16 Relativistie Energy and Momentum 16-1 Relativity and the philosophers In this chapter we shall continue to discuss the principle of relativity of 16-1 Relativity and the philosophers Einstein and Poincare,as it affects our ideas of physics and other branches of human thought. 16-2 The twin paradox Poincare made the following statement of the principle of relativity:“Accord- 16-3 Transformation of velocities ing to the principle of relativity,the laws of physical phenomena must be the same 16-4 Relativistic mass for a fixed observer as for an observer who has a uniform motion of translation relative to him,so that we have not,nor can we possibly have,any means of 16-5 Relativistic energy discerning whether or not we are carried along in such a motion." When this idea descended upon the world,it caused a great stir among philos- ophers,particularly the“cocktail-party philosophers,"who say,“Oh,it is very simple:Einstein's theory says all is relative!"In fact,a surprisingly large number of philosophers,not only those found at cocktail parties(but rather than embarrass them,we shall just call them“cocktail-party philosophers'"),will say,“That all is relative is a consequence of Einstein,and it has profound influences on our ideas."In addition,they say“It has been demonstrated in physics that phenomena depend upon your frame of reference."We hear that a great deal.but it is difficult to find out what it means.Probably the frames of reference that were originally referred to were the coordinate systems which we use in the analysis of the theory of relativity.So the fact that“things depend upon your frame of reference"is supposed to have had a profound effect on modern thought.One might well wonder why,because,after all,that things depend upon one's point of view is so simple an idea that it certainly cannot have been necessary to go to all the trouble of the physical relativity theory in order to discover it.That what one sees depends upon his frame of reference is certainly known to anybody who walks around, because he sees an approaching pedestrian first from the front and then from the back;there is nothing deeper in most of the philosophy which is said to have come from the theory of relativity than the remark that "A person looks different from the front than from the back."The old story about the elephant that several blind men describe in different ways is another example,perhaps,of the theory of rela- tivity from the philosopher's point of view. But certainly there must be deeper things in the theory of relativity than just this simple remark that“A person looks different from the front than from the back."Of course relativity is deeper than this,because we can make definite predictions with it.It certainly would be rather remarkable if we could predict the behavior of nature from such a simple observation alone. There is another school of philosophers who feel very uncomfortable about the theory of relativity,which asserts that we cannot determine our absolute velocity without looking at something outside,and who would say,"It is obvious that one cannot measure his velocity without looking outside.It is self-evident that it is meaningless to talk about the velocity of a thing without looking outside;the physicists are rather stupid for having thought otherwise,but it has just dawned on them that this is the case.If only we philosophers had realized what the prob- lems were that the physicists had,we could have decided immediately by brain- work that it is impossible to tell how fast one is moving without looking outside, and we could have made an enormous contribution to physics.These philosophers are always with us,struggling in the periphery to try to tell us something,but they never really understand the subtleties and depths of the problem. 16-1
Our inability to detect absolute motion is a result of experiment and not a result of plain thought, as we can easily illustrate. In the first place, Newton believed that it was true that one could not tell how fast he is going if he is moving with uniform velocity in a straight line. In fact, Newton first stated the principle of relativity, and one quotation made in the last chapter was a statement of New ton's. Why then did the philosophers not make all this fuss about"all is relative, r whatever, in Newton,s time? Because it was not until Maxwells theory of lectrodynamics was developed that there were physical laws that suggested tha one could measure his velocity without looking outside; soon it was found experi- mentally that one could not Now, is it absolutely, definitely, philosophically necessary that one should not be able to tell how fast he is moving without looking outside? One of the consequences of relativity was the development of a philosophy which saic You can only define what you can measure! Since it is self-evident that one can not measure a velocity without seeing what he is measuring it relative to, therefor it is clear that there is no meaning to absolute velocity. The physicists should have realized that they can talk only about what they can measure. But that is the whole problem: whether or not one can define absolute velocity is the same as the problem of whether or not one can detect in an experiment, without looking outside, whether he is moving. In other words, whether or not a thing is measurable is not something to be decided a priori by thought alone, but something that can be decided only by experiment. Given the fact that the velocity of light is 186, 000 mi/ sec, one will find few philosophers who will calmly state that it is self-evident that if light goes 186,000 mi/sec inside a car, and the car is going 100,000 mi /sec, that the light Iso goes 186,000 mi/ sec past an observer on the ground. That is a shocking fact to them; the very ones who claim it is obvious find, when you give them a specific fact, that it is not obvious Finally, there is even a philosophy which says that one cannot detect any motion except by looking outside. It is simply not true in physics. True, one can lot perceive a unform motion in a straight line, but if the whole room were ro tating we would certainly know it, for everybody would be thrown to the wall- there would be all kinds of"centrifugal"effects. That the earth is turning on its axis can be determined without looking at the stars, by means of the so-called Foucault pendulum, for example. Therefore it is not true that"all is relative" it is only uniform velocity that cannot be detected without looking outside. Uniform rotation about a fixed axis can be. when this is told to a philosopher he is very upset that he did not really understand it, because to him it seems impossible that one should be able to determine rotation about an axis without looking outside the philosopher is good enough, after some time he may come back and say, I understand. We really do not have such a thing as absolute rotation; we are really rotating relative to the stars, you see. And so some influence exerted by the stars on the object must cause the centrifugal force. Now, for all we know, that is true; we have no way, at the present time, of telling whether there would have been centrifugal force if there were no stars and nebulae around. We have not been able to do the experiment of removing all the nebulae and then dmit that the philosopher may be right. He comes back, therefore, in delight and says,"It is absolutely necessary that the world ultimately turn out to be this way absolute rotation means nothing; it is only relative to the nebulae. Then we say line, relative to the nebulae should produce no effects inside a car?", ty in a straight him, Now, my friend, is it or is it not obvious that uniform velocit Now that the motion is no longer absolute, but is a motion relative to the nebulae, it becomes a mysterious question, and a question that can be answered only by experiment. What, then, are the philosophic influences of the theory of relativity? If we limit ourselves to influences in the sense of what kind of new ideas and suggestions are made to the physicist by the principle of relativity, we could describe some of them as follows. The first discovery is, essentially, that even those ideas which have been held for a very long time and which have been very accurately verified might be wrong. It was a shocking discovery, of course, that Newton's laws are
wrong, after all the years in which they seemed to be accurate. Of course it is clear, not that the experiments were wrong, but that they were done over only a limited range of velocities, so small that the relativistic effects would not have been evident. But nevertheless, we now have a much more humble point of view of our physical laws-everything can be wrong Whe secondly, if we have a set of"strange"ideas, such as that time goes slower n one moves, and so forth, whether we like them or do not like them is an relevant question. The only relevant question is whether the ideas are consistent with what is found experimentally. In other words, the"strange ideas "need only agree with experiment, and the only reason that we have to discuss the be havior of clocks and so forth is to demonstrate that although the notion of the time dilation is strange, it is consistent with the way we measure time. Finally, there is a third suggestion which is a little more technical but which has turned out to be of enormous utility in our study of other physical laws, and that is to look at the symmetry of the laws or, more specifically, to look for the ways in which the laws can be transformed and leave their form the same. When we discussed the theory of vectors, we noted that the fundamental laws of motion are not changed when we rotate the coordinate system, and now we learn that they are not changed when we change the space and time variables in a particular way, given by the Lorentz transformation. So this idea of studying the patterns or operations under which the fundamental laws are not changed has proved to be a eful one 16-2 The twin paradox To continue our discussion of the lorentz transformation and relativistic effects, we consider a famous so-called"paradox "of Peter and Paul, who are supposed to be twins, born at the same time. When they are old enough to drive a space ship, Paul fies away at very high speed. Because Peter, who is left on the ground, sees Paul going so fast, all of Paul's clocks appear to go slower, his heart beats go slower, his thoughts go slower, everything goes slower, from Peters point of view. Of course, Paul notices nothing unusual, but if he travels around and about for a while and then comes back, he will be younger than Peter, the man on the ground! That is actually right; it is one of the consequences of the theory of relativity which has been clearly demonstrated. Just as the mu-mesons last longer when they are moving, so also will Paul last longer when he is moving. This is called a"paradox " only by the people who believe that the principle of relativity means that all motion is relative; they say, "Heh, heh, heh, from the point of view of Paul, can,t we say that Peter was moving and should therefore appear to age more slowly? By symmetry, the only possible result is that both should be the same age when they meet. But in order for them to come back together and make the comparison, Paul must either stop at the end of the trip and make a comparison of clocks or, more simply, he has to come back, and the one who comes back must be the man who was moving, and he knows this, because he had to turn tround. When he turned around, all kinds of unusual things happened in his space ship-the rockets went off, things jammed up against one wall, and so on- while Peter felt nothin So the way to state the rule is to say that the man who has felt the accelerations, pho has seen things fall against the walls, and so on, is the one who would be the younger; that is the difference between them in an"absolute "sense, and it is certainly correct. When we discussed the fact that moving mu-mesons live longer. example their straight-line motion in the ut we also make mu-mesons in a laboratory and cause them to go in a curve with a magnet, and even under this accelerated motion, they last exactly as much longer as they do when they are moving in a straight line. Although no one has arranged an experiment explicitly so that we can get rid of the paradox, one could compare a mu-meson which is left standing with one that had gone around a complete rcle, and it would surely be found that the one that went around the circle lasted longer. Although we have not actually carried out an experiment using a complete
