Chapter 9

The Space Time Perspective

I. Introduction

So far we have sketched the development of our concepts of space and time from a historical perspective. Our aim has been to reconstruct the views of the ancient and cl

assical writers with as much fidelity to the original as is appropriate for our purposes. The net result has been that, despite a hint or two that our concepts of spatiality and temporality are closely intertwined, the theory of space and time seems to be an uneasy alliance between two treatments of separate but accidentally related features of nature. This point is brought home by Newton's theory of space and time which is the foundation of the classical view. For Newton, Absolute Space and Absolute Time each exist more or less independently of one another and it is, at best, a happy accident that they function as an integrative framework for human observers.

From a contemporary point of view, space and time are not conceived of as two distinct entities more or less arbitrarily juxtaposed together, but rather as one integrated space-time framework. Minkowski, a pioneer in relativity theory, said: "The views of space and time which I wish to lay before you... are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."

What we want to do is to develop a point of view which will exhibit the unitary nature of space-time and at the same time allow us to reconstruct the views of previous thinkers in terms of this framework. The resulting framework will then allow us to compare the modern theories of relativity with the earlier views. The aim is to exhibit the transitions from Aristotle to Newton to the Special Theory of Relativity to the General Theory of Relativity as a series of conceptually natural shifts.

Let us begin again from scratch and re-examine the early views from a modern perspective. One of the characteristics of modern science is that it rests, to an extent unparalleled by earlier theories, on experience and observation. The ideas of space and time, on the other hand, are fairly abstract and require a certain degree of theoretical sophistication in order to be grasped. One evidence for this is the failure of the mythopoeic thinkers (or of Aristotle) to develop a generalized notion of spatiality. Of course, other Greek thinkers did employ a general notion of spatiality, but it was different in key respects from our own. The point is that spatiality is not an obvious notion and there are no overriding empirical pressures to develop one concept of spatiality rather than another. The concepts of space and time developed by various thinkers at various times are, to a large extent, influenced by the theoretical concerns of the systems in which they are embedded. Despite this, if we go back and re-examine the observational basis for the views of space and time developed by mythopoeic man, Aristotle and Newton, the basis for a unifying picture emerges.

Mythopoeic thinkers did not develop, as far as we know, any general conceptions of spatiality and temporality. The observational basis of the mythopoeic views of space and time seems to have been emotively significant ritual events. The advent of spring and the rites of planting serve to mark off a significant moment. The autumn chill and the harvest moon mark off another significant moment, and so on. In the myths of eternal recurrence, the implicit standard of sameness of place and sameness of time is sameness of ritual significance. Thus, even though there is no general sense of space as an expanse or time as a linear progression from earlier to later there is a common standard of sameness and difference of moments and places.

For Aristotle, it is clear that he takes the observational basis of our idea of time to be the constant regular motion of the heavens. The regular rotation of the fixed stars serves as a series of events which mark off constant successive temporal entities. By this time, emotive significance is not a primary principle of temporal individuation and so there is no temptation for Aristotle to argue that the return of the heavens to their original position is (or signals) the return to an original moment. The idea of historical progression has given rise to the concept of a linear progression of time.

The observational basis of Aristotle's doctrine of places is not so easily found. It seems that the doctrine of natural-places arises, in part, from observations of falling heavy bodies (rocks) and rising light bodies (fire, steam). The contrast between the circular motions of the heavenly bodies and the rectilinear motions of terrestrial objects reinforces the idea that the heavenly positions are of a different character from earthly positions. Again the observational key seems to be objects moving, starting and stopping, i.e., processes and events.

For Newton, as the Scholium makes clear, only relative spaces and relative times are readily grounded in experience and observations. The move to absolute space and absolute time requires an abstractive leap although Newton does try to indirectly ground these notions in observation via the "bucket experiment." The observational basis for Newton's view of space and time emerges as the same as that of Aristotle and the mythopoeic thinkers -- namely, the observation of processes and events. The central concept of modern space-time theories also turns out to be the concept of an event. We can think of processes, such as the diurnal revolution of the 'fixed stars' as a series of events, e.g., first the heavens have a certain orientation with respect to a fixed landmark, then as they revolve, different stages in the total process can be identified with the different events which are the coming into a certain new orientation with respect to the fixed landmark. A moment's reflection should convince you that any process can be analogously thought of as a sequence of appropriately chosen events. All of this suggests that by choosing events to be the primitive elements of our conceptual picture, we should be able to develop a framework common to all theories of space and time and in terms of which the various alternatives can be reconstructed. This constitutes the primacy of events.

What, then, do we mean by an event? Intuitively an event is an occurrence or happening of some sort. The rising of the moon, the fall of Rome, an explosion, the death of Napoleon, the writing of the first draft of this chapter, the absorption of a photon by my eyeball, the change in electric potential in a capacitor, a supernova, etc., all are examples of events.

However, we want to focus on an organization of the set of events which treats "point" events as fundamental. Consider the set of all such point events, past, present, and future. Call this big set E.

