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Chapter 1 The Space and Time of Relativity
Chapter 2 Relativistic Mechanics
Two great theories underlie almost all of modern physics, both of them discovered during the first 25 years of the twentieth century. The first of these, relativity, was pioneered mainly by one person, Albert Einstein, and is the subject of Part I of this book (Chapters 1 and 2). The second, quantum theory, was the work of many physicists, including Bohr, Einstein, Heisenberg, Schrödinger, and others; it is the subject of Part II. In Parts III and IV we describe the applications of these great theories to several areas of modern physics.
Part I contains just two chapters. In Chapter 1 we describe how several of the ideas of relativity were already present in the classical physics of Newton and others, and then describe how Einstein's careful analysis of the relationship between different reference frames, taking account of the observed invariance of the speed of light, changed our whole concept of space and time.Then, in Chapter 2 we describe how the new ideas about space and time required a radical revision of Newtonian mechanics and a redefinition of the basic ideas — mass, momentum, energy, and force — on
which mechanics is built. In the final section of Chapter 2, we briefly describe general relativity, which is the generalization of relativity to include gravity and accelerated reference frames.
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The Space and Time of Relativity
1.2 The Relativity of Orientation and Origin
1.3 Moving Reference Frames
1.4 Classical Relativity and the Speed of Light
;1.5 The Michelson-Morley Experiment
1.6 The Postulates of Relativity
1.7 Measurement of Time
1.8 The Relativity of Time; Time Dilation
1.9 Evidence for Time Dilation
1.10 Length Contraction
1.11 The Lorentz Transformation
1.12 Applications of the Lorentz Transformation
1.13 The Velocity-Addition Formula
;1.14 The Doppler Effect
Problems for Chapter 1
;Sections marked with a star can be omitted without significant loss of continuity.
Most physical measurements are made relative to a chosen reference system. If we measure the time of an event as t = 5 seconds, this must mean that t is 5 seconds relative to a chosen origin of time, t = 0. If we state that the position of a projectile is given by a vector r = (x, y, z), we
must mean that the position vector has components x, y, z relative to a system of coordinates with a definite orientation and a definite origin r = 0. If we wish to know the kinetic energy K
of a car speeding along a road, it makes a big difference whether we measure K relative to a
reference frame fixed on the road or to one fixed on the car. (In the latter case K = 0, of course.) A little reflection should convince you that almost every measurement requires the specification
of a reference system relative to which the measurement is to be made. We refer to this fact as
the relativity of measurements.
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The theory of relativity is the study of the consequences of this relativity of measurements. It is perhaps surprising that this could be an important subject of study. Nevertheless, Einstein showed, starting with his first paper on relativity in 1905, that a careful analysis of how measurements depend on coordinate systems revolutionizes our whole understanding of space and time, and requires a radical revision of classical, Newtonian mechanics.
In this chapter we discuss briefly some features of relativity as it applies in the classical theories of Newtonian mechanics and electromagnetism and then describe the
Michelson-Morley experiment, which (with the support of numerous other, less direct experiments) shows that something is wrong with the classical ideas of space and time. We then state the two postulates of Einstein's relativity and show how they lead to a new picture of space and time in which both lengths and time intervals have different values when measured in any two reference frames that are moving relative to one another. In Chapter 2 we show how the revised notions of space and time require a revision of classical mechanics. We shall find that the resulting relativistic mechanics is usually indistinguishable from Newtonian mechanics when applied to bodies moving with normal terrestrial speeds, but is entirely different when applied to bodies with speeds that are a substantial fraction of the speed of light, c. In
particular, we shall find that no body can be accelerated to a speed greater than c, and that mass
2is a form of energy, in accordance with the famous relation . Emc=
Einstein's theory of relativity is really two theories. The first, called the special theory of relativity, is ―special‖ in that its primary focus is restricted to unaccelerated frames of
reference and excludes gravity. This is the theory that we shall be studying in Chapters 1 and 2 and applying to our later discussions of radiation, nuclear, and particle physics.
The second of Einstein's theories is the general theory of relativity, which is ―general‖ in that it includes accelerated frames of reference and gravity. Einstein found that the study of accelerated reference frames led naturally to a theory of gravitation, and general relativity turns out to be the relativistic theory of gravity. In practice, general relativity is needed only in areas where its predictions differ significantly from those of Newtonian gravitational theory. These include the study of the intense gravity near black holes, of the large-scale universe, and
12of the effect the earth's gravity has on extremely accurate time measurements (one part in 10
or so). General relativity is an important part of modern physics; nevertheless, it is an advanced topic and, unlike special relativity, is not required for the other topics we treat in this book. Therefore, we have given only a brief description of general relativity in the final, and optional, section of Chapter 2.
