi-tion diagram in Fig. 1.4(b). (Relative to the earth,
the light has velocity u per-pendicular to v; relative to the
ether, it travels with speed c in the directionshown.) This speed
is
Since the speed is the same on the return journey, the total
time for the roundtrip on path 2 is
(1.7)
where we have again used the binomial approximation (1.6), this
time with
Comparing (1.5) and (1.7), we see that the waves traveling along
the twoarms take slightly different times to return to M, the
difference being
(1.8)
If this difference were zero, the two waves would arrive in step
and in-terfere constructively, giving a bright resultant signal.
Similarly, if were anyinteger multiple of the lights period, (where
is the wavelength),they would interfere constructively. If were
equal to half the period,
(or or ), the two waves would be exactly out of stepand would
interfere destructively. We can express these ideas more
compactlyif we consider the ratio
(1.9)
This is the number of complete cycles by which the two waves
arrive out ofstep; in other words, N is the phase difference,
expressed in cycles. If N is aninteger, the waves interfere
constructively; if N is a half-odd integer
the waves interfere destructively.The phase difference N in
(1.9) is the phase difference due to the earths
motion relative to the supposed ether frame. In practice, it is
impossible to besure that the two interferometer arms have exactly
equal lengths, so there willbe an additional phase difference due
to the unknown difference in lengths.Tocircumvent this
complication, Michelson and Morley rotated their interferom-eter
through observing the interference as they did so.This rotation
wouldnot change the phase difference due to the different arm
lengths, but it shouldreverse the phase difference due to the
earths motion (since arm 2 would now
90,
AN = 12 , 32 , 52 , B ,
N =tT
=lb2>cl>c =
lb2
l
2.5T, 1.5T,t = 0.5Tt
lT = l>c tt
t = t1 - t2 Llc
b2
n = - 12 .
t2 =2l3c2 - v2 = 2lc31 - b2 L 2lc A1 + 12 b2 B
u = 3c2 - v2M2
x = b2.n = -1
11 - x2n L 1 - nx
TAYL01-001-045.I 12/10/02 1:50 PM Page 11
-
Albert Einstein(18791955,GermanSwissAmerican)
Like all scientific theories, relativitywas the work of many
people.Nevertheless, Einsteins contribu-tions outweigh those of
anyoneelse by so much that the theory isquite properly regarded as
his. Aswe will see in Chapter 4, he alsomade fundamental
contributionsto quantum theory, and it was forthese that he was
awarded the1921 Nobel Prize in physics. Theexotic ideas of
relativity and thegentle, unpretentious persona ofits creator
excited the imaginationof the press and public, and Ein-stein
became the most famous sci-entist who ever lived. Asked whathis
profession was, the aged Ein-stein once answered,photograph-ers
model.
12 Chapter 1 The Space and Time of Relativity
be along v and arm 1 across it). Thus, as a result of the
rotation, the phasedifference N should change by twice the amount
(1.9),
(1.10)
This implies that the observed interference should shift from
bright to darkand back to bright again times. Observation of this
shift would confirmthat the earth is moving relative to the ether
frame, and measurement of would give the value of and hence the
earths velocity
In their experiment of 1887, Michelson and Morley had an arm
length(This was accomplished by having the light bounce back and
forth
between several mirrors.) The wavelength of their light was
andas we have seen, was expected to be of order Thus the
shiftshould have been at least
(1.11)
Although they could detect a shift as small as 0.01, Michelson
and Morleyobserved no significant shift when they rotated their
interferometer.
Michelson and Morley were disappointed and shocked at their
result,and it was almost 20 years before anyone drew the right
conclusion from it that light has the same speed c in all
directions in all inertial frames, the ideathat Einstein adopted as
one of the postulates of his theory of relativity.
1.6 The Postulates of Relativity
We have seen that the classical ideas of space and time had led
to twoconclusions:
1. The laws of Newtonian mechanics hold in an entire family of
referenceframes, any one of which moves uniformly relative to any
other.
2. There can be only one reference frame in which light travels
at the samespeed c in all directions (and, more generally, in which
all laws of electro-magnetism are valid).
The MichelsonMorley experiment and numerous other experiments in
thesucceeding hundred years have shown that the second conclusion
is false.Light travels with speed c in all directions in many
different reference frames.
Einsteins special theory of relativity is based on the
acceptance of thisfact. Einstein proposed two postulates, or
axioms, expressing his convictionthat all physical laws, including
mechanics and electromagnetism, should bevalid in an entire family
of reference frames. From these two postulates, hedeveloped his
special theory of relativity.
Before we state the two postulates of relativity, it is
convenient to ex-pand the definition of an inertial frame to be any
reference frame in which allthe laws of physics hold.
An inertial frame is any reference frame (that is, system of
coordinates andtime t) where all the laws of physics hold in their
simplest form.
x, y, z
N =2lb2
lL
2 * 111 m2 * 110-422590 * 10-9 m
L 0.4
10-4.b = v>cl = 590 nm;
l L 11 m.
v = bc.b,N
N
N =2lb2
l
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Section 1.6 The Postulates of Relativity 13
Notice that we have not yet said what all the laws of physics
are; to a largeextent, Einstein used his postulates to deduce what
the correct laws of physicscould be. It turns out that one of the
laws that survives from classical physicsinto relativity is Newtons
first law, the law of inertia. Thus our newly definedinertial
frames are in fact the familiar unaccelerated frames where a body
onwhich no forces act moves with constant velocity. As before, a
reference frameanchored to the earth is an inertial frame (to the
extent that we ignore thesmall accelerations due to the earths
rotation and orbital motion); a referenceframe fixed to a rapidly
rotating turntable is not an inertial frame.
Notice also that in defining an inertial frame, we have
specified that thelaws of physics must hold in their simplest form.
This is because one cansometimes modify physical laws so that they
hold in noninertial frames as well.For example, by introducing a
fictitious centrifugal force, one can arrangethat the laws of
statics are valid in a rotating frame. It is to exclude this kind
ofmodification that we have added the qualification in their
simplest form.
The first postulate of relativity asserts that there is a whole
family ofinertial frames.
FIRST POSTULATE OF RELATIVITYIf S is an inertial frame and if a
second frame moves with constant velocity rela-tive to S, then is
also an inertial frame.
