General Relativity from A to B 2/13/81 Edition

By Robert Geroch

The University of Chicago Press

Copyright © 1978 The University of Chicago
All rights reserved.
ISBN: 978-0-226-28864-2

Contents

Preface,
Introduction,
A. THE SPACE-TIME VIEWPOINT,
1. Events and Space-Time: The Basic Building Blocks,
2. The Aristotelian View: A "Personalized" Framework,
3. The Galilean View: A Democratic Framework,
4. Difficulties with the Galilean View,
B. GENERAL RELATIVITY,
5. The Interval: The Fundamental Geometrical Object,
6. The Physics and Geometry of the Interval,
7. Einstein's Equation: The Final Theory,
8. An Example: Black Holes,
Conclusion,
Index,

CHAPTER 1

Events and Space-Time: The Basic Building Blocks

The notion of an event is the basic building block of the theory. It will dominate all that follows.

By an event we mean an idealized occurrence in the physical world having extension in neither space nor time. For example, "the explosion of a firecracker" or "the snapping of one's fingers" would represent an event. (By contrast, "a particle" would not represent an event, for it has "extension in time"; "a long piece of rope" has "extension in space.") By "occurrence in the physical world" we mean that an event is to be regarded as a part of the world in which we live, not as a construct in some theory. Of course, there are many events around: some occurred long ago, some are occurring now, and others will (presumably) occur in the future. What is meant by "idealized ... having extension in neither space nor time" requires more explanation. Consider the explosion of a firecracker. The explosion lasts for some finite time (say, one-tenth of a second), and so this occurrence has extension in time; the explosion takes place over some finite region of space (say, one-quarter of an inch), so it has extension in space. If, however, we used a smaller and faster-burning firecracker, these "extensions" would be smaller. An event is to be an idealization of this situation in the limit of a "very small, very fast-burning" firecracker. (The situation is similar to that which would arise from the statement: "A point on the blackboard is an idealized chalk mark having extension neither up-down nor right-left." This analogy goes a little deeper: events will shortly become "points" of an appropriate space.)

We regard two events as being "the same" if they coincide, that is, if they "occur at the same place at the same time." That is to say, we are not now concerned with how an event is marked — by firecracker or finger-snap — but only with the thing itself.

Is one to regard events as "existing" even if there is nobody there to mark them with finger-snap or otherwise (for example, in a dark, empty closet at 3 A.M.)? It is part of what we wish to mean by an event that the answer is to be yes. Perhaps it would have been better to say originally "An event is an idealized potential occurrence...." As a general rule, failure in physics to attribute "existence" to things not directly perceived leads to various difficulties of the "If a tree falls in the forest and nobody is there to hear it, does it still make a sound?" variety. Failure to do so in the present case would, as far as I can see, make further development of the theory virtually impossible. This is not to say that such questions are uninteresting or unimportant. Rather, it has become the custom in physics to relegate them to others by the practice of being liberal in bestowing "existence."

Are events real? What are they really like? These questions are dealt with (more accurately, avoided) by means of another custom. Physics does not, at least in my opinion, deal with what is "real" or with what something is "really like." The reason, I suppose, is some combination of (1) One does not know how to effectively attack such questions. (2) One does not know what sort of thing would represent an answer. (3) These questions are too hard. In any case, one conventionally deals with relationships between things which one does not (or perhaps cannot) understand on a deeper level. One does, of course, sometimes come to understand some basic concept more deeply. (For example, space and time were basic concepts in Newtonian gravitation. With general relativity, one does feel a sense of deeper understanding.) Perhaps it is true to say that one has found from experience that deeper insight into the basic concepts of a theory comes most often, not from a frontal attack on those concepts, but rather from working upward into the theory itself.

Relationships between events — that is what we are after. Virtually everything we say hereafter can be resolved, directly or indirectly, into some statement of such a relationship.

We wish to discover the "correct" theory of the relationship between events. It is instructive to arrive at the final theory indirectly, through a sequence of preliminary attempts. We begin then with the rather naive view of everyday experience, a view which will subsequently be found to be inappropriate.

According to the Aristotelian view, an event is naturally characterized by giving its position in space together with the time of its occurrence.

We can make this view more explicit. Let there be set up, within a room, a Cartesian coordinate system x, y, z. That is to say, each position in space is to be described by three real numbers: the value of x, the value of y, and the value of z. For example, the "value of x" might be the distance of that position from one side wall, the "value of y" the distance from the front wall, and the "value of z" the distance from the floor. Our coordinate system permits, then, "numerical location of positions." The position described by x = 12, y = 3, z = 9 is that located 12 feet from the side wall, 3 feet from the front wall, and 9 feet from the floor. Now let the room be filled solidly — wall to wall, floor to ceiling — with people. Each person always maintains his same position within the room. Each person can describe his fixed position, then, by giving the appropriate values of x, y, and z. Let those values be printed on a small badge which each person wears. Next let there be distributed, to each of our subjects, an accurate watch. These watches are all synchronized (for example, by having another person communicate with each person and compare his watch with theirs).