1.2 The Relativity of Orientation and Origin
In your studies of classical physics you probably did not pay much attention to the relativity of measurements. Nevertheless, the ideas were present and, whether or not you were aware of it, you probably exploited some aspects of relativity theory in solving certain problems. Let us illustrate this claim with two examples.
In problems involving blocks sliding on inclined planes, it is well known that one can choose coordinates in various ways. One could, for example, use a coordinate system S with
origin O at the bottom of the slope and with axes Ox horizontal, Oy vertical, and Oz across the
slope, as shown in Fig. 1.1(a). Another possibility would be a reference frame S' with origin O'
at the top of the slope and axes O'x' parallel to the slope, O'y' perpendicular to the slope, and
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O'z' across it, as in Fig. 1.1(b). The solution of any problem relative to the frame S may look
quite different from the solution relative to S', and it often happens that one choice of axes is
much more convenient than the other. (For some examples, see Problems 1.1 to 1.3.) On the other hand, the basic laws of motion, Newton's laws, make no reference to the choice of origin and orientation of axes, and are equally true in either coordinate system. In the language of relativity theory, we can say that Newton's laws are invariant, or unchanged, as we shift our
attention from frame S to S', or vice versa. It is because the laws of motion are the same in either coordinate system that we are free to use whichever system is more convenient. [FIG. 1.1 = old 1.1 XX]
The invariance of the basic laws when we change the origin or orientation of axes is true in all of classical physics — Newtonian mechanics, electromagnetism, and thermodynamics.
It is also true in Einstein's theory of relativity. It means that in any problem in physics one is free to choose the origin of coordinates and the orientation of axes in whatever way is most expedient. This freedom is very useful and we often exploit it. However, it is not especially interesting in our study of relativity, and we shall not have much occasion to discuss it further. 1.3 Moving Reference Frames
As a more important example of relativity, we consider next a question involving two reference frames that are moving relative to one another. Our discussion will raise some interesting questions about classical physics, questions that were satisfactorily answered only when Einstein showed that the classical ideas about the relation between moving reference frames needed revision.
Let us imagine a student standing still in a train that is moving with constant velocity v
along a horizontal track. If the student drops a ball, where will the ball hit the floor of the train? One way to answer this question is to use a reference frame S fixed on the track, as shown in
Fig. 1.2(a). In this coordinate system the train and student move with constant velocity v to the
right. At the moment of release, the ball is traveling with velocity v and it moves, under the
influence of gravity, in the parabola shown. It therefore lands to the right of its starting point (as measured in the ground-based frame S). However, while the ball is falling the train is
moving, and a straightforward calculation shows that the train moves exactly as far to the right as does the ball. Thus the ball hits the floor at the student's feet, vertically below his hand. [FIG. 1.2 = old 1.2 XX]
Simple as this solution is, one can reach the same conclusion even more simply by using a reference frame S' fixed to the train, as in Fig. 1.2(b). In this coordinate system the train and
-vstudent are at rest (while the track moves to the left with constant velocity ). At the
moment of release the ball is at rest (as measured in the train-based frame S'). It therefore falls
straight down and naturally 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 x', y', z') is entitled to use
Newton's laws of motion and hence to predict that a 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 that Newton's laws of motion hold for an observer on the ground (using coordinates x, y, z), does it follow that Newton's laws also hold for an observer in the train (using x', y', z')? Equivalently, are Newton's laws invariant as we pass from the ground-based 08/30/11 250891973.doc
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frame S to the train-based frame S' ? Within the framework of classical physics, the answer to this question is ―yes,‖ as we now show.
Since Newton's laws refer to velocities and accelerations, let us first consider the velocity of the ball. We let u denote the ball's velocity relative to the ground-based frame S and
u' the ball's velocity relative to the train-based S'. Since the train moves with constant velocity
v relative to the ground, we naturally expect that
? (1.1) uuv=+.