We can reword this postulate to say that the laws of physics are
invariant as wechange from one reference frame to a second frame,
moving uniformly rela-tive to the first. This property is familiar
from classical mechanics, but inrelativity it is postulated for all
the laws of physics.
The first postulate is often paraphrased as follows: There is no
suchthing as absolute motion. To understand what this means,
consider a frame attached to a rocket moving at constant velocity
relative to a frame S anchoredto the earth. The question we want to
ask is this: Is there any scientific sense inwhich we can say that
is really moving and that S is really stationary (or,perhaps, the
other way around)? If the answer were yes, we could say that Sis
absolutely at rest and that anything moving relative to S is in
absolute mo-tion. However, the first postulate of relativity
guarantees that this is impossi-ble:All laws observable by an
earthbound scientist in S are equally observableby a scientist in
the rocket any experiment that can be performed in S canbe
performed equally in Thus no experiment can possibly show
whichframe is really moving. Relative to the earth, the rocket is
moving; relative tothe rocket, the earth is moving; and this is as
much as we can say.
Yet another way to express the first postulate is to say that
among thefamily of inertial frames, all moving relative to one
another, there is nopreferred frame. That is, physics singles out
no particular inertial frame asbeing in any way more special than
any other frame.
The second postulate identifies one of the laws that holds in
all inertialframes.
SECOND POSTULATE OF RELATIVITYIn all inertial frames, light
travels through the vacuum with the same speed,
in any direction.c = 299,792,458 m>s
S.S;
S
S
SS
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14 Chapter 1 The Space and Time of Relativity
O
S
FIGURE 1.5The chief observer at O distributesher helpers, each
with an identicalclock, throughout S.
This postulate is, of course, the formal expression of the
MichelsonMorley re-sult. We can say briefly that it asserts the
universality of the speed of light c.
The second postulate flies in the face of our normal experience.
Nev-ertheless, it is now a firmly established experimental fact. As
we explore theconsequences of the two postulates of relativity, we
are going to encounterseveral unexpected effects that may be
difficult to accept at first. All ofthese effects (including the
second postulate itself) have the subtle proper-ty that they become
important only when bodies travel at speeds reasonablyclose to the
speed of light. Under ordinary conditions, at normal
terrestrialspeeds, these effects simply do not show up. In this
sense, none of the sur-prising consequences of Einsteins relativity
really contradicts our everydayexperience.
1.7 Measurement of Time
Before we begin exploring the consequences of the relativity
postulates, weneed to say a word about the measurement of time. We
are going to find thatthe time of an event may be different when
measured from different frames ofreference. This being the case, we
must first be quite sure we know what wemean by measurement of time
in a single frame.
It is implicit in the second postulate of relativity, with its
reference tothe speed of light, that we can measure distances and
times. In particular,we take for granted that we have access to
several accurate clocks. Theseclocks need not all be the same; but
when they are all brought to the samepoint in the same inertial
frame and are properly synchronized, they mustof course agree.
Consider now a single inertial frame S, with origin O and axes
Weimagine an observer sitting at O and equipped with one of our
clocks. Usingher clock, the observer can easily time any event,
such as a small explosion, inthe immediate proximity of O since she
will see (or hear) the event the mo-ment it occurs. To time an
event far away from O is harder, since the light (orsound) from the
event has to travel to O before our observer can sense it. Toavoid
this complication, we let our observer hire a large number of
helpers,each of whom she equips with an accurate clock and assigns
to a fixed, knownposition in the coordinate system S, as shown in
Fig. 1.5. Once the helpers arein position, she can check that their
clocks are still synchronized by havingeach helper send a flash of
light at an agreed time (measured on the helpersclock); since light
travels with the known speed c (second postulate), she cancalculate
the time for the light to reach her at O and hence check the
setting ofthe helpers clock.
With enough helpers, stationed closely enough together, we can
be surethere is a helper sufficiently close to any event to time it
effectively instanta-neously. Once he has timed it, he can, at his
leisure, inform everyone else of theresult by any convenient means
(by telephone, for example). In this way anyevent can be assigned a
time t, as measured in the frame S.
When we speak of an inertial frame S, we will always have in
mind a sys-tem of axes Oxyz and a team of observers who are
stationed at rest through-out S and equipped with synchronized
clocks. This allows us to speak of theposition and the time t of
any event, relative to the frame S.r = 1x, y, z2
x, y, z.
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Section 1.8 The Relativity of Time; Time Dilation 15
1.8 The Relativity of Time; Time Dilation
We are now ready to compare measurements of times made by
observers intwo different inertial frames, and we are going to find
that, as a consequence ofthe relativity postulates, times measured
in different frames inevitably dis-agree. To this end, we imagine
the familiar two frames, S anchored to theground and anchored to a
train moving at constant velocity v relative to theground.We
consider a thought experiment (or gedanken experiment fromthe
German) in which an observer at rest on the train sets off a
flashbulb onthe floor of the train, vertically below a mirror
mounted on the roof, a height habove. As seen in the frame (fixed
in the train), a pulse of light travelsstraight up to the mirror,
is reflected straight back, and returns to its startingpoint on the
floor. We can imagine a photocell arranged to give an audiblebeep
as the light returns. Our object is to find the time, as measured
in eitherframe, between the two events the flash as the light
leaves the floor and thebeep as it returns.
Our experiment, as seen in the frame is shown in Fig. 1.6(a).
Since is an inertial frame, light travels the total distance at
speed c. Therefore, thetime for the entire trip is
(1.12)
This is the time that an observer in frame will measure between
the flashand the beep, provided of course, that his clock is
reliable.
The same experiment, as seen from the inertial frame S, is shown
inFig. 1.6(b). In this frame the light travels along the two sides
and of thetriangle shown. If we denote by the time for the entire
journey, as measuredin S, the time to go from A to B is During this
time the train travels a dis-tance and the light, moving with speed
c, travels a distance (Note that this is where the postulates of
relativity come in; we have taken thespeed of light to be c in both
S and ) The dimensions of the right triangleS.
c t>2.v t>2, t>2.t
BCAB
S
t =2hc
2hSS,
S
S
v
A D
h
BS
Flash Beep
(a) (c)
S
vt/2
ct/2
v
A D C
BS
Flash
(b)
S
Beep
FIGURE 1.6(a) The thought experiment asseen in the train-based
frame (b) The same experiment as seenfrom the ground-based frame
S.Notice that two observers areneeded in this frame. (c)
Thedimensions of the triangle ABD.
