General Relativity from A to B

Robert Geroch

• Editorial: University of Chicago Press
• Año: 2013
• ISBN:
  9780226190792
• Categoría: Biological Sciences
• Categoría: Science General

"This beautiful little book is certainly suitable for anyone who has had an introductory course in physics and even for some who have not."—Joshua N. Goldberg, Physics Today "An imaginative and convincing new presentation of Einstein's theory of general relativity.... The treatment is masterful, continual emphasis being placed on careful discussion and motivation, with the aim of showing how physicists think and develop their ideas."— Choice

B

General Relativity

5

The Interval: The Fundamental Geometrical Object

We decided (or at least asserted) in chapter 4 that the Galilean view needs modification. We will now set down the basic ground rules.

We have already gotten into difficulties once with the Aristotelian view. We rushed in too quickly, allowing ourselves to use any sort of instruments or ideas with abandon; we imposed structure right and left, finding out only later (with the Galilean view) that some of it wasn’t “intrinsic to space-time.” It is tempting to try to avoid making the same mistakes again by adopting an ultracautious attitude. We shall be highly skeptical, not accepting anything until it is soundly grounded in many observations; we shall proceed very slowly and very carefully in very small steps. Such an attitude (unfortunately, if you wish) seems never to get one anywhere. Had it been adopted at the beginning of this century, we would probably not have relativity today. What is needed, experience has shown, is the right mix of caution and daring. One must at times be rash, accepting (perhaps temporarily) ideas with very little observational basis; one must at other times be ultracautious, examining “obvious” notions with care. The art (and it is an art) consists of making judicious choices of what is to be in the first category and what in the second. It is these “judicious choices,” then, that we must now make.

I cannot hope to make what ultimately turned out to be the “right” choices seem natural and reasonable. There is, to put it dramatically, an imaginative leap grounded in judgment and experience and little else. The choice, however, is this: We shall retain the broad, qualitative features of space-time. That is to say, we shall retain the notion of an event, and the idea of assembling those events into space-time. Points still represent events, world-lines still represent particles, two-dimensional surfaces still represent ropes, and so on. Intersections of world-lines still represent collisions of the particles, a world-line meeting the world-region of the earth still represents the meeting of the particle and the earth, a world-line on a two-dimensional surface still represents a bead sliding on a wire. All else, for the moment at least, we discard. We don’t have clocks, we don’t have “spatial distance between two events,” we don’t have “events occurred at the same time,” we don’t have “speed of the particle,” and we certainly don’t have any Aristotelian setups. In short, we retain the general idea of space-time as the way to represent the world, and discard all the detailed, geometrical, numerical information. In the transition from the Aristotelian to the Galilean view, we switched from a rigid picture of space-time to a “sliding-pack-of-cards” picture. We are now going to the extreme in this direction. It is as though space-time were drawn on a rubber sheet, which can be stretched, pulled, and bent. All that we retain, all that we care about for the moment, are the broad, qualitative features of the various points, lines, surfaces, and so on drawn on the sheet. (True, any actual drawing of space-time will involve geometrical relationships that we do not wish to regard as meaningful at this point. It’s the same problem as that in the passage from Aristotelian to Galilean—and the solution is the same. One must just get used to viewing space-time diagrams as though they were printed on a rubber sheet, subject to being stretched and pulled without changing anything.)

The choices of the previous paragraph do not sound as revolutionary as they were because we have prejudiced the situation by our treatment, in the first four chapters, of prerelativistic space-time. Even so, the choices, I would like to claim, are a bit shocking. Originally, recall, events, space-time, and so on were introduced merely as an aid in picturing what we thought of as the “real geometrical structure” of space and time. These notions were just a convention we all agreed to adopt to communicate with each other. Now, however, these notions are brought to the fore, with the (previously) “more natural” ideas, such as spatial distances and elapsed times, suppressed.