We shall refer to this equation as the classical velocity-addition formula. It reflects our
common sense ideas about space and time, and asserts that velocities obey ordinary vector addition. Although it is one of the central assumptions of classical physics, equation (1.1) is one of the first victims of Einstein's relativity. In Einstein's relativity the velocities u and u' do not
satisfy (1.1), which is only an approximation (although a very good approximation) that is valid when all speeds are much less than the speed of light, c. Nevertheless, we are for the moment
discussing classical physics, and we therefore assume for now that the classical velocity-addition formula is correct.
Now let us examine Newton's three laws, starting with the first (the law of inertia): A body on which no external forces act moves with constant velocity. Let us assume that this law holds in the ground-based frame S. This means that if our ball is isolated from all outside
?forces, its velocity u is constant. Since and the train's velocity v is constant, it uuv=-
follows at once that u' is also constant, and Newton's first law also holds in the train-based frame S'. We shall find that this result is also valid in Einstein's relativity; that is, in both classical physics and Einstein's relativity, Newton's first law is invariant as we pass between two frames whose relative velocity is constant.
Newton's second law is a little more complicated. If we assume that it holds 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 F', m', a' are the Fa=m
corresponding quantities measured relative to the train-based frame S'. We shall do this by
arguing that each of F', m', a' is in fact equal to the corresponding quantity F, m, and a.
The proof that F = F' depends, to some extent, on how one has chosen to define force. Perhaps the simplest procedure is to define forces by their effect on a standard calibrated spring balance. Since observers in the two frames S and S' will certainly agree on the reading of the
balance, it follows that any force will have the same value as measured in S and S'; that is, F =
Within the domain of classical physics it is an experimental fact that any technique for measuring mass (for example, an inertial balance) will produce the same result in either reference frame; that is, m = m'.
Finally, we must look at the acceleration. The acceleration measured in S is
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where t is the time as measured by ground-based observers. Similarly, the acceleration measured in S' is
?du? (1.2) a=?dt
where t' is the time measured by observers on the train. Now, it is a central assumption of classical physics that time is a single universal quantity, the same for all observers; that is, the times t and t' are the same, or t = t'. Therefore, we can replace (1.2) by
we can simply differentiate with respect to t and find that
dv? (1.3) aa=-dt
?or, since v is constant, . aa=
??We have now argued that , m' = m, and . Substituting into the FF=aa=
equation , we immediately find that Fa=m
That is, Newton's second law is also true for observers using the train-based coordinate frame S'.
The third law,
(action force) = – (reaction force),
is easily treated. Since any given force has the same value as measured in S or S', the truth of
Newton's third law in S immediately implies its truth in S'.
We have now established that if Newton's laws are valid in one reference frame, they are also valid in any second frame that moves with constant velocity relative to the first. This shows why we could use the normal rules of projectile motion in a coordinate system fixed to the moving train. More generally, in the context of our newfound interest in relativity, it establishes an important property of Newton's laws: If space and time have the usual properties assumed in classical physics, then Newton's laws are invariant as we transfer our attention from one coordinate frame to a second one moving with constant velocity relative to the first.
Newton's laws would not still hold in a coordinate system that was accelerating.
Physically, this is easy to understand. If our train were accelerating forward, then just to keep the ball at rest (relative to the train) would require a force; that is, the law of inertia would not
?hold in the accelerating train. To see the same thing mathematically, note that if uuv=-
?aand v is changing, then u' is not constant even if u is. Further, the acceleration as given by
(1.3) is not equal to a, since is not zero; so our proof of the second law for the train's ddtv/
frame S' also breaks down. In classical physics, the unaccelerated frames in which Newton's 08/30/11 250891973.doc
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laws hold (including the law of inertia) are often called inertial frames. In fact, one
convenient definition (good in both classical and relativistic mechanics) of an inertial frame is just that it is a frame where the law of inertia holds. The result we have just proved can be rephrased to say that an accelerated frame is noninertial.
1.4 Classical Relativity and the Speed of Light
Although Newton's laws are invariant as we change from one unaccelerated frame to another (if we accept the classical view of space and time), the same is not true of the laws of electromagnetism. We can show this by separately examining each law — Gauss's law,
Faraday's law, and so on — but the required calculations are complicated. A simpler
procedure is to recall that the laws of electromagnetism demand that, in a vacuum, light signals
*and all other electromagnetic waves travel in any direction with speed
18 c== 3.0010 m/s,emoo
where are the permittivity and permeability of the vacuum. Thus if the em and 00
electromagnetic laws hold in a frame S, light must travel with the same speed c in all directions,
as seen in S.
Let us now consider a second frame S' traveling relative to S and imagine a pulse of
light moving in the same direction as S', as shown on the left of Fig. 1.3. The pulse has speed c
relative to S. Therefore, by the classical velocity-addition formula (1.1), it should have speed
cv- as seen from S'. Similarly, a pulse traveling in the opposite direction would have speed c + v as seen from S', and a pulse traveling in any other, oblique direction would have a
cv-different speed, intermediate between and c + v. We see that in the frame S' the speed of
light should vary between cv- and c + v according to its direction of propagation. Since the
laws of electromagnetism demand that the speed of light be exactly c, we conclude that these
laws — unlike those of mechanics — could not be valid in the frame S'.
[FIG.1.3 = old1.3 XX]
The situation just described was well understood by physicists toward the end of the nineteenth century. In particular, it was accepted as entirely obvious that there could be only one frame, called the ether frame, in which light traveled at the same speed, c, in all directions.
The name ―ether frame‖ derived from the belief that light waves must propagate through a medium, in much the same way that sound waves were known to propagate in the air. Since light propagates through a vacuum, physicists recognized that this medium, which no one had ever seen or felt, must have unusual properties. Borrowing the ancient name for the substance of the heavens, they called it the ―ether.‖ The unique reference frame in which light traveled at
speed c was assumed to be the frame in which the ether was at rest. As we shall see, Einstein's relativity implies that neither the ether, nor the ether frame, actually exists. [NEWTON BIO XX]
Our picture of classical relativity can be quickly summarized. In classical physics we take for granted certain ideas about space and time, all based on our everyday experiences. For example, we assume that relative velocities add 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 Newton's laws should be valid in a whole family 08/30/11 250891973.doc
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of reference frames, any one of which moves uniformly relative to any other. On the other hand, we have seen that there could be no more than one reference frame, called the ether frame, relative to which the electromagnetic laws hold, and in which light travels through the vacuum with speed c in all directions.
It should perhaps be emphasized that although this view of nature turned out to be wrong, it was nevertheless perfectly logical and internally consistent. One might argue on philosophical or aesthetic grounds (as Einstein did) that the difference between classical mechanics and classical electromagnetism is surprising and even unpleasing, but theoretical arguments alone could not decide whether or not the classical view is correct. This question could only be decided by experiment. In particular, since classical physics implied that there was a unique ether frame where light travels at speed c in all directions, there had to be some
experiment that showed whether or not this was so. This was exactly the experiment that Michelson, later assisted by Morley, performed between the years 1880 and 1887, as we now describe.
If one assumed the existence of a unique ether frame, it seemed clear that as the earth orbits around the sun, it must be moving relative to the ether frame. In principle, this motion relative to the ether frame should be easy to detect. One would simply have to measure the speed (relative to the earth) of light traveling in various directions. If one found different speeds in different directions, one would conclude that the earth is moving relative to the ether frame, 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 would have to conclude that at the time of the measurements the earth happened to be at rest relative to the ether frame. In this case one should probably repeat the experiment a few months later, by which time the earth would be at a different point on its orbit and its velocity relative to the ether frame should surely be nonzero.
In practice, this experiment is extremely difficult because of the enormous speed of light:
8 . c= 310 m/s
If our speed relative to the ether is v, then the observed speed of light should vary between
cv- and c + v. Although the value of v is unknown, it should on average be of the same order
as the earth's orbital velocity around the sun,
(or possibly more if the sun is also moving relative to the ether frame). Thus the expected
4change in the observed speed of light due to the earth's motion is about 1 part in 10 . This was
too small a change to be detected by direct measurement of the speed of light at that time.
To avoid the need for such direct measurements, Michelson devised an interferometer in which a beam of light was split into two beams by a partially reflecting surface; the two beams traveled along perpendicular paths and were then reunited to form an interference pattern; this pattern was sensitive to differences in the speed of light in the two perpendicular directions and so could be used to detect any such differences. By 1887, Michelson and Morley had built an interferometer (described below) that should have been able to detect differences in the speed of
4light much smaller than the part in 10 expected. To their surprise and chagrin, they could
detect absolutely no difference at all.