S.
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16 Chapter 1 The Space and Time of Relativity
are therefore as shown in Fig. 1.6(c).Applying Pythagorass
theorem, wesee that*
or, solving for
(1.13)
where we have again used the ratio
of the speed to the speed of light c. The time is the time that
observers inS will measure between the flash and the beep
(provided, again, that theirclocks are reliable).
The most important and surprising thing about the two answers
(1.12)and (1.13) is that they are not the same. The time between
the two events,the flash and the beep, is different as measured in
the frames S and Specifically,
(1.14)
We have derived this result for an imagined thought experiment
involving aflash of light reflected back to a photocell. However,
the conclusion applies toany two events that occur at the same
place on the train: Suppose, for instance,that we drop a knife on
the table and a moment later drop a fork. In principle,at least, we
could arrange for a flash of light to occur at the moment the
knifelands, and we could position a mirror to reflect the light
back to arrive just asthe fork lands. The relation (1.14) must then
apply to these two events (thelanding of the knife and the landing
of the fork). Now the falling of the knifeand fork cannot be
affected by the presence or absence of a flashbulb and pho-tocell;
thus neither of the times or can depend on whether we actuallydid
the experiment with the light and the photocell. Therefore, the
relation(1.14) holds for any two events that occur at the same
place on board the train.
The difference between the measured times and is a direct
conse-quence of the second postulate of relativity. (In classical
physics ofcourse.) You should avoid thinking that the clocks in one
of our frames mustsomehow be running wrong; quite the contrary, it
was an essential part of ourargument that all the clocks were
running right. Moreover, our argumentmade no reference to the kind
of clocks used (apart from requiring that theybe correct). Thus the
difference (1.14) applies to all clocks. In other words,time itself
as measured in the two frames is different. We will discuss
theexperimental evidence for this surprising conclusion
shortly.
Several properties of the relationship (1.14) deserve comment.
First, ifour train is actually at rest then and (1.14) tells us
that
That is, there is no difference unless the two frames are in
relativet = t.b = 01v = 02,
t = t,tt
tt
t =t31 - b2
S.
tv
b =v
c
t =2h3c2 - v2 = 2hc 131 - b2t,
a c t2
b2 = h2 + av t2
b2ABD
*Here we are taking for granted that the height h of the train
is the same as measuredin either frame, S or We will prove that
this is correct in Section 1.10.S.
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Section 1.8 The Relativity of Time; Time Dilation 17
motion. Further, at normal terrestrial speeds, and thus
thedifference between and is very small.
Example 1.1
The pilot of a jet traveling at a steady sets a buzzer in the
cockpit togo off at intervals of exactly 1 hour (as measured on the
plane). What wouldbe the interval between two successive buzzes as
measured by two observerssuitably positioned on the ground? (Ignore
effects of the earths motion; thatis, consider the ground to be an
inertial frame.)
The required interval between two buzzes is given by (1.14),
withand Thus
We have to be a bit careful in evaluating this time. The number
in thedenominator is so close to 1 that most calculators cannot
tell the difference.(It takes 12 significant figures to distinguish
from 1.) In this situa-tion the simplest and best course is to use
the binomial approximation,
which is an excellent approximation, provided x is small.[This
important approximation was already used in (1.6) and is discussed
inProblems 1.121.14 and in Appendix B.] In the present case,
setting and we find
The difference between the two measured times is or
1.8nanoseconds. (A nanosecond, or ns, is ) It is easy to see why
classicalphysicists had failed to notice this kind of
difference!
The difference between and gets bigger as increases. In
modernparticle accelerators it is common to have electrons and
other particles withspeeds of and more. If we imagine repeating our
thought experimentwith the frame attached to an electron with Eq.
(1.14) gives
Differences as large as this are routinely observed by particle
physicists, as wediscuss in the next section.
If we were to put (that is, ) in Eq. (1.14), we would get
theabsurd result, and if we put (that is, ), we would getan
imaginary answer. These ridiculous results suggest (correctly) that
mustalways be less than c.
v 6 c
vb 7 1v 7 ct = t>0; b = 1v = c
t =t41 - 10.9922 L 7t
b = 0.99,S0.99c
vtt
10-9 s.5 * 10-13 hour,
= 1.0000000000005 hours = 11 hour2 * A1 + 12 * 10-12 B
t = t11 - b22-1>2 L t A1 + 12 b2 Bn = -1>2, x = b
2
11 - x2n L 1 - nx,1 - 10-12
t =t31 - b2 = 1 hour31 - 10-12
b = v>c = 10-6,t = 1 hour
300 m>s
ttb V 1;v V c
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18 Chapter 1 The Space and Time of Relativity
This is one of the most profound results of Einsteins
relativity: The speed ofany inertial frame relative to any other
inertial frame must always be less than c.In other words, the speed
of light, in addition to being the same in all iner-tial frames,
emerges as the universal speed limit for the relative motion
ofinertial frames.
The factor that appears in Eq. (1.14) crops up in so
manyrelativistic formulas that it is traditionally given its own
symbol,
(1.15)
Since is always smaller than c, the denominator in (1.15) is
always less thanor equal to 1 and hence
(1.16)
The factor equals 1 only if The larger we make the larger
becomes; and as approaches c, the value of increases without
limit.
In terms of Eq. (1.14) can be rewritten
(1.17)
That is, is always greater than or equal to This asymmetry may
seemsurprising, and even to violate the postulates of relativity
since it suggests aspecial role for the frame In fact, however,
this is just as it should be. In ourexperiment the frame is special
since it is the unique inertial frame wherethe two events the flash
and the beep occurred at the same place. Thisasymmetry was implicit
in Fig. 1.6, which showed one observer measuring (since both events
occurred at the same place in ) but two observers mea-suring (since
the two events were at different places in ). To emphasizethis
asymmetry, the time can be renamed and (1.17) rewritten as
(1.18)
The subscript 0 on indicates that is the time indicated by a
clock that isat rest in the special frame where the two events
occurred at the same place.This time is often called the proper
time between the events. The time ismeasured in any frame and is
always greater than or equal to the proper time
For this reason, the effect embodied in (1.18) is often called
time dilation.The proper time is the time indicated by the clock on
the moving
train (moving relative to S, that is); is the time shown by the
clocks at reston the ground in frame S. Since the relation (1.18)
can be looselyparaphrased to say that a moving clock is observed to
run slow.