Figure 9-1

The above examples are all collections or aggregates of the "events" that we wish to study. Our emphasis will be on those events in terms of which we can conceptually reconstruct, at least in principle, the less primitive kinds of events listed above. Thus we would describe Napoleon as a bundle of primitive events. If we think of the primitive events as point (firecracker-like) happenings than the description of Napoleon would necessitate using a rather fat group of primitive events. A similar remark can be made for the other "events" mentioned above. In our quest to reconstruct the models of space and time we will always use E and consider that we are dealing with events that are not aggregates of other events. Somehow we must distill out only those kinds of events that can be used unambiguously. The "fat" subsets do not serve our purpose.

Given this set, we can, for whatever purposes we may have, select out certain members of this big set and examine their relationship to each other. Call the first event el, and the second event e2. We can then consider the subset of E which consists of just el, and e2, i.e., Sl = {el, e2}. We then may wish to determine what, if any, interesting relationships exist between the members of Sl, or we can form other collections of events, call them S2, S3 . . . etc. and then ask what, if any, interesting relationships exist between Si and Sj for some i and j. For example, there is one set, call it Sn which contains el and is such that all the other events in Sn are events involving Napoleon. There is another set, call it Se, which contains all events occurring on Elbe. el is also a member of that set, so Sn and Se have one member in common. (See the diagram below).

Figure 9-2

This is a rather trivial fact about Sn and Se but the general point should be clear. Given E, it is possible to consider subsets of E and the relations between them. To consider the relationships which held between members of E and between subsets of E we have to organize and order the members and subsets of E. How we organize and order these elements (which are either events or collections of events) will be a function of what we are interested in knowing about E, its members, and its subsets.

When we organize and order the members and subsets of E we are structuring the set E in various ways. The set E itself is just a collection of events; to organize the events in a particular way is to impose a structure on E. What we want to show now is that the different theories of space and time can be shown to be the result of imposing different structures on this collection of events. Having reconstructed, in particular, the Aristotelian and Newtonian theories of space and time along these lines, the conceptual continuity between these earlier treatments and contemporary relativity theory should be made clear.

It is important to note that this attempt to impose various structures on the interrelations among primitive events is really at the heart of the spirit of modern science and, indeed, of rational thought. Physicists strive to discover the fundamental laws of the universe. From our viewpoint this means they attempt to find structures on the set of primitive events E that are fundamental in that these "laws" represent "universally" valid statements about E. The evolution of the conceptual foundations of space and time has always been intertwined with these questions of the physical laws of nature. We shall see in what follows the close interaction between the structures that we end up imposing (or discovering!) for spatial and temporal structure and the phenomena that constitute the universe. Hence we will see that attempts to organize events in patterns will induce us to focus on some events as more important than others. (The prime example will be the growing importance of light as a fundamental stuff). II. Aristotle Revisited.

Before we begin, a general methodological remark is in order. The following reconstruction of Aristotle's position makes no claims to historical accuracy. We are not now, as we were earlier, interested in what Aristotle did say, rather we are interested in what he would have said had he had the benefit of the event point of view without the knowledge of the intervening science that gave rise to it. We are, in effect, conceptually reorganizing Aristotle's view around the concept of an event. To clearly distinguish our Aristotle from the historical Aristotle, we will call the view to be here developed a Neo-Aristotelian position.

Recall the general features of Aristotle's cosmology. The earth is envisioned to be at rest in the center of a three dimensional sphere, the outer layer of which rotates uniformly around the pole star. Every object has its natural place and its natural motion is to move toward that natural place. Earth and water tend to move down toward the center of the earth and air and fire tend to move up toward the sphere of fixed stars. The sphere of fixed stars is composed of a fifth element whose natural motion is circular. The following two dimensional diagram captures the essential elements of the Aristotelian world view:

Figure 9-3

The view and the neo-Aristotelian view to be based on it assumes the existence of three privileged positions and two privileged motions.

PL: The pole star is fixed.

P2: The earth is fixed.

P3: The distance to the sphere of fixed stars is fixed, and finite.

P4: Unconstrained heavy (light) objects move to the center of the earth (to the sphere of fixed stars).

P5: The sphere of fixed stars rotates uniformly about the pole star.

Time: The uniform rotation of the fixed stars which is the same for all observers serves as a privileged continuous motion which defines a universal time function. In an Aristotelian or neo-Aristotelian universe it makes perfectly good sense to talk about it being, say, 12 noon on the day of the vernal equinox in the year 200 (measured from some,Le conventional time base) at the same instant throughout the universe. The fact that a universal time function exists means these moments of time can be operationally defined in the following way. According to Aristotle, the motion of the heavens, in theory, is uniform. That means that an observer in a specified location, say Athens, looking in a specified direction in the night sky should see the heavens rotate in a uniform and regular manner above him. If this were so, one could use the motion of the sphere as a perfect clock, marking off units in terms of complete orbits. Alas, nature is not so amenable to the desires of man. The observed motions of the planets, in particular, are not so regular. The total observed motion of the heavens turns out to be very complicated. Aristotle was convinced, in principle, that the complicated motion could be shown to be the resultant of a finite series of perfectly regular circular motions, but he wisely left the details to the practicing astronomers. For eighteen hundred years astronomers tried to realize Aristotle's vision, which assumed that the natural motions of heavenly bodies must be circular. It was only with Kepler that the program of constructing the orbits of the planets from circular motions was finally abandoned.