[MICHELSON BIO XX]
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The Michelson-Morley and similar experiments have been repeated many times, at different times of year and with ever-increasing precision, but always with the same final
*result. With hindsight it is easy to draw the right conclusion from their experiment: Contrary to all expectations, light always travels with the same speed in all directions relative to an earth-based reference frame, even though the earth has different velocities at different times of year. In other words, light travels at the same speed c in all directions in many different
inertial frames, and the notion of a unique ether frame with this property must be abandonned.
This conclusion is so surprising that it was not taken seriously for nearly 20 years. Rather, several ingenious alternative theories were advanced that explained the Michelson-Morley result but managed to preserve the notion of a unique ether frame. For example, in the ―ether-drag‖ theory, it was suggested that 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 same way that the earth does drag its atmosphere with it). If this were the case, an earth-bound observer would automatically be at rest relative to the ether, and Michelson and Morley would naturally have found that light had the same speed in all directions at all times of year. Unfortunately, this neat explanation of the Michelson-Morley result requires that light from the stars would be bent as it entered the earth's envelope of ether. Instead, astronomical observations show that light from any star continues to move in a straight line as it arrives at
The ether-drag theory, like all other alternative explanations of the Michelson-Morley result, has been abandoned because it fails to fit all the facts. Today, nearly all physicists agree that Michelson and Morley's failure to detect our motion relative to the ether frame was because there is no ether frame. The first person to accept this surprising conclusion and to develop its consequences into a complete theory was Einstein, as we describe, starting in Section 1.6.
;1.5 The Michelson-Morley Experiment
;More than a hundred years later, the Michelson-Morley experiment remains the simplest and cleanest evidence that light travels at the same speed in all directions in all inertial frames — what became the second postulate of
relativity. Naturally, we think you should know a little of how this historic experiment worked. Nevertheless, if you are pressed for time, you can omit this section without loss of continuity.
Figure 1.4 is a simplified diagram of Michelson's interferometer. Light from the source hits the half-silvered mirror M and splits, part traveling to the mirror M, and part to M . The two 12
beams are reflected at M and M2, and return to M, which sends part of each beam on to the 1
observer. In this way the observer receives two signals, which can interfere constructively or destructively depending on their phase difference.
[FIG.1.4 = old 1.4 XX]
To calculate this phase difference, suppose for a moment that the two arms of the interferometer, from M to M, and M to M , have exactly the same length l, as shown. In this 12
case any phase difference must be due to the different 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 earth's velocity v. In this case the light travels from M to M with speed c + v (relative to the 1
cv-interferometer) and back from M to M with speed . Thus the total time for the round 1
trip on path 1 is
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lllc2 (1.4) t=+=221cvcvcv+--
It is convenient to rewrite this in terms of the ratio
-4which we have seen is expected to be very small, . In terms of ，, (1.4) becomes b:10
212ll2 . (1.5) t=?1b()21cc1-b
In the last step we have used the binomial approximation (discussed in Appendix B and in
Problems 1.12 – 1.14),
n , (1.6) ()11-?xnx
which holds for any number n and any x much smaller than 1. (In the present case,
2.) nxb=-=1 and
The speed of light traveling from M to M2 is given by the velocity-addition diagram in Fig. 1.4(b). (Relative to the earth the light has velocity u perpendicular to v; relative to the
ether it travels with speed c in the direction shown.) This speed is
22 . ucv=-
Since the speed is the same on the return journey, the total time for the round trip on path 2 is
222lll21 (1.7) t==?1b()22222ccvc--1b
1where we have again used the binomial approximation (1.6), this time with . n=-2
Comparing (1.5) and (1.7), we see that the waves traveling along the two arms take
slightly different times to return to M, the difference being
l2 . (1.8) D=- tttb12c
DtIf this difference were zero, the two waves would arrive in step and interfere constructively,
Dtgiving a bright resultant signal. Similarly, if were any integer multiple of the light's
Dtperiod, (where is the wavelength), they would interfere constructively. If Tc=l/l
D=tT0.5were equal to half the period, (or 1.5T, or 2.5T, . . . ), the two waves would be exactly out of step and would interfere destructively. We can express these ideas more
compactly if we consider the ratio
22Dtlclbb/ . (1.9) N===Tcll/
This is the number of complete cycles by which the two waves arrive out of step; in other
words, N is the phase difference, expressed in cycles. If N is an integer, the waves interfere
135constructively; if N is a half-odd integer (), the waves interfere destructively. N=,,,...222