Finally, we should reemphasize the fundamental symmetry between
anytwo inertial frames. We chose to conduct our thought experiment
with theflash and beep at one spot on the train (frame ), and we
found that
However, we could have done things the other way around: If
aground-based observer (at rest in S) had performed the same
experiment witha flash of light and a mirror, the flash and beep
would have occurred in thesame spot on the ground; and we would
have found that The greatmerit of writing the time-dilation formula
in the form (1.18), ist = g t0 ,
t t.
t 7 t.S
t0 t,t
t0t0 .
t
t0t0
t = g t0 t0
t0tSt
St
SS.
t.t
t = g t t
g,gv
gv,v = 0.g
g 1
v
g =131 - b2 = 141 - 1v>c22
g.1>31 - b2
TAYL01-001-045.I 12/10/02 1:50 PM Page 18
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Section 1.9 Evidence for Time Dilation 19
*The test was actually carried out twice once flying east and
once west with satis-factory agreement in both cases. The results
quoted here are from the more decisivewestward flight. For more
details, see J. C. Hafele and R. E. Keating, Science, vol. 177,p.
166 (1972). Since the accuracy of this original experiment has been
questioned, weshould emphasize that the experiment has been
repeated many times, with improvedaccuracy, and there is now no
doubt at all that the observations support the predictionsof
relativity.An alternative characterization is the mean life which
differs from by a constantfactor. We will define both of these more
carefully in Chapter 17.
t1>2t,
that it avoids the problem of remembering which is frame S and
which thesubscript 0 always identifies the proper time, as measured
in the frame inwhich the two events were at the same spot.
1.9 Evidence for Time Dilation
In his original paper on relativity, Einstein predicted the
effect that is nowcalled time dilation. At that time there was no
evidence to support the predic-tion, and many years were to pass
before any was forthcoming. The first tests,using the unstable
particle called the muon as their clock, were carried out in1941.
(See Problem 1.27.)
It was only with the advent of super-accurate atomic clocks that
testsusing man-made clocks became possible. The first such test was
carried out in1971. Four portable atomic clocks were synchronized
with a reference clock atthe U.S. Naval Observatory in Washington,
D.C., and all four clocks were thenflown around the world on a jet
plane and returned to the Naval Observatory.The discrepancy between
the reference clock and the portable clocks aftertheir journey was
predicted (using relativity) to be
(1.19)
while the observed discrepancy (averaged over the four portable
clocks) was*
(1.20)
We should mention that the excellent agreement between (1.19)
and(1.20) is more than a test of the time difference (1.18),
predicted by special rel-ativity. Gravitational effects, which
require general relativity, contribute alarge part of the predicted
discrepancy (1.19). Thus this beautiful experimentis a confirmation
of general, as well as special, relativity.
Much simpler tests of time dilation and tests involving much
larger dila-tions are possible if one is prepared to use the
natural clocks provided by un-stable subatomic particles. For
example, the charged meson, or pion, is aparticle that is formed in
collisions between rapidly moving atomic nuclei (aswe discuss in
detail in Chapter 18). The pion has a definite average
lifetime,after which it decays or disintegrates into other
subatomic particles, and onecan use this average life as a kind of
natural clock.
One way to characterize the life span of an unstable particle is
the half-life
the average time after which half of a large sample of the
particles in ques-tion will have decayed. For example, the
half-life of the pion is measured to be
(1.21)t1>2 = 1.8 * 10-8 s
t1>2 ,
p
273 ; 7 ns
275 ; 21 ns
S;
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20 Chapter 1 The Space and Time of Relativity
This means that if one starts at with pions, then after halfof
them will have decayed and only will remain. After a further
half of those will have decayed and only will remain.After
another only will remain. And so on. In general, aftern half-lives,
the number of particles remaining will be
At particle-physics laboratories, pions are produced in large
numbers incollisions between protons (the nuclei of hydrogen atoms)
and various othernuclei. It is usually convenient to conduct
experiments with the pions at a gooddistance from where they are
produced, and the pions are therefore allowed tofly down an
evacuated pipe to the experimental area. At the Fermilab
nearChicago the pions are produced traveling very close to the
speed of light, atypical value being
and the distance they must travel to the experimental area is
about Let us consider the flight of these pions, first from the
(incorrect) classical viewwith no time dilation and then from the
(correct) relativistic view.
As seen in the laboratory, the pions time of flight is
(1.22)
A classical physicist, untroubled by any notions of relativity
of time, wouldcompare this with the half-life (1.21) and calculate
that
That is, the time needed for the pions to reach the experimental
area is 183half-lives. Therefore, if is the original number of
pions, the number tosurvive the journey would be
and for all practical purposes, no pions would reach the
experimental area.This would obviously be an absurd way to do
experiments with pions, and it isnot what actually happens.
In relativity, we now know, times depend on the frame in which
they aremeasured, and we must consider carefully the frames to
which the times T and
refer. The time T in (1.22) is, of course, the time of flight of
the pions asmeasured in a frame fixed in the laboratory, the lab
frame. To emphasize this,we rewrite (1.22) as
(1.23)
On the other hand, the half-life refers to time as seen bythe
pions; that is, is the half-life measured in a frame anchored to
the pions,the pions rest frame. (This is an experimental fact: The
half-lives quoted byphysicists are the proper half-lives, measured
in the frame where the particlesare at rest.) To emphasize this, we
write (temporarily)
(1.24)t1>21p rest frame2 = 1.8 * 10-8 s
t1>2t1>2 = 1.8 * 10-8 s
T1lab frame2 = 3.3 * 10-6 s
t1>2
N =N0
2183L 18.2 * 10-562N0
N0
T L 183t1>2
T =Lv
L103 m
3 * 108 m>s = 3.3 * 10-6 s
L = 1 km.
v = 0.9999995c
N0>2n.t = n t1>2 ,N0>81.8 * 10-8 s,
N0>4N0>21.8 * 10-8 s,N0>2
1.8 * 10-8 sN0t = 0
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Section 1.10 Length Contraction 21
We see that the classical argument here used two times, T and
mea-sured in different inertial frames. A correct argument must
work consistentlyin one frame, for example the lab frame. The
half-life measured in the labframe is given by the time-dilation
formula as times the half-life (1.24). With
it is easy to see that
and hence that
(1.25)
Comparing (1.23) and (1.25) we see that
That is, the pions flight down the pipe lasts only one-fifth of
the relevant half-life. In this time very few of the pions decay,
and almost all reach the experi-mental area. (The number that
survive is ) That this isexactly what actually happens in all
particle-physics laboratories is powerfulconfirmation of the
relativity of time, as first predicted by Einstein in 1905.
Example 1.2
The particle ( is the Greek capital L and is pronounced lambda.)
is anunstable subatomic particle that decays into a proton and a
pion
with a half-life of If several lambdas arecreated in a nuclear
collision, all with speed on average how farwill they travel before
half of them decay?
The half-life as measured in the laboratory is (since is the
prop-er half-life, as measured in the rest frame). Therefore, the
desired distanceis With
and the required distance is
Notice how even with speeds as large as 0.6 c, the factor is not
very muchlarger than 1, and the effect of time dilation is not
dramatic. Notice also thata distance of a few centimeters is much
easier to measure than a time oforder thus measurement of the range
of an unstable particle is oftenthe easiest way to find its
half-life.
1.10 Length Contraction
The postulates of relativity have led us to conclude that time
depends on thereference frame in which it is measured. We can now
use this fact to show thatthe same must also apply to distances:
The measured distance between two
10-10 s;
g
distance = vgt1>2 = 11.8 * 108 m>s2 * 1.25 * 11.7 * 10-10
s2 = 3.8 cm
g =131 - b2 = 1.25b = 0.6,vgt1>2 .
t1>2gt1>2
v = 0.6 c,t1>2 = 1.7 * 10-10 s.1 : p + p2
N = N0>20.2 L 0.9N0 .
T1lab frame2 L 0.2t1>21lab frame2
= 1.8 * 10-5 s = 1000 * 11.8 * 10-8 s2
t1>21lab frame2 = gt1>21p rest frame2
g = 1000
b = 0.9999995,g
t1>2
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22 Chapter 1 The Space and Time of Relativity
*We are taking for granted that the speed of S relative to is
the same as that of relative to S.This follows from the basic
symmetry between S and as required by thepostulates of
relativity.
SSS
events depends on the frame relative to which it is measured.We
will show thiswith another thought experiment. In the analysis of
this thought experiment, itwill be important to recognize that,
even in relativity, the familiar kinematicrelation
is valid in any given inertial frame (with all quantities
measured in that frame),since it is just the definition of velocity
in that frame.
We imagine again our two frames, S fixed to the ground and fixed
to atrain traveling at velocity v relative to the ground; and we
now imagine ob-servers in S and measuring the length of the train.
For an observer in thismeasurement is easy since he sees the train
at rest and can take all the time heneeds to measure the length
with an accurate ruler. For an observer Q on theground, the
measurement is harder since the train is moving. Perhaps the
sim-plest procedure is to time the train as it passes Q [Fig.
1.7(a)]. If and arethe times at which the front and back of the
train pass Q and if then Q can calculate the length l (measured in
S) as
(1.26)
To compare this answer with we note that observers on the train
couldhave measured by a similar procedure. As seen from the train,
the observerQ on the ground is moving to the left with speed* and
observers on the traincan measure the time for Q to move from the
front to the back of the train asin Fig. 1.7(b). (This would
require two observers on the train, one at the frontand one at the
back.) If this time is
(1.27)
Comparing (1.26) and (1.27), we see immediately that since the
times and are different, the same must be true of the lengths l and
To calculatel.t
t
l = v t
t,
v,l
l,
l = v t
t = t2 - t1 ,t2t1
l
S,S
S
distance = velocity * time
vS
Observer Q
(a)
(b)
S l vt
l vt
v
S S
Observer Q
FIGURE 1.7(a) As seen in S, the train moves adistance to the
right. (b) Asseen in the frame S andobserver Q move a distance to
the left.
v tS,v t
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Section 1.10 Length Contraction 23
the difference, we need to relate and using the time-dilation
formula. Inthe present experiment the two events of interest, Q
opposite the trainsfront and Q opposite the trains back, occur at
the same place in S (where Qis at rest). Therefore, the
time-dilation formula implies that
Comparing (1.26) and (1.27), we see that
(1.28)
The length of the train as measured in S is less than (or equal
to) thatmeasured in
Like time dilation, this result is asymmetric, reflecting the
asymmetry ofour experiment: The frame is special since it is the
unique frame where themeasured object (the train) is at rest. [We
could, of course, have done the ex-periment the other way around;
if we had measured the length of a house atrest in S, the roles of
l and in (1.28) would have been reversed.] To empha-size this
asymmetry and to avoid confusion as to which frame is which, it is
agood idea to rewrite (1.28) as
(1.29)
where the subscript 0 indicates that is the length of an object
measured in itsrest frame, while l refers to the length measured in
any frame.The length canbe called the objects proper length. Since
the effect implied by (1.29) isoften called length contraction (or
Lorentz contraction, or LorentzFitzgeraldcontraction, after the two
physicists who first suggested some such effect). Theeffect can be
loosely described by saying that a moving object is observed to
becontracted.
Evidence for Length Contraction
Like time dilation, length contraction is a real effect that is
well establishedexperimentally. Perhaps the simplest evidence comes
from the same experi-ment as that discussed in connection with time
dilation, in which unstablepions fly down a pipe from the collision
that produces them to the experimen-tal area. As viewed from the
lab frame, we saw that time dilation increases thepions half-life
by a factor of from to In the example discussed, itwas this
increase that allowed most of the pions to complete the journey to
theexperimental area before they decayed.
