By the year 1900, most of physics seemed to be encompassed in the two great theories of Newtonian mechanics and Maxwell's theory of electromagnetism. Unfortunately, there were inconsistencies between the two theories that seemed irreconcilable. Although many physicists struggled with the problem, it took the genius of Einstein to see that the inconsistencies were concerned not merely with mechanics and electromagnetism, but with our most elementary ideas of space and time. In the special theory of relativity, Einstein resolved these difficulties and profoundly altered our conception of the physical universe.
Readers looking for a concise, well-written explanation of one of the most important theories in modern physics need search no further than this lucid undergraduate-level text. Replete with examples that make it especially suitable for self-study, the book assumes only a knowledge of algebra. Topics include classical relativity and the relativity postulate, time dilation, the twin paradox, momentum and energy, particles of zero mass, electric and magnetic fields and forces, and more.
About the Author
James H. Smith was Professor of Physics at the University of Illinois.
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Introduction to Special Relativity
By James H. Smith
Dover Publications, Inc.Copyright © 1993 James H. Smith
All rights reserved.
Classical Relativity and the Relativity Postulate
1-1 THE POSTULATE OF RELATIVITY
LIFE ON A LARGE OCEAN LINER goes on very much as it does on shore. People swim in the pool, play shuffleboard on deck, eat in the dining room. Even on a jet airplane going at 600 mi/hr it takes no extra effort to eat dinner. Coffee pours just as it does for someone at rest on the earth. The point is that, although we think of these vehicles as being in rapid motion, it seems to make no difference in the behavior of commonplace objects when such behavior is referred to the moving system — we sometimes say: when such behavior is viewed from the moving frame of reference. We are not surprised at this; we expect it. In fact, if we stop to think a moment, we see that any other situation would be very peculiar indeed. For the speeds of these vehicles we have just mentioned are trivial compared to the speed of the earth about the sun or the sun through the galaxy or the galaxy. ... In fact, we soon run into difficulty. We do not know how to measure a velocity at all unless we refer it to something, i.e., to some frame of reference. It would be very peculiar if the mechanical behavior of objects depended on the speed with which we were moving, i.e., if the laws of physics were different for observers moving with different speeds. It would mean, for instance, that a billiard player would have to adjust his style of play to the season of the year, because the motion of the earth about the sun requires that its velocity differ by 60 km/sec at 6-month intervals. It is our common experience that no such adjustment is necessary, and that no experiment, however precise, has shown any such difference at all. That is all the postulate of relativity means.
It does, of course, go somewhat beyond our ordinary experience because it says that all experiments will give results independent of the velocity of the frame of reference in which the measurements are made. All of our experience is limited — and that includes laboratory experience in physics. All that experience can say is that nobody has yet performed an experiment that contradicts this postulate of relativity; tomorrow someone may do so. We will then have to revise our opinions. But for now we take the postulate at its face value, without reservation, and admit that no experiment can be performed which will give different results when performed in two laboratories moving uniformly with respect to one another.
The word "experiment" is used in the first postulate in a somewhat restricted sense. To pursue the analogy of billiards it means the making of a particular shot in a particular way. The "results" of the "experiment" are simply the consequences of the shot. The experiment consists of setup and results, both related to a particular frame of reference. The first postulate says that if the setup is made in the same way, the results will be the same whether the billiard table is "fixed" on the earth or carried in a speeding plane. On the other hand, measuring the "speed of the earth" is not an experiment in this sense. There is no particular set of initial conditions; there are no consequences to be determined. Clearly, the speed of the earth depends on the particular frame of reference in which the measurement is made.
1-2 FRAMES OF REFERENCE
In the last section we used the term frame of reference, and we implied the measurement of motion with respect to a frame of reference. We will be using this term frequently, and in this section we will formulate our ideas somewhat more precisely.
Although the term is usually applied to the entire situation in which a particular experiment is performed, it is probably helpful to think of a frame of reference as the coordinate system with respect to which measurements are made. To say that an automobile is moving at 60 mi/hr in the frame of reference of the earth, or simply with respect to the earth, implies that the automobile passed one point fixed on the earth at one instant and passed another point fixed on the earth 60 mi distant from the first, one hour later. When we say that a man walks down the aisle of a jet airliner at 3 ft/sec, we imply that the measurement was made in the frame of reference of the airliner, i.e., with respect to a coordinate system fixed to the airliner. With respect to a coordinate system fixed to the ground, the man is possibly moving 900 ft/sec. We see, then, that even in this very simple situation, the description of the motion of an object (the man) depends on the frame of reference from which it is viewed (the plane or the earth).
Suppose that there are two coordinate systems moving with respect to one another with a speed υ. We call one the O system and the other the O' system. The whole O' system is moving to the right with a speed υ along the positive x-axis of the O system. Conversely, the O system is moving toward negative x' as measured in the O' system. Figure 1-1 shows the system at several different times. Figures 1-1a, b, and c have been drawn as if the O system remained fixed on the page and Figures 1-1d, e, and f as if the O' system remained fixed, but it must be clearly understood that the sequence depicted by Figures 1-1a, b, and c is the same as that depicted by Figures 1-1d, e, and f. The only thing with physical content is that O and O' are moving apart with the speed υ. Figure 1-1c looks just exactly like Figures 1-1f.
Now let us suppose that some object A starts in the O system and moves from the origin (x = 0) at the time t = 0 and later is found to be at the point (x, y) at the time t. This is shown in Figures 1-2a and 1-2b. Furthermore, suppose the origin of the coordinate system O' happens to coincide with the origin of O at t = 0 but is moving along the x-axis with the speed υ so that by the time t it has advanced a distance υ t. This is shown in Figures 1-3a and 1-3b. It would be equally correct to say that O had moved a distance υ t toward negative x' measured with respect to O'. The object A is found a distance x' from the O' origin at the time t, and a glance at Figure 1-3b shows clearly that x = x' + υ t. Since the relative motion of the coordinate systems occurred along the x-axis, it is equally clear that y = y' and if a third coordinate were shown, z = z'. Although it is, for present purposes, meaningless to distinguish them, we will explicitly state that clocks in both coordinate systems read alike and therefore t = t'. Summarizing, we have
1-3 INERTIAL FRAMES OF REFERENCE
In what we have said so far it seems that any frame of reference is equivalent to any other. That is certainly not so. Coffee may pour in a smoothly riding jet plane just like coffee on the ground, but if the ride is bumpy, allowances must be made. No fancy apparatus is necessary. One's stomach is an excellent indicator. The difference between a smooth ride and a bumpy one is clearly one of acceleration. All experiments, so reads the first postulate, give the same results in uniformly moving coordinate systems. It follows that no experiment will detect uniform motion. Even one's stomach detects accelerated motion.
For instance, take Netwon's First Law, the Law of Inertia: "A body at rest will remain at rest, or a body in uniform motion will remain in uniform motion unless acted on by a force." When that motion is measured with respect to a bumpy jet airliner, the law simply is not true. It is not true on a merry-go-round. Put a marble on the floor of a merry-go-round and it will not remain at rest with respect to the merry-go-round. It will immediately accelerate toward the outside. What does it mean, then, to say that Newton's First Law is true? When we say that it is true, we simply mean that there are coordinate systems where it is true. For most purposes the earth is such a coordinate system. Sensitive measurements can, nevertheless, detect accelerations due to its rotation. Astronomical observations lead us to believe that the law is more nearly true when referred to the coordinate system of the fixed stars. This discussion could lead us far afield. We will simply assume that there exist coordinate systems where Newtonian mechanics works. We call such frames of reference inertial frames of reference or sometimes just inertial frames. In such frames of reference certain laws of physics hold. What our experience tells us, and the relativity postulate states explicitly, is that in all frames of reference moving uniformly with respect to an inertial frame, the same laws of physics hold. In this book we shall be almost entirely concerned with such inertial frames. How the laws of physics must be formulated in noninertial, i.e., accelerated, systems is the province of the more complex general theory of relativity.
1-4 THE CONSERVATION OF MOMENTUM IN DIFFERENT FRAMES OF REFERENCE
Our discussion of experiments in different frames of reference has, up to now, been rather general and qualitative. We have said that the relativity postulate arose largely from important ideas in classical mechanics, and it is the purpose of this section to explore this connection in more detail. We are going to use, as a simple example, an experiment in conservation of momentum performed in a laboratory fixed on the earth and in a second laboratory moving uniformly past it at a speed υ in the positive x-direction with respect to the earth. This second "laboratory" might be a speeding train.
We will assume that the laboratory fixed on the earth is an inertial frame of reference, i.e., that the familiar laws of mechanics hold there. Among these is the law of conservation of momentum. We will show that if momentum is conserved on the earth, it is also conserved on the train.
Suppose that an observer on the train, Figure 1-4a and b, performs a simple collision experiment between objects of masses m1 and m2. For simplicity, suppose all velocities are in the x-direction. The object m1 has a velocity u1' before the collision and U1' after it; m2 has velocities u2' and U2'. The primes indicate that these velocities are to be measured with respect to the train. Since we claim to be in doubt about the validity of conservation of momentum on the train, we do not claim to know the relations between the velocities before and after the collision directly. We do, however, know that momentum is conserved in the frame of reference of the ground. We therefore attempt to describe the same experiment in that frame of reference.
To an observer on the ground watching this experiment, Figure 1-4c and d, the object m1 has a velocity
u1 = u1' + υ (1-2)
Equation 1-2 is a familiar one, but is proved in Exercise 1 from Equation 1-1. Since momentum is known to be conserved in the frame of reference of the ground
m1u1 + m2u2 = m1U1 + m2U2 (1-3)
Substitution of Equation 1-2 into Equation 1-3 gives
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where equations similar to Equation 1-2 have been used for the other velocities. Canceling the terms containing υ,
m1u1' + m2u2' = m1U1' + m2U2' (1-4)
This last equation expresses conservation of momentum in the frame of reference of the train.
Proving that momentum was conserved on the train if it was conserved on the ground, i.e., if momentum was conserved in one frame of reference then in all frames moving uniformly with respect to it, depended primarily on two things:
1. The form of the dependence of momentum on velocity, namely momentum = mu.
2. The relation between the velocities of an object as seen by two different observers, i.e., u = u' + υ which, in turn, depends ultimately on Equation 1-1 relating the positions of an object as seen by two observers.
1-5 ABSOLUTE VELOCITY VS. RELATIVE VELOCITY
In the last section we saw that an experimenter on a train would find that momentum was conserved even though the train was moving rapidly. In fact, we said it could be shown that any mechanics experiment he could perform would give the same result as it would if performed on the earth. The importance of this idea for mechanics was well understood. Einstein recognized it as a general property of nature applying to all types of experiments, not merely those of mechanics.
Now if the results of any experiment performed on the uniformly moving train will give the same results as an identical experiment performed on the earth, then it is clear that there is no way for the experimenter on the train to tell whether he is moving — at least by the performance of an experiment done entirely on board the train. This leads to an alternative statement of the relativity postulate which is often convenient to remember and which is somewhat closer to the popular conception of relativity:
No experiment can be performed which will detect an absolute velocity through space.
Of course, the simplest experiment in the world, looking out the window, will detect relative motion between the train and the earth, but that is all. The man riding on the train is unable to tell whether he is moving or the earth is moving past him. Even the jiggles and joggles are not enough to assure us that the train is moving because they would still be there if, in some cosmic joke, a giant were pulling the "rug" out from beneath the train!
1-6 THE RELATIVE SPEED OF TWO OBSERVERS AS MEASURED BY EACH OF THEM
There is a corollary to the first postulate which is probably so self- evident that it needs no proof, and yet it is so important for some of our future considerations that this section is devoted to a discussion of it. It is this:
If the uniform speed of an observer O' relative to an observer O is υ, then the speed of O relative to O' is also υ.
This only means that if a man standing beside the highway sees a car going down the highway at 60 mi/hr, the driver of the car sees the first man passing at 60 mi/hr.
As a further example, suppose two trains are passing each other, and a man on train A has a device for measuring the speed of train B past train A. (See Figure 1-5.) He finds υA (i.e., υA is the reading of the device on train A). He has arranged for an identical device to be placed aboard train B and for the information to be telemetered back to him. The result of this experiment is that the remote device reports υB (the reading of the device on train B).
Excerpted from Introduction to Special Relativity by James H. Smith. Copyright © 1993 James H. Smith. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
1. Classical Relativity and the Relativity Postulate
2. Light Waves and the Second Postulate
3. Time Dilation: Proper and Improper Time
4. Length Measurements
5. Velocity and Acceleration
6. The Twin Paradox
7. The Lorentz Transformation and Notation
8. Proper- or Four-Velocity
9. Momentum and Energy
10. Particles of Zero Mass
11. Center-of-Mass and Particle Systems
13. Electric and Magnetic Fields and Forces
Appendix A. Approximate Calculations in Relativity
Appendix B. A Summary of Relativistic Formulas
Appendix C. A Table of Particles