circle, it is really not necessary, of course, because everything fits together all right This may not satisfy those who insist that every single fact be demonstrated directly but we confidently predict the result of the experiment in which Paul goes in a complete circle. 16-3 Transformation of velocities The main difference between the relativity of Einstein and the relativity of Newton is that the laws of transformation connecting the coordinates and times between relatively moving systems are different. The correct transformation law that of lorentz. is (16.1) / c2 These equations correspond to the relatively simple case in which the relative motion of the two observers is along their common x-axes. Of course other direc- tions of motion are possible, but the most general Lorentz transformation is rather complicated, with all four quantities mixed up together. We shall continue to use this simpler form, since it contains all the essential features of relativity. Let us now discuss more of the consequences of this transformation. First, it is interesting to solve these equations in reverse. That is, here is a set of linear quations, four equations with four unknowns, and they can be solved in reverse, for x, y, z, I in terms of x', y, z, t,. The result is very interesting, since it tells us how a system of coordinates"at rest"looks from the point of view of one that is moving. " Of course, since the motions are relative and of uniform velocity, the man who is "moving"can say, if he wishes, that it is really the other fellow who is moving and he himself who is at rest. And since he is moving in the opposite direction, he should get the same transformation, but with the opposite sign of velocity. That is precisely what we find by manipulation, so that is consistent If it did not come out that way, we would have real cause to worry (16.2) t= rt ux/c2 tivity. We recall that one of the original puzzles was that light travels at 186,000 mi/sec in all systems, even when they are in relative motion. This is a special case of the more general problem exemplified by the following. Suppose that an object inside a space ship is going at 100,000 mi /sec and the space ship itself is going at 100,000 mi/sec; how fast is the object inside the space ship moving from the point of view of an observer outside? We might want to say 200, 000 mi/sec, which is faster than the speed of light. This is very unnerving, because it is not supposed to be going faster than the speed of light! The general problem is as follows Let us suppose that the object inside the ship from the point of view of the man inside, is moving with velocity v, and that the space ship itself has a velocity u with respect to the ground. We want to know with what velocity vr this object is moving from the point of view of the man on the ground. This is, of course, still Thich the mo in the x-directid
transformation for velocities in the y-dir or for any angle; these can be orked out as needed. Inside velocity is vr, which means that the displacement x is equal to the velocity times the time Now we have only to calculate what the position and time are from the point of view of the outside observer for an object which has the relation(16.2)between x'and r,. So we simply substitute (16.)into(16. 2), and obtain x=√1-v2/C2 (164) But here we find x expressed in terms of r. In order to get the velocity as seen by side, we must divide his distance by his time, not by the other man's time! So we must also calculate the time as seen from the outside, which is (16.5) Now we must find the ratio of x to t, which is +Ur (166) the square roots having cancelled. This is the law that we seek: the resultant ve- locity, the"summing "of two velocities, is not just the algebraic sum of two veloc- ities(we know that it cannot be or we get in trouble), but is"corrected"by Now let us see what happens. Suppose that you are moving inside the space ship at half the speed of light, and that the space ship itself is going at half the speed light. Thus u is yc and v is 2c, but in the denominator uv is one-fourth, so that So, in relativity,"half"and“half" does not make“one," it makes only“4/5.” Of course low velocities can be added quite easily in the familiar way, because so long as the velocities are small compared with the speed of light we can forget about the(1 uu/c2)factor; but things are quite diferent and quite interesting at high velocity. et us take a limiting case. Just for fun, suppose that inside the space ship the man was observing light itself. In other words, v c, and yet the space ship is moving. How will it look to the man on the ground? The answer will be 1+ uc/c Therefore, if ething is moving at the speed of light inside the ship. to be moving at the speed of light from the point of view of the man too! This is good, for it is, in fact, what the Einstein theory of designed to do in the first place-so it had better work Of course. there are cases in which the motion is not in the direction of the uniform translation. For example, there may be an object inside the ship which is just moving"upward"with the velocity v,, with respect to the ship, and the ship is moving"horizontally "Now, we simply go through the same thing, only using ys instead of x's, with the result so that if vr,= 0 yy= y= wyr, (167)