Suppose, however, that we viewed the same experiment from the
pionsrest frame. In this frame the pions are stationary and there
is no time dilationto increase their half-life. So how do they
reach the experimental area? Theanswer is that in this frame the
pipe is moving, and length contraction reducesits length by the
same factor from L to Thus observers in this framewould say it is
length contraction that allows the pions to reach the experi-mental
area. Naturally, the number of pions completing the journey is
thesame whichever frame we use for the calculation.
L>g.g,
gt1>2 .t1>2g,
l l0 ,l0
l0
l =l0g
l0
l
S
S.
l =lg
l
t = g t
tt
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24 Chapter 1 The Space and Time of Relativity
Example 1.3
A space explorer of a future era travels to the nearest
star,Alpha Centauri, ina rocket with speed The distance from earth
to the star, as mea-sured from earth, is light years. What is this
distance as measured bythe explorer, and how long will she say the
journey to the star lasts? [A lightyear is the distance traveled by
light in one year, which is just c multiplied by1 year, or
kilometers. In many problems it is better to write it as
since the c often cancels out, as we will see.]The distance is
the proper distance between earth and
the star (which we assume are relatively at rest). Thus the
distance as seenfrom the rocket is given by the length-contraction
formula as
If then so
We can calculate the time T for the journey in two ways: As seen
fromthe rocket, the star is initially away and is approaching
withspeed Therefore,
(1.30)
(Notice how the factors of c conveniently cancel when we use
andmeasure speeds as multiples of c.)
Alternatively, as measured from the earth frame, the journey
lasts for atime
But because of time dilation, this is times T(rocket frame),
which istherefore
in agreement with (1.30), of course.Notice how time dilation (or
length contraction) allows an appreciable
saving to the pilot of the rocket. If she returns promptly to
earth, then as a re-sult of the complete round trip she will have
aged only 3.8 years, while hertwin who stayed behind will have aged
8.8 years. This surprising result, some-times known as the twin
paradox, is amply verified by the experiments dis-cussed in Section
1.9. In principle, time dilation would allow explorers to
T1rocket frame2 = T1earth frame2g
= 1.9 years
g
T1earth frame2 = L1earth frame2v
=4 c # years
0.9c= 4.4 years
c # years
=1.7 c # years
0.9c= 1.9 years
T1rocket frame2 = L1rocket frame2v
v = 0.9c.1.7 c # years
L1rocket frame2 = 4 c # years2.3
= 1.7 c # years
g = 2.3,b = 0.9,
L1rocket frame2 = L1earth frame2g
L = 4 c # years1 c # year,
9.46 * 1012
L = 4v = 0.9 c.
TAYL01-001-045.I 12/10/02 1:50 PM Page 24
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Section 1.11 The Lorentz Transformation 25
*Note that our previous two thought experiments were asymmetric,
requiring two ob-servers in one of the frames, but only one in the
other.
make in one lifetime trips that would require hundreds of years
as viewedfrom earth. Since this requires rockets that travel very
close to the speed oflight, it is not likely to happen soon! See
Problem 1.22 for further discussionof this effect.
Lengths Perpendicular to the Relative Motion
We have so far discussed lengths that are parallel to the
relative velocity,such as the length of a train in its direction of
motion. What happens tolengths perpendicular to the relative
velocity, such as the height of the train?It is fairly easy to show
that for such lengths, there is no contraction or ex-pansion.To see
this, consider two observers, Q at rest in S and at rest in and
suppose that Q and are equally tall when at rest. Now, let us
assumefor a moment that there is a contraction of heights analogous
to the lengthcontraction (1.29). If this is so, then as seen by
will be shorter as herushes by. We can test this hypothesis by
having hold up a sharp knife ex-actly level with the top of his
head; if is shorter, Q will find himselfscalped (or worse) as the
knife goes by.
This experiment is completely symmetric between the two frames S
andThere is one observer at rest in each frame, and the only
difference is the
direction in which each sees the other moving.* Therefore, it
must also be truethat as seen by it is Q who is shorter. But this
implies that the knife willmiss Q. Since it cannot be true that Q
is both scalped and not scalped, we havearrived at a contradiction,
and there can be no contraction. By a similar argu-ment, there can
be no expansion, and, in fact, the knife held by simplygrazes past
Qs scalp, as seen in either frame. We conclude that lengths
per-pendicular to the relative motion are unchanged; and the
Lorentz-contractionformula (1.29) applies only to lengths parallel
to the relative motion.
1.11 The Lorentz Transformation
We are now ready to answer an important general question: If we
know the co-ordinates and time t of an event, as measured in a
frame S, how can wefind the coordinates and of the same event as
measured in a sec-ond frame Before we derive the correct
relativistic answer to this question,we examine briefly the
classical answer.
We consider our usual two frames, S anchored to the ground and
an-chored to a train traveling with velocity v relative to S, as
shown in Fig. 1.8. Be-cause the laws of physics are all independent
of our choice of origin andorientation, we are free to choose both
axes Ox and along the same line,parallel to v, as shown. We can
further choose the origins of time so that
at the moment when passes O. We will sometimes refer to
thisarrangement of systems S and as the standard
configuration.S
Ot = t = 0
O x
S
S ?tx, y, z,
x, y, z
Q
Q,
S:
QQQ, Q
QS,Q
v
S S
xvt x
y y
P
OO
FIGURE 1.8In classical physics the coordinatesof an event are
related as shown.
TAYL01-001-045.I 12/10/02 1:50 PM Page 25
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Galileo Galilei(15641642, Italian)
Considered by some the father ofmodern science, Galileo
under-stood the importance of experi-ment and theory and was a
masterof both. Although he did not dis-cover the telescope, he
improvedit and was the first to use it as atool of astronomy,
discovering themountains on the moon, phases ofVenus, moons of
Jupiter, stars ofthe Milky Way, and sunspots androtation of the
sun. Among hismany contributions to mechanics,he established the
law of inertiaand proved that gravity acceleratesall bodies equally
and that the peri-od of a small-amplitude pendulumis independent of
the amplitude.He understood clearly that thelaws of mechanics hold
in all unac-celerated frames, arguing that in-side an enclosed
cabin it would beimpossible to detect the uniformmotion of a ship.