The operationalization of a uniform time function turns out to be an enormously complicated business. For the sake of our neo-Aristotelian picture, however, we shall assume that the universe is more sympathetic to the will of man and that a uniform time function can be observationally defined more or less along the lines suggested above.

Using the motion of the heavens as his guide, we may suppose that an observer at Athens constructs a master clock from which one can read off Athens standard time. The claim we are making is that this time is the standard for all observers, no matter where they are in the universe. To facilitate the discussion, we shall assume that every observer, no matter where he is in the universe, is supplied with a clock identical to the master clock at Athens. Each observer needs his own clock, because the rotation of the Heavens, which serves as the natural standard for the Athens observer, will not appear the same to all the observers scattered throughout the universe.

In addition to assuming that each observer has a clock identical in construction to the master clock, we also assume that the rate at which the clocks tick is not dependent on their state of motion. This means that if a copy of the master clock is produced in Athens, and while at Athens ticks at the same rate as the master clock, then it will tick at the same rate when it is sent to its observer wherever he may be in the universe. We assume that the clocks are synchronized with the master clock as they are produced in Athens. They are then sent to various observation posts through- out the universe. Since they were initially synchronized with the master clock and since their ticking rate is independent of their state of motion, we expect that they will remain synchronized when they are at their observation stations at varying distances from Athens. However, once the copies are separated from the master clock, we cannot compare them directly to make sure that they are still ticking at the same rate and still reading the same time. The standard procedure for determining that clocks at a distance from one another are synchronized is to send a message or signal from one clock to the other. If the signal travels at a finite speed we must correct for the time it takes the signal to travel from the master clock to the distant clock. This requires that we know how far the clocks are from one another and how fast the signal travels. It turns out that to know this requires that one already have synchronized the distant clock with the master clock. This difficulty can be avoided by assuming that an infinitely fast signal can be sent from the master clock to all its copies. Since the signal travels infinitely fast, it takes no time to travel any distance, and so if a signal is sent from the master clock to to, all the other clocks will receive the message instantaneously and should be set to read t . Once all the clocks are so synchronized, then all observers will agree that if some event occurs at Athens at tl according to the master clock, it occurs at tl according to their local clock as well. Each event will, thus, be associated with a unique time coordinate that all observers will agree upon. That is what we mean by saying that there is a universal time function.

We have spent so much time defining the universal time function in order to bring out the point that the existence of a universal time function rests on two crucial assumptions: (l) clocks can be constructed which tick at rates which are independent of their states of motion, and

(2) infinitely fast signals are, in principle if not in practice, possible.

Both assumptions, we shall see, must be abandoned (along with the concept of a universal time function) when we come to formulate Einstein's theories of relativity. Both, however, are implicit assumptions of the Newtonian view of space and time, which still retains the idea of such a universal time function.

Let us return to the construction of the Aristotelian universe from an event point of view. If you think about the 2-dimensional diagram of the Aristotelian universe given above you will realize that it is a spatial diagram where each point in the circle represents a place in the universe. In attempting to reconstruct the Aristotelian view on an event basis, we want to develop a method which allows us to graph, not places, but events. Recall our set of events E. To say that a universal time function exists is to say that we can use different moments of time to decompose E into a sequence of subsets each of which contains only events which occur at the time in question.

Since the same time exists throughout the universe, these subsets will be uniquely determined. For example, call the set of events occurring at 12 noon on the day of the vernal equinox in the year 200, VE12,200. Then, for any event e, e is either an element of VE12,200 or it is not. Each such set, which contains only events occurring at a specific time, is called a simultaneity-slice (i.e., all the events in VE12,200 for example, are simultaneous with one another). Thus, the set of all events E can be represented as a sequence of simultaneity-slices. Given the view that time is a linear progression (a view Aristotle held) it seems natural to order this sequence of slices in terms of earlier-later relation. The net effect is to suggest that a graph of events in a neo-Aristotelian view should look something like:

Figure 9-4

The upward point arrow represents the sequence of temporal moments with the earlier moments of time lower on the page. Of course, the arrow extends indefinitely towards the past and indefinitely towards the future (recall that, for Aristotle, time has no beginning and no end). To conserve paper we only show a section of the arrow.

The set E now has much more structure than when we started. But, recall, to say that el and e2 are on the same slice is only to say that they occur at the same time, but whether they occurred close to one another or at opposite ends of the universe is not evident from the diagram as it now stands. We need to impose some more structure, in particular we want to organize the events within a given slice in a useful fashion.

Since the center of the earth is presumed fixed at the center of the universe, it serves as a convenient reference point. Actually because the universe is isotropic about the earth, the center of the earth is a distinguished point. If we ideally think of all events occurring on the earth at a given time to be represented by those points {ei,i = on earth}, then the fact that the earth remains fixed suggests that we represent the earth's center on our diagram by a vertical line parallel to the T-axis. Since the earