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Feynman Physics Lectures V1 Ch17 Space-Time

Feynman Physics Lectures V1 Ch17 Space-Time - 17 Space Time...

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Unformatted text preview: 17 Space- Time 17—1 The geometry of space-time The theory of relativity shows us that the relationships of positions and times as measured in one coordinate system and another are not what we would have expected on the basis of our intuitive ideas. It is very important that we thoroughly understand the relations of space and time implied by the Lorentz transformation, and therefore we shall consider this matter more deeply in this chapter. The Lorentz transformation between the positions and times (x, y, z, t) as measured by an observer “standing still,” and the corresponding coordinates and time (x’, y’, z’, 1’) measured inside a “moving” space ship, moving with velocity u are , _ x — ut x/l — u2/c2 ’ y, 2, k: H (17.1) t —— ux/c2 — V1 — u2/c2. Let us compare these equations with Eq. (11.5), which also relates measurements in two systems, one of which in this instance is rotated relative to the other: “a | I x = xcosa + ysin6, I y = ycos 6 — xsin 6, (17.2) 2’ = z. In this particular case, Moe and Joe are measuring with axes having an angle 9 between the x’- and x-axes. In each case, we note that the “primed” quantities are “mixtures” of the “unprimed” ones: the new x’ is a mixture of x and y, and the new y’ is also a mixture of x and y. An analogy is useful: When we look at an object, there is an obvious thing we might call the “apparent width,” and another we might call the “depth.” But the two ideas, width and depth, are not fundamental properties of the object, because if we step aside and look at the same thing from a different angle, we get a different width and a different depth, and we may develop some formulas for computing the new ones from the old ones and the angles involved. Equations (17.2) are these formulas. One might say that a given depth is a kind of “mixture” of all depth and all width. If it were impossible ever to move, and we always saw a given object from the same position, then this whole business would be irrelevant—we would always see the “true” width and the “true” depth, and they would appear to have quite different qualities, because one appears as a subtended optical angle and the other involves some focusing of the eyes or even intuition; they would seem to be very different things and would never get mixed up. It is because we can walk around that we realize that depth and width are, somehow or other, just two differ- ent aspects of the same thing. Can we not look at the Lorentz transformations in the same way? Here also we have a mixture—of positions and the time. A difference between a space measure- ment and a time measurement produces a new space measurement. In other words, in the space measurements of one man there is mixed in a little bit of the time, as seen by the other. Our analogy permits us to generate this idea: The “reality” of 17—1 17-1 The geometry of space-time 17—2 Space-time intervals 17—3 Past, present, and future 17—4 More about four-vectors 17—5 Four-vector algebra Ci SLOW x, x Fig. 17—1. Three particle paths in space-time: (a) a particle at rest at x = x0; (b) a particle which starts at x = x0 and moves with constant speed; (c) a particle which starts at high speed but slows down. (0) NOT CORRECT (b) CORRECT Fig. 17—2. Two views of a disinfe- grating particle. an object that we are looking at is somehow greater (speaking crudely and intui- tively) than its “width” and its “depth” because they depend upon how we look at it; when we move to a new position, our brain immediately recalculates the width and the depth. But our brain does not immediately recalculate coordinates and time when we move at high speed, because we have had no effective experience of going nearly as fast as light to appreciate the fact that time and space are also of the same nature. It is as though we were always stuck in the position of having to look at just the width of something, not being able to move our heads appreci- ably one way or the other; if we could, we understand now, we would see some of the other man’s time—we would see “behind,” so to speak, a little bit. Thus we shall try to think of objects in a new kind of world, of space and time mixed together, in the same sense that the objects in our ordinary space-world are real, and can be looked at from different directions. We shall then consider that objects occupying space and lasting for a certain length of time occupy a kind of a “blob” in a new kind of world, and that we look at this “blob” from different points of View when we are moving at different velocities. This new world, this geometrical entity in which the “blobs” exist by occupying position and taking up a certain amount of time, is called space-time. A given point (x, y, z, t) in space-time is called an event. Imagine, for example, that we plot the x-positions horizontally, y and z in two other directions, both mutually at “right angles” and at “right angles” to the paper (1), and time, vertically. Now, how does a moving particle, say, look on such a diagram? If the particle is standing still, then it has a certain x, and as time goes on, it has the same x, the same x, the same x; so its “path” is a line that runs parallel to the t-axis (Fig. 17—1 a). On the other hand, if it drifts outward, then as the time goes on x increases (Fig. 17—1 b). So a particle, for ex- ample, which starts to drift out and then slows up should have a motion something like that shown in Fig. 17—1(c). A patricle, in other words, which is permanent and does not disintegrate is represented by a line in space-time. A particle which disintegrates would be represented by a forked line, because it would turn into two other things which would start from that point. What about light? Light travels at the speed c, and that would be represented by a line having a certain fixed slope (Fig. 17—1 d). Now according to our new idea, if a given event occurs to a particle, say if it suddenly disintegrates at a certain space-time point into two new ones which follow some new tracks, and this interesting event occurred at a certain value of x and a certain value of t, then we would expect that, if this makes any sense, we just have to take a new pair of axes and turn them, and that will give us the new I and the new x in our new system, as shown in Fig. l7—2(a). But this is wrong, because Eq. (17.1) is not exactly the same mathematical transformation as Eq. (17.2). Note, for example, the difference in sign between the two, and the fact that one is written in terms of cos 0 and sin 6, while the other is written with algebraic quanti- ties. (Of course, it is not impossible that the algebraic quantities could be written as cosine and sine, but actually they cannot.) But still, the two expressions are very similar. As we shall see, it is not really possible to think of space-time as a real, ordinary geometry because of that difference in sign. In fact, although we shall not emphasize this point, it turns out that a man who is moving has to use a set of axes which are inclined equally to the light ray, using a special kind of projection parallel to the x’- and t’-axes, for his x’ and t’, as shown in Fig. 17—2(b). We shall not deal with the geometry, since it does not help much; it is easier to work with the equations. 17-2 Space-time intervals Although the geometry of space-time is not Euclidean in the ordinary sense, there is a geometry which is very similar, but peculiar in certain respects. If this idea of geometry is right, there ought to be some functions of coordinates and time which are independent of the coordinate system. For example, under ordinary rotations, if we take two points, one at the origin, for simplicity, and the other one somewhere else, both systems would have the same origin, and the distance from 1 7—2 here to the other point is the same in both. That is one property that is inde- pendent of the particular way of measuring it. The square of the distance is x2 + y2 + 22. Now what about space-time? It is not hard to demonstrate that we have here, also, something which stays the same, namely, the combination czt2 — x2 — y2 — z2 is the same before and after the transformation: c2tr2 _ 12 12 x — y — z’2 = cztz — x2 — y2 — 22. (17.3) This quantity is therefore something which, like the distance, is “real” in some sense; it is called the interval between the two space-time points, one of which is, in this case, at the origin. (Actually, of course, it is the interval squared, just as x2 + y2 + 22 is the distance squared.) We give it a different name because it is in a different geometry, but the interesting thing is only that some signs are reversed and there is a c in it. Let us get rid of the c; that is an absurdity if we are going to have a wonderful space with x’s and y’s that can be interchanged. One of the confusions that could be caused by someone with no experience would be to measure widths, .say, by the angle subtended at the eye, and measure depth in a different way, like the strain on the muscles needed to focus them, so that the depths would be measured in feet and the widths in meters. Then one would get an enormously complicated mess of equations in making transformations such as (17.2), and would not be able to see the clarity and simplicity of the thing for a very simple technical reason, that the same thing is being measured in two dilTerent units. Now in Eqs. (17.1) and (17.3) nature is telling us that time and space are equivalent; time becomes space; they should be measured in the same units. What distance is a “second”? It is easy to figure out from (17.3) what it is. It is 3 X 108 meters, the distance that light would go in one second. In other words, if we were to measure all distances and times in the same units, seconds, then our unit of distance would be 3 X 108 meters, and the equations would be simpler. Or another way that we could make the units equal is to measure time in meters. What is a meter of time? A meter of time is the time it takes for light to go one meter, and is therefore 1/3 X 10‘8 sec, or 3.3 billionths of a second! We would like, in other words, to put all our equations in a system of units in which c = 1. If time and space are measured in the same units, as suggested, then the equations are obviously much simplified. They are x’ = x — ut , m y’ = y, z, = 2’ (17.4) ,I = £19.. V1 — u2 t'2 _ x’2 __ y’2 _ 2,2 ___ t2 _ x2 _ yz _ 22' (175) If we are ever unsure or “frightened” that after we have this system with c = l we shall never be able to get our equations right again, the answer is quite the opposite. It is much easier to remember them without the c’s in them, and it is always easy to put the c’s back, by looking after the dimensions. For instance, in V1 — u2, we know that we cannot subtract a velocity squared, which has units, from the pure number 1, so we know that we must divide u2 by c2 in order to make that unitless, and that is the way it goes. The difl‘erence between space-time and ordinary space, and the character of an interval as related to the distance, is very interesting. According to formula (17.5), if we consider a point which in a given coordinate system had zero time, and only space, then the interval squared would be negative and we would have an imaginary interval, the square root of a negative number. Intervals can be either real or imaginary in the theory. The square of an interval may be either positive or negative, unlike distance, which has a positive square. When an interval is imaginary, we say that the two points have a space-like interval between them 17—3 LIGHT - CONE LIGHT-CONE Fig. 17—3. The space-time surrounding a point at the origin. reg ion (instead of imaginary), because the interval is more like space than like time. On the other hand, if two objects are at the same place in a given coordinate system, but differ only in time, then the square of the time is positive and the distances are zero and the interval squared is positive; this is called a time-like interval. In our diagram of space-time, therefore, we would have a representation something like this: at 45° there are two lines (actually, in four dimensions these will be “cones,” called light cones) and points on these lines are all at zero interval from the origin. Where light goes from a given point is always separated from it by a zero interval, as we see from Eq. (17.5). Incidentally, we have just proved that if light travels with speed c in one system, it travels with speed c in another, for if the interval is the same in both systems, i.e., zero in one and zero in the other, then to state that the propagation speed of light is invariant is the same as saying that the interval 18 zero. 17—3 Past, present, and future The space-time region surrounding a given space-time point can be separated into three regions, as shown in Fig. 17—3. In one region we have space-like inter- vals, and in two regions, time-like intervals. Physically, these three regions into which space-time around a given point is divided have an interesting physical relationship to that point: a physical pbject or a signal can get from a point in region 2 to the event 0 by moving along at a speed less than the speed of light. Therefore events in this region can affect the point 0, can have an influence on it from the past. In fact, of course, an object at P on the negative t-axis is precisely in the “past” with respect to 0; it is the same space-point as 0, only earlier. What happened there then, affects 0 now. (Unfortunately, that is the way life is.) An- other object at Q can get to 0 by moving with a certain speed less than c, so if this object were in a space ship and moving, it would be, again, the past of the same space-point. That is, in another coordinate system, the axis of time might go through both 0 and Q. So all points of region 2 are in the “past” of 0, and any- thing that happens in this region can affect 0. Therefore region 2 is sometimes called the aflective past, or affecting past; it is the locus of all events which can affect point 0 in any way. Region 3, on the other hand, is a region which we can affect from 0, we can “hit” things by shooting “bullets” out at speeds less than c. So this is the world whose future can be affected by us, and we may call that the aflective future. Now the interesting thing about all the rest of space-time, i.e., region 1, is that we can neither affect it now from 0, nor can it affect us now at 0, because nothing can go faster than the speed of light. Of course, what happens at R can affect us later; that is, if the sun is exploding “right now,” it takes eight minutes before we know about it, and it cannot possibly affect us before then. What we mean by “right now” is a mysterious thing which we cannot define and we cannot affect, but it can affect us later, or we could have affected it if we had done something far enough in the past. When we look at the star Alpha Centauri, we see it as it was four years ago; we might wonder what it is like “now.” “Now” means at the same time from our special coordinate system. We can only see Alpha Centauri by the light that has come from our past, up to four years ago, but we do not know what it is doing “now”; it will take four years before what it is doing “now” can affect us. Alpha Centauri “now” is an idea or concept of our mind; it is not something that is really definable physically at the moment, because we have to wait to observe it; we cannot even define it right “now.” Furthermore, the “now” depends on the coordinate system. If, for example, Alpha Centauri were moving, an observer there would not agree with us because he would put his axes at an angle, and his “now” would be a diflerent time. We have already talked about the fact that simultaneity is not a unique thing. There are fortune tellers, or people who tell us they can know the future, and there are many wonderful stories about the man who suddenly discovers that he has knowledge about the affective future. Well, there are lots of paradoxes pro- duced by that because if we know something is going to happen, then we can make 17—4 sure we will avoid it \by doing the right thing at the right time, and so on. But actually there is no fortune teller who can even tell us the present! There is no one who can tell us what is really happening right now, at any reasonable distance, because that is unobseryable. We might ask ourselves this question, which we leave to the student to try to answer: Would any paradox be produced if it were suddenly to become possible to know things that are in the space-like intervals of region 1? 17—4 More about four-vectors Let us now return to our consideration of the analogy of the Lorentz trans- formation and rotations of the space axes. We have learned the utility of collecting together other quantities which have the same transformation properties as the coordinates, to form what we call vectors, directed lines. In the case of ordinary rotations, there are many quantities that transform the same way as x, y, andz under rotation: for example, the velocity has three components, an x, y, and z—component; when seen in a "different coordinate system, none of the components is the same, instead they are all transformed to new values. But, somehow or other, the velocity “itself” has a greater reality than do any of its particular components, and we represent it by a directed line. We therefore ask: Is it or is it not true that there are quantities which transform, or which are related, in a moving system and in a nonmoving system, in the same way as x, y, z, and t? From our experience with vectors, we know that three of the quantities, like x, y, 2, would constitute the three components of an ordinary space-vector, but the fourth quantity would look like an ordinary scalar under space rotation, because it does not change so long as we do not go into a moving coordinate system. Is it possible, then, to associate with some of our known “three-vectors” a fourth object, that we could call the “time component,” in such a manner that the four objects together would “rotate” the same way as position and time in space-time? We shall now show that there is, indeed, at least one such thing (there are many of them, in fact): the three components of momentum, and the energy as the time component, transform together to make what we call a “four— vector.” In demonstrating this, since it is quite inconvenient to have to write c’s everywhere, we shall use the same trick concerning units of the energy, the mass, and the momentum, that we used in Eq. (17.4). Energy and mass, for example: differ only by a factor c2 which is merely a question of units, so we can say energy is the mass. Instead of having to write the c2, we put E = m, and then, of course, if there were any trouble we would put in the right amounts of c so that the units would straighten out in the last equation, but not in the intermediate ones. Thus our equations for energy and momentum are E=m=m/\/l—-1)2, W (17.6) mv = mov/x/l - 122. P Also in these units, we have E2 — p2 = mg. (17.7) For example, if we measure energy in electron volts, what does a mass of l electron volt mean? It means the mass whose rest energy is 1 electron volt, that is, moc2 is one electron volt. For example, the rest mass of an electron is 0.511 X 106 ev. Now what would the momentum and energy look like in a new coordinate system? To find out, we shall have to transform Eq. (17.6), which we can do because we know how the velocity transforms. Suppose that, as we measure it, an object has a velocity 2), but we look upon the same object from the point of view of a space ship which itself is moving with a velocity u, and in that system we use a prime to designate the corresponding thing. In order to simplify things at first, we shall take the case that the velocity v is in the direction of u. (Later, we can do the more general case.) What is v’, the velocity as seen from the space ship? It is the 1 7—5...
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