Shall we just stop here and call it a “view”? It would certainly be a very simple view—one almost immune from any conflicts with observations. (I would be hard-pressed even to imagine any observation which wouldn’t fit into this broad framework.) Alas, we cannot. The problem is precisely that what we have so far is so immune from conflicts with observations. Indeed, it doesn’t assert much of anything about the world at all. We all have at least some intuitive notions about spatial distances, elapsed times, and so on, and these must, in some way or other, be brought into our view if it is to be other than empty. A theory of physics in general, and a world-view in particular, must make at least some commitment about the physical world if it is to be worth anything at all.

The conclusion, then, is that we must somehow work into this broad framework some hard, numerical, geometrical information about space-time. We need something to replace the old spatial distances, elapsed times, and so forth. Here is where we wish to be ultracautious. The problem (it is now known) is that we were entirely too cavalier in describing, for example, the original Aristotelian setup. We allowed ourselves “watches” without further discussion, and we implicitly assumed a number of properties of these “watches” (some of which, as we shall see shortly, are just not true in our world). We allowed ourselves “meter sticks” (used to measure the distances recorded on the badges) with implicitly assumed properties. We in effect allowed ourselves to “just know” about space-time around us without a careful physical prescription for what is to be done.

This time, we wish to be more careful, in at least the following senses. First, we shall not allow ourselves to use any old measuring instrument we happen to stumble across; rather, we shall restrict consideration to some minimum number. Second, we shall require that these instruments—whichever ones we select—shall first be properly represented within our space-time framework. We do not accept instruments that we don’t even understand well enough to draw in space-time. Third, we require a clear, explicit statement of those properties that our instruments (now represented in space-time) are to be assumed to have. We may, of course, suppose what properties we want and need, but we do require that such suppositions be explicit.

The paragraph above is of little help in deciding what instruments we should allow ourselves. For this, we must again try to make a careful judgment (translated, lucky guess). After mulling the matter over a bit, one might hit on the following idea. What sorts of instruments are actually needed to obtain geometrical information about space-time? One needs essentially two different types of instruments. One of these instruments must go out into space-time (that is, away from our world-line), collect or respond to what space-time is like, and carry that information back to us. Without such an instrument, we should forever be confined, in terms of our knowledge, to our own world-line. A second instrument would then also be needed to “record” or “make numerical” the information brought back. Without this second instrument we would only be able to sense space-time away from our world-line, not say anything concrete about it. This idea vastly simplifies our search for instruments, for we know not only the number (two), but also, in general terms, what the two instruments are to be doing for us. We now turn to the specific choice of instruments.

What shall we use for the instrument which reacts to space-time away from our world-line and brings information back? Let’s try a few possibilities. Consider first a meter stick. It’s a whole meter long, and so it certainly “sticks out into space-time away from our world-line.” There are, however, two unpleasant features about meter sticks. The first is that, in terms of space-time, a meter stick is a rather complicated object: It’s represented by a two-dimensional surface. (Of course, the markings on the meter stick would be lines ruled on this surface.) The second feature is perhaps even worse. How do we go about reading the marks on the meter stick? The world-line, say, of the mark “82 centimeters” on the meter stick will be out in space-time away from our own world-line (fig. 31). If we are going to invent some mechanism to tell us where that mark is, then we might as well use that mechanism to tell us about space-time. (It will not do, of course, to go over to the “82-centimeter” mark to see where it is, because if our world-line gets over there then we are there, and so we are directly reading things on our own world-line, without the need of any meter stick.) Meter sticks, in short, are not an entirely satisfactory way to determine what space-time is like. (Recall, now, their extensive use for the Aristotelian setup.) Consider, then, the possibility of using particles (say, thrown by the observer) to react to space-time. One could arrange for the particles to be bounced back to the observer to tell him what space-time is like away from his world-line (fig. 32). Particles, indeed, are much better than meter sticks for our purposes (and, in fact, they could perfectly well be used, albeit at the expense of somewhat more complicated constructions). The problem with particles is that they can have all sorts of different speeds when thrown out, which is just another complication. The solution, perhaps by now obvious, is to use light-pulses. Light w...

Índice

• Cover
• Copyright
• Title Page
• Contents
• Preface
• Introduction
• A. The Space-Time Viewpoint
• B. General Relativity
• Conclusion
• Index