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

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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

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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

TAYL01-001-045.I 12/10/02 1:50 PM Page 23

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

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

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

TAYL01-001-045.I 12/10/02 1:51 PM Page 26

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,

TAYL01-001-045.I 12/10/02 1:51 PM Page 27

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

TAYL01-001-045.I 12/10/02 1:51 PM Page 28

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,

TAYL01-001-045.I 12/10/02 1:51 PM Page 30

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