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Galilean Relativity
1.1. Relativity and Common Sense
A child walks along the floor of a moving train. Passengers on the train measure the child's speed and find it to be 1 meter per second. When ground-based observers measure the speed of the same child, they obtain a different value; observers on an airplane flying overhead obtain still another. Each set of observers obtains a different value when measuring the same physical quantity. Finding the relation between those values is a typical problem in relativity.
There is nothing at all startling about these observations; relativity was not invented by Albert Einstein. Einstein's work did, however, drastically change the way such phenomena are understood; the term "relativity" as used today generally refers to Einstein's theory.
The study of relativity began with the work of Galileo Galilei around 1630; Isaac Newton also made important contributions. The ideas described in this chapter, universally accepted until 1900, are known as "Galilean relativity."
Galilean relativity is fully consistent with the intuitive notions that we call "common sense." In the example above, if the train moves at 30 meters per second (m/sec) in the same direction as the child, common sense suggests that ground-based observers should find the child's speed to be 31 m/sec; Galilean relativity gives precisely that value. Einstein's theory, as we shall see, gives a different result.
In the case of the child, the difference between the two theories is minute. The speed measured by ground observers according to Einstein's relativity differs from the Galilean value 31 m/sec only in the fourteenth decimal place; no measurement could possibly detect such a tiny difference. This result is characteristic of Einsteinian relativity: its predictions are indistinguishable from those of Galilean relativity whenever the observers, as well as all objects under observation, move slowly relative to one another. That realm is generally called the nonrelativistic limit, although Galilean or Newtonian limit would be a more apt designation. "Slowly" here means at a speed much less than the speed of light.
The speed of light plays a central role in Einstein's theory; whenever any speed in the problem approaches that value, Einsteinian relativity departs dramatically from that of Galileo and Newton. Because the speed of light is so great, however, most commonly observed phenomena are adequately described by Galilean relativity.
The "special" theory of relativity, which is the principal subject of this book, is restricted to observers who move uniformly, that is, at constant speed in the same direction. If observers move with changing speeds, or along curved paths, the problem of relating their measurements is much more complicated. Einstein addressed that problem as well, in his "general" theory of relativity. Because the general theory involves quite advanced mathematics, I can give only a descriptive treatment in chapter 8. The special theory, in contrast, requires only elementary algebra and geometry and can be presented with full rigor.
Many of the conclusions of special relativity run counter to our intuition concerning the nature of space and time. Before Einstein, no one doubted that time is absolute. Newton put it as follows in his Principia: "Absolute, true, and mathematical time, of itself and from its own nature, flows equably without relation to anything external."
Special relativity obliges us to abandon the absolute nature of time. We shall see, for example, that the time order of two events can depend on the relative motion of the observers who view them. One set of observers may find that a certain event A occurred before another event B, whereas according to a second set of observers, who are moving relative to the first, B occurred before A. This result is surely difficult to accept.
In some cases, a reversal of time ordering would be truly bizarre. Suppose that at event A a moth lands on the windshield of a moving car; the car clock reads 12:00. At event B another moth lands; the car clock now reads 12:05. For the driver of the car, the order of those events is a direct sensory experience: she can see both events happen right in front of her and can assert with confidence that A happened first. If observers on the ground were to claim that event B happened first, they would be denying that sensory experience; moreover, the car clock would according to them be running backward! (It would read 12:05 before it reads 12:00.)
As we shall see, special relativity implies that moving clocks run slow. That is itself a strange result, but clocks running backward would be too much to swallow. No such disaster arises, however. In the case of the moths, event A happens first according to all observers. A reversal of time ordering can occur only for events spaced so far apart that no single observer (and no single clock) can be present at both. The order of such events is not a direct sensory experience for anyone; it can be determined only by comparing the readings of two distinct clocks, one present at event A and the other present at B. If two sets of observers disagree on the order of those events, no one's sensory experience is contradicted and no one sees any clock running backward. The proof of this assertion, given in chapter 5, depends on the fact that nothing can travel faster than light, one of the important consequences of special relativity.
A logical requirement of any theory is causality. If event A is the cause of event B, A must occur before B: the cause must precede the effect. We will see in chapter 5 that special relativity is consistent with the causality requirement. Whenever a cause-and-effect relation exists between two events, their time order is absolute: all observers agree on which one happened first.
Figure 1.1 shows a hypothetical experiment to illustrate the relativistic reversal of time ordering. Event A takes place in San Francisco and event B in New York. According to clocks at rest at those locations, A occurs before B. The same events are monitored by observers on spaceships moving from west to east at equal speeds; one ship is over San Francisco when event A occurs, and the other is over New York when event B occurs. Special relativity predicts that if the ships are moving fast enough, their clocks can show event B happening before A. Notice that no single clock is present at both events; the relevant times in the problem are recorded by four distinct clocks, two on the ground and two on the spaceships.
I hasten to add that no such experiment has ever been performed. The fastest available rockets travel a few kilometers per second, only about one hundred thousandth the speed of light. At that speed, the events of figure 1.1 would have to be separated in time by less than a millionth of a second if a reversal of time order were to be detectable. Moreover, the speeds of the two spaceships would have to be equal to within a very small tolerance. The experiment is just too hard to carry out. But we can be confident that if faster rockets were available and if other technical requirements were met, the effect could be detected.
The evidence that confirms special relativity comes principally from atomic and subatomic physics. In many experiments particles move at speeds close to that of light, and the effects of special relativity are dramatic. Particles are created and annihilated in accord with the famous Einstein relation E = mc2. No understanding of such phenomena, or of the kinematics of high-energy particle reactions, would be possible without relativity. Thus Einstein's theory is confirmed daily in every high-energy physics laboratory. Particle reactions are not within the realm of everyday experience, however; in the latter realm, everything moves fairly slowly and relativistic effects are not manifested. If the speed of light were much smaller, the effects of special relativity would be more prominent and our intuition concerning the nature of time would be quite different.
The preceding discussion is intended to provide a taste of what is to come and to encourage the reader to approach relativity with an open mind. I am not suggesting that any conclusion contrary to one's intuition be accepted uncritically, even though the context may be restricted to unfamiliar phenomena. On the contrary, any such conclusion must be vigorously challenged. Before abandoning ideas that appear to be self-evident, one must be satisfied that the experimental evidence is sound and the logical arguments are compelling.
1.2. Events, Observers, and Frames Of Reference
I begin by defining some important terms. In relativity an event is any occurrence with which a definite time and a definite location are associated; it is an idealization in the sense that any actual event is bound to have a finite extent both in time and in space.
A frame of reference consists of an array of observers, all at rest relative to one another, stationed at regular intervals throughout space. A rectangular coordinate system moves with the observers, so that the x, y, and z coordinates of each observer are constant in time. The observers carry clocks that are synchronized: each clock has the same reading at the same time.
Each observer records all events that occur at her location. Each event has four coordinates: three space coordinates and a time. By definition, the space coordinates are the coordinates of the observer who detected the event and the time of the event is the reading of her clock when it occurs.
A second frame of reference consists of another array of observers, all at rest relative to one another and all moving at the same velocity relative to the first set. They have their own coordinate system and their own (synchronized) clocks, and they also record the coordinates of events. The coordinates of a given event in two frames of reference are, in general, different. The central problem of relativity is just to determine the relation between the two sets of coordinates; this turns out to be not so simple a matter as it first appears.
Throughout this book, whenever observations in two frames of reference are being compared, one frame will be called S and the other S'. Coordinates measured in frame S' will be designated by primed symbols, and those measured in frame S will be designated by unprimed symbols. Events will be labeled E1, E2, E3, and so on. Thus, x'1, y'1, z'1, and t'1 denote the coordinates of event E1 measured in frame S'; x'2, y'2, z'2, and t2 denote the coordinates of event E2 measured in frame S, and so on.
As an illustration, let us return to the problem of the child walking on a train. Figure 1.2 shows the child's motion as seen in two frames of reference, one fixed on the train (sketches [a] and [b]) and one fixed on the ground (sketches [c] and [d].) S is the ground frame and S' the train frame. The two sets of axes are parallel to one another. The train's motion as seen from the ground is taken to be in the x direction and the floor of the car is in the x-y plane. Since the child has no motion in the z direction, the figure has been simplified by omitting the z and z' axes.
In figure 1.2a, the child is just passing a train observer labeled H'; this is event E1 The space coordinates of E1 in S' are x'1 = 2, y'1 = 1, z'1 = 0; its time coordinate t'1 is the reading of the clock held by H' as the child passes her. Some time later, as shown in figure 1.2b, the child passes a second train observer, labeled J'; this is event E2. The space coordinates of E2 are x'2 = 2, y'2 = 4, z'2 = 2; its time coordinate t'2 is the reading of the clock held by J'.
Figure 1.2c shows event E1 as seen in the ground frame. The child is just passing ground observer B. The space coordinates of E1 in S are x1 = 2, y1 = 1, z1 = 0; its time coordinate is read off B's clock. Figures 1.2a and 1.2c should be thought of as being superposed: the positions of ground observer B, train observer H', and the child all coincide when E1 occurs.
Figure 1.2d similarly shows E2 as seen in frame S; the child is now passing ground observer Q. The space coordinates of E2 in frame S are x2 = 5, y2 = 4, z2 = 0. The positions of Q, J', and the child all coincide at E2. Notice that B and H', whose positions coincided at E1, no longer coincide at E2. AS seen from the ground, all the train observers have moved to the right during the interval between the two events. (As seen from the train, all the ground observers have moved an equal distance to the left.)
Inspection of the figures reveals that the length of the child's path measured in the ground frame is greater than that measured in the train frame. The child's speed in the ground frame is correspondingly greater (provided the elapsed time is the same in both frames, which is true in Galilean relativity).
The notion of a frame of reference as an (essentially infinite) array of observers is not intended to be a literal description of how measurements are carried out. It would be impractical, to say the least, to station observers throughout all space in the manner prescribed. But there is no reason in principle why that could not be done. In what follows, every event is assumed to be monitored by observers on the scene.
1.3. The Principle of Relativity And Inertial Frames
The principle of relativity was first enunciated by Galileo in 1632. Galileo's argument is clear and graphically put.
Salviatus: Shut yourself up with some friend in the main cabin below decks on some large ship and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle which empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something toward your friend, you need throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully (though there is no doubt that when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still. In jumping, you will pass on the floor the same spaces as before, nor will you make larger jumps toward the stern than toward the prow, ... despite the fact that during the time that you are in the air the floor under you will be going in a direction opposite to your jump.... Finally the butterflies and flies will continue their flights indifferently toward every side, nor will it ever happen that they are concentrated toward the stern, as if tired out from keeping up with the course of the ship, from which they will have been separated during long intervals by keeping themselves in the air.... The cause of all these correspondences of effects is the fact that the ships' motion is common to all the things contained in it.
Galileo is asserting, in effect, that the laws of nature are the same in any two frames of reference that move uniformly with respect to one another. If identical experiments are carried out by two sets of observers, with identical initial conditions, all the results will be the same. It follows that there is no way to determine by means of experiments carried out in a given frame of reference whether the frame is at rest or is moving uniformly. Only the relative velocity between frames can be measured. This set of assertions is called the principle of relativity.
Excerpted from Understanding Relativity by Leo Sartori. Copyright © 1996 The Regents of the University of California. Excerpted by permission of UNIVERSITY OF CALIFORNIA PRESS.
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Anonymous
Posted November 15, 2004
My favorite book on Relativity- definitely the one I have spent the most time reading. It has some math and tackles most of special relativity and, to a lesser extent, general relativity. It has just enough math to challenge you (as long as you can remember some of algebra!) but never gets too hard. Furthermore, through examples and simple math it shows just how radical and how much of an original thinker Einstein truly was- how he took some basic ideas and extended them to uncover many 'truths' that had eluded those before him.
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Overview
Sartori covers general relativity and cosmology, but focuses on Einstein's theory. He tracks ...