Imagine, then, the arrangement sketched above. We use this arrangement to characterize events as follows. Let some event be chosen, marked, say, by the explosion of a firecracker. Since our subjects are packed solidly, one of them will be in the immediate vicinity of the explosion. Let that person write on a slip of paper the three numbers (x, y, and z values) which appear on his badge, and also a fourth number, the time, according to his watch, at which the explosion was experienced. This slip of paper is then passed forward to a moderator desirous of knowing our characterization of this particular event.

Of what does our characterization consist? Of four numbers, the values of x, y, z, and t (time). The first three numbers give the "position of the event in space"; the fourth gives the "time of its occurrence." We are here characterizing events, then, according to the Aristotelian view.

Why did we go on and on, taking the trouble to be so explicit and so careful about such a simple idea? There are several reasons. The characterization of physical phenomena (such as events) is supposed to be grounded on a more or less explicit set of instructions for actually carrying out the characterization experimentally in the physical world. Normally, it is pretty clear what is to be done, and the instructions need not be given in great detail. Here, however, our concern is the structure of space and time itself, and care in saying exactly what we mean is not an empty exercise. A more important reason is that, as we shall see later, implicit in the construction above are certain assumptions about the way space and time operate. These various assumptions, it will turn out, are simply not true in our world. It is convenient, therefore, to have the present characterization in sufficient detail that we can later pick out these assumptions.

We claimed earlier to be concerned with relationships between events, not "views." What relationships, then, are implied by the Aristotelian view? Let there be given two fixed events. We ask whether, in the Aristotelian view, each of the following makes sense.

Do the two events have the same position in space? Since an event is here characterized by giving its position in space together with the time of its occurrence, this question does make sense. In terms of the explicit formulation, let the first event be characterized by values x0y0z0t0 and the second event by x0', y0', z0', t0'. We may say that these two events have the "same position" provided x0 = x0', y0 = y0', and z0 = z0'.

Do the two events occur at the same time? Again the question makes sense — explicitly, we may say that the two events "occur at the same time" provided t0 = t0'. (Clearly, two events are the same provided they have both the same position and the same time.)

What is the distance between the two events? This question also makes sense. Since an event is characterized by, among other things, its position in space, we can simply compute the "distance in space between those two positions." Explicitly, the computation would be as follows. We first compute x0 - x0', the difference between the x values of the two events; then y0 - y0', the difference between the y values; then z0 - z0'. The distance between the two events would then be given, according to the Pythagorean theorem, by the following equation: (distance)2 = (x0 - x0')2 + (y0 - y0')2 + (z0 - z0')2.

What is the elapsed time between the two events? This question makes sense. Explicitly, the elapsed time would be given by the following equation: (elapsed time) = t0 - t0'.

It is not shocking that these questions all make sense, for we are used to addressing them in everyday life. "We are now in the exact position where the Titanic sank." "It is now just six weeks since Carter was elected." It is in this sense, then, that the Aristotelian view is the popular one.

Finally, we can give a few additional examples of everyday notions which, within the Aristotelian view, make sense. "Is this particle at rest?" is a sensible question, for we can answer it by finding the position of the particle at various successive times. If that position is always the same, no matter what the time, we may say that the particle is at rest. "What distance did this particle travel between one time and some time later?" is sensible. At each instant of time, we consider the event "the particle at that instant," and associate with that event, by the Aristotelian view, a position in space. Computing the distance between these successive positions in space, and taking their sum, we obtain the total distance traveled by the particle. "What is the speed of this particle?" Since we attach meaning to the distance traveled by a particle, we also attach meaning to its speed, the number computed by dividing distance traveled by elapsed time.

Not a single one of all the notions above will make sense, in their present generality, in relativity theory.

Here and hereafter, we shall denote by M the set of all possible events in our universe: all those events that have occurred in the past, all those occurring now, and all that will occur in the future; those in this room, in our solar system, in other galaxies. This one enormous set M will be called space-time.

A point of M, then, represents an event. A region of M, on the other hand, represents some collection of events, for example, the collection "all events which occurred within this room between 10:30 and 11:30 on 8 January 1976." As an illustration, we will now describe in terms of space-time the idealization involved in the original description of an event. Let a firecracker explode, and consider the collection of all events internal to the explosion itself. They would correspond to some region, as shown in figure 1. If, instead, a smaller, faster-burning fire cracker had been used, the corresponding region would be smaller in space-time. The idealization, then, involves the "collapse of these regions down to a single point of space-time."

Space-time would not be very interesting if it served merely as a repository for events. Its interest stems, rather, from the fact that many other, considerably more complicated things in the world can also be described within its framework. We shall see many examples of this later; we will give just one here. Let there be one particle, which we wish to describe in terms of space-time. Now a particle could not be described by a single point of M (that is, by a single event), for a particle has "extension in time." The appropriate description is in terms of a certain collection of events, namely the following. Consider the collection of all events which occur in the immediate presence of the particle (that is, for events marked by a firecracker, those for which the particle is internal to the explosion). This is the set of events which would be described if one continually followed the particle around throughout its life, snapping one's fingers on it. The resulting collection of events would be described by a line drawn in space-time (the line so drawn that it passes through precisely the events described above). This single line, called the world-line of the particle, completely describes everything one could want to know about the particle, for it tells us all the events experienced by the particle, that is, "where the particle is at all times." A particle, then, is not a "point" from the viewpoint of space-time — it is a line.

As an extension of this example, let us now consider two particles, A and B. Each particle is represented in terms of space-time by its world-line. Suppose that these two lines happened to intersect at some point p, as shown in figure 2. How is this to be interpreted physically? Well p, as a point of space-time, represents an event. The essential feature of p is that it lies on both world-lines. This means that the event represented by p is directly experienced by both particle A and particle B. In other words, both particles were there at that event. This is what we would call a physical collision of two particles. Intersection of world-lines thus corresponds to collision. If the world-lines of two particles do not intersect, the particles never collided.

We are trying to discover a theory of relationships between events. What role does space-time play in this endeavor? Since the events are represented as points of M, relationships between events are relationships between the various points of M. A set of such relationships is a kind of internal structure imposed on M. Our goal, then, is to find what structure we can on space-time M.

CHAPTER 2

The Aristotelian View: A "Personalized" Framework

In chapter 1, we introduced, on the one hand, some elements of the Aristotelian view, and, on the other, space-time. The former is a particular attitude about what natural relationships there are between events, the latter is a more or less attitude-independent assemblage of events. Our first goal in the present chapter is to combine these two. That is, we wish to express the relationships between events according to the Aristotelian view as structure within space-time. We then give, using this "Aristotelianized space-time," several examples of how one translates back and forth between geometrical constructs in space-time and various goings-on in the physical world.

What is it that the Aristotelian view provides? It provides a characterization of each event by four real numbers, the values of x, y, z, and t. What is space-time? It is the collection of all possible events. Thus, the incorporation of the Aristotelian view into space-time is just the introduction, in space-time, of a certain coordinate system. (A coordinate system in the plane permits us to associate, with each point of the plane, two real numbers, the value of x and the value of y. Here, our coordinate system in space-time M permits us to associate, with each point of M [event], four real numbers, the values of x, y, z, and t.)

When we say "The plane is two-dimensional" we mean essentially that, to locate a point of the plane requires the specification of two real numbers (the values of x and y). Similarly for "Physical space is three-dimensional." By the same token, then, we are to regard space-time as four-dimensional, for the location of a point in space-time (event) now requires the specification of four numbers. One remark should be made in connection with this four-dimensionality. The view has for some reason come to be widely held that "the fourth dimension" is a deep and mysterious thing which permits extraordinary happenings in the world, and which only a few people can really understand. We emphasize that this is just not true. We now already "have four dimensions." On the other hand, we have not yet introduced a single statement about the way the physical world operates that was not known to all of us since childhood. True, we have perhaps been more careful and precise in our discussion than we might have been previously, yet the fact remains that, with no additional contributions whatever to our basic fund of physical information, we have arrived at a description in terms of four dimensions. If you like, "four dimensions" is just a convenient way of describing the world and thinking about the world, nothing more. Is the "fourth dimension" real? It should now be clear, from these remarks and from the discussion of "reality" in chapter 1, that physics will not answer such a question, and that the attitude of physicists will be that such a question is not germane. There is the physical world, and then there is our description of it. As long as our description is reasonably clear and reasonably accurate, there will be no objections. We can change our description every Friday morning if we wish. Nature doesn't care about our descriptions; She just keeps rolling along. If these days we choose to describe Her in terms of a space-time, and if that space-time has four dimensions, then, as long as that description is reasonably clear and reasonably accurate, that's fine and that's the end of it. Tomorrow's description may have two dimensions or nineteen dimensions. All of us, I can assure you, now understand "the fourth dimension" as well as anybody.

(Continues...)Excerpted from General Relativity from A to B 2/13/81 Edition by Robert Geroch. Copyright © 1978 The University of Chicago. Excerpted by permission of The University of Chicago Press.
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