This argumentappeared in his Dialogue on the TwoChief World Systems
and was usedto show that the earth could per-fectly well be moving
in orbitaround the sun without our beingaware of it in everyday
life. Forpublishing this book, he was foundguilty of heresy by the
Holy Officeof the Inquisition, and his bookwas placed on the Index
of Prohib-ited Books from which it wasnot removed until 1835.
26 Chapter 1 The Space and Time of Relativity
Now consider an event, such as the explosion of a small
firecracker, thatoccurs at position and time t as measured in S.
Our problem is to calcu-late, in terms of (and the velocity ) the
coordinates of thesame event, as measured in accepting at first the
classical ideas of spaceand time. First, since time is a universal
quantity in classical physics, we knowthat Next, from Fig. 1.8 it
is easily seen that and (and, similarly, although the z coordinate
is not shown in the figure).Thus, according to the ideas of
classical physics,
(1.31)
These four equations are often called the Galilean
transformation afterGalileo Galilei, who was the first person known
to have considered the invari-ance of the laws of motion under this
change of coordinates. They transformthe coordinates of any event
as observed in S into the correspondingcoordinates as observed
in
If we had been given the coordinates and wanted to findwe could
solve the equations (1.31) to give
(1.32)
Notice that the equations (1.32) can be obtained directly from
(1.31) by ex-changing with and replacing by This is because
therelation of S to is the same as that of to S except for a change
in the signof the relative velocity.
The Galilean transformation (1.31) cannot be the correct
relativistic rela-tion between and (For instance, we know from time
dila-tion that the equation cannot possibly be correct.) On the
other hand,the Galilean transformation agrees perfectly with our
everyday experience andso must be correct (to an excellent
approximation) when the speed is smallcompared to c. Thus the
correct relation between and willhave to reduce to the Galilean
relation (1.31) when is small.
To find the correct relation between and we consid-er the same
experiment as before, which is shown again in Fig. 1.9. We
havenoted before that distances perpendicular to v are the same
whether measuredin S or Thus
(1.33)y = y and z = z
S.
x, y, z, t,x, y, z, tv>c x, y, z, tx, y, z, t
v
t = tx, y, z, t.x, y, z, t,
SS-v.vx, y, z, tx, y, z, t
t = t z = z y = y x = x + vt
x, y, z, t,x, y, z, t
S.x, y, z, tx, y, z, t
t = t z = z y = y x = x - vt
z = zy = yx = x - vtt = t.
Sx, y, z, tvx, y, z, t,
x, y, z,
v
S
Both measured in S Measured in S
S
xvt x
y y
P
OO
FIGURE 1.9The coordinate is measured in
The distances x and aremeasured at the same time t in theframe
S.
vtS.x
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Section 1.11 The Lorentz Transformation 27
exactly as in the Galilean transformation. In finding it is
important to keepcareful track of the frames in which the various
quantities are measured; in ad-dition, it is helpful to arrange
that the explosion whose coordinates we are dis-cussing produces a
small burn mark on the wall of the train at the point where it
occurs. The horizontal distance from the origin to the mark at as
measured in is precisely the desired coordinate Meanwhile, the
samedistance, as measured in S, is (since x and are the horizontal
dis-tances from O to and O to at the instant t, as measured in S).
Thusaccording to the length-contraction formula (1.29),
or(1.34)
This gives in terms of x and t and is the third of our four
required equations.Notice that if is small, and the relation (1.34)
reduces to the first of theGalilean relations (1.31), as
required.
Finally, to find in terms of and t, we use a simple trick. We
canrepeat the argument leading to (1.34) but with the roles of S
and reversed.That is, we let the explosion burn a mark at the point
P on a wall fixed in S, andarguing as before, we find that
(1.35)
[This can be obtained directly from (1.34) by exchanging with
and re-placing by ] Equation (1.35) is not yet the desired result,
but we can com-bine it with (1.34) to eliminate and find Inserting
(1.34) in (1.35), we get
Solving for we find that
or, after some algebra (Problem 1.37),
(1.36)
This is the required expression for in terms of x and t. When is
muchsmaller than 1, we can neglect the second term, and since we
get in agreement with the Galilean transformation, as required.
Collecting together (1.33), (1.34), and (1.36), we obtain our
required fourequations.
(1.37) t = g t - vxc2
z = z y = y x = g1x - vt2
t L t,g L 1,v>ct
t = g t - vxc2
t = gt -g2 - 1gv
x
t
x = g3g1x - vt2 + vt4t.x
-v.vx, tx, t
x = g1x + vt2
Sx, y, z,t
g L 1vx
x = g1x - vt2
x - vt =xg
OPvtx - vt
x.S,P,OP
x,
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-
Hendrik Lorentz(18531928, Dutch)
Lorentz was the first to writedown the equations we now callthe
Lorentz transformation, al-though Einstein was the first to
in-terpret them correctly. He alsopreceded Einstein with the
lengthcontraction formula (though, again,he did not interpret it
correctly).He was one of the first to suggestthat electrons are
present inatoms, and his theory of electronsearned him the 1902
Nobel Prizein physics.
28 Chapter 1 The Space and Time of Relativity
These equations are called the Lorentz transformation, or
LorentzEinsteintransformation, in honor of the Dutch physicist
Lorentz, who first proposedthem, and Einstein, who first
interpreted them correctly. The Lorentz trans-formation is the
correct relativistic modification of the Galilean transforma-tion
(1.31).
If one wants to know in terms of one can simply ex-change the
primed and unprimed variables and replace by in the nowfamiliar
way, to give
(1.38)
These equations are sometimes called the inverse Lorentz
transformation.The Lorentz transformation expresses all the
properties of space and
time that follow from the postulates of relativity. From it, one
can calculate allof the kinematic relations between measurements
made in different inertialframes. In the next two sections we give
some examples of such calculations.
1.12 Applications of the Lorentz Transformation
In this section we give three examples of problems that can
easily be analyzedusing the Lorentz transformation. In the first
two we rederive two familiar re-sults; in the third we analyze one
of the many apparent paradoxes of relativity.
Example 1.4
Starting with the equations (1.37) of the Lorentz
transformation, derive thelength-contraction formula (1.29).
Notice that the length-contraction formula was used in our
derivationof the Lorentz transformation.Thus this example will not
give a new proof oflength contraction; it will, rather, be a
consistency check on the Lorentztransformation, to verify that it
gives back the result from which it wasderived.
Let us imagine, as before, measuring the length of a train
(frame )traveling at speed relative to the ground (frame S). If the
coordinates of theback and front of the train are and as measured
in the trainsproper length (its length as measured in its rest
frame) is
(1.39)
To find the length l as measured in S, we carefully position two
observers onthe ground to observe the coordinates and of the back
and front of thetrain at some convenient time t. (These two
measurements must, of course,be made at the same time t.) In terms
of these coordinates, the length l asmeasured in S is (Fig.
1.10)
l = x2 - x1 .
x2x1
l0 = l = x2 - x1
S,x2 ,x1v
S
t = g t + vxc2
z = z y = y x = g1x + vt2
-v,vz, t,x, y,x, y, z, t
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-
Now, consider the following two events, with their coordinates
as measuredin S.
Event Description Coordinates in S
1 Back of train passes first observer2 Front of train passes
second observer
We can use the Lorentz transformation to calculate the
coordinates of eachevent as observed in
Event Coordinates in
12
(We have not listed the times and since they dont concern us
here.) Thedifference of these coordinates is
(1.40)
(Notice how the times and cancel out because they are equal.)
Since thetwo differences in (1.40) are respectively and l, we
conclude that
or
as required.
Example 1.5
Use the Lorentz transformation to rederive the time-dilation
formula (1.18).In our discussion of time dilation we considered two
events, a flash and
a beep, that occurred at the same place in frame
xflash = xbeep
S,
l =l0g
l0 = gll = l0
t2t1
x2 - x1 = g1x2 - x12
t2t1
x2 = g1x2 - vt22x1 = g1x1 - vt12
S
S.
x2 , t2 = t1x1 , t1
Section 1.12 Applications of the Lorentz Transformation 29
v
S
x1 t1 t2 t1x2
FIGURE 1.10If the two observers measure and at the same time
then l = x2 - x1 .
(t1 = t2),x2x1
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30 Chapter 1 The Space and Time of Relativity
The proper time between the two events was the time as measured
in
To relate this to the time
as measured in S, it is convenient to use the inverse Lorentz
transformation(1.38), which gives
and
If we take the difference of these two equations, the
coordinates anddrop out (since they are equal) and we get the
desired result,
Example 1.6
A relativistic snake of proper length 100 cm is moving at speed
tothe right across a table. A mischievous boy, wishing to tease the
snake, holdstwo hatchets 100 cm apart and plans to bounce them
simultaneously on thetable so that the left hatchet lands
immediately behind the snakes tail. Theboy argues as follows: The
snake is moving with Therefore, itslength is contracted by a
factor
and its length (as measured in my rest frame) is 80 cm. This
implies thatthe right hatchet will fall 20 cm in front of the
snake, and the snake will beunharmed. (The boys view of the
experiment is shown in Fig. 1.11.) Onthe other hand, the snake
argues thus: The hatchets are approaching mewith and the distance
between them is contracted to 80 cm. SinceI am 100 cm long, I will
be cut in pieces when they fall. Use the Lorentztransformation to
resolve this apparent paradox.
Let us choose two coordinate frames as follows: The snake is at
rest inframe with its tail at the origin and its head at Thetwo
hatchets are at rest in frame S, the left one at the origin and
theright one at x = 100 cm.
x = 0x = 100 cm.x = 0S
b = 0.6,
g =131 - b2 = 121 - 0.36 = 54
b = 0.6.
v = 0.6c
t = tbeep - tflash = g1tbeep - tflash2 = g t0xflash
xbeep
tflash = g tflash - vxflashc2
tbeep = g tbeep - vxbeep
c2
t = tbeep - tflash
t0 = t = tbeep - tflash
S,
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Section 1.12 Applications of the Lorentz Transformation 31
xL 0
v
x 80 cm
t 0
xR 100 cm
FIGURE 1.11As seen in the boys frame S, thetwo hatchets
bouncesimultaneously (at ) 100 cmapart. Since the snake is 80 cm
long,it escapes injury.
t = 0
As observed in frame S, the two hatchets bounce simultaneously
atAt this time the snakes tail is at and his head must therefore
be
at [You can check this using the transformation with and you
will find that as required.] Thus,as observed in S, the experiment
is as shown in Fig. 1.11. In particular, theboys prediction is
correct and the snake is unharmed. Therefore, the snakesargument
must be wrong.
To understand what is wrong with the snakes argument, we must
ex-amine the coordinates, especially the times, at which the two
hatchetsbounce, as observed in the frame The left hatchet falls at
and
According to the Lorentz transformation (1.37), the coordinates
ofthis event, as seen in are
and
As expected, the left hatchet falls immediately beside the
snakes tail, at timeas shown in Fig. 1.12(a).
On the other hand, the right hatchet falls at and Thus, as seen
in it falls at a time given by the Lorentz transformation as
We see that, as measured in the two hatchets do not fall
simultaneously.Since the right hatchet falls before the left one,
it does not necessarily haveto hit the snake, even though they were
only 80 cm apart (in this frame). Infact, the position at which the
right hatchet falls is given by the Lorentztransformation as
and, indeed, the hatchet misses the snake, as shown in Fig.
1.12(b).The resolution of this paradox and many similar paradoxes
is seen to be
that two events which are simultaneous as observed in one frame
are not neces-sarily simultaneous when observed in a different
frame. As soon as one recog-nizes that the two hatchets fall at
different times in the snakes rest frame, thereis no longer any
difficulty understanding how they can both miss the snake.
xR = g1xR - vtR2 = 54 1100 cm - 02 = 125 cm
S,
tR = g tR - vxRc2
= 54
0 - 10.6c2 * 1100 cm2c2
= -2.5 nsS,
xR = 100 cm.tR = 0tL = 0,
xL = g1xL - vtL2 = 0.
tL = g tL - vxLc2
= 0S,
xL = 0.tL = 0S.
x = 100 cm,t = 0,x = 80 cmx = g1x - vt2;x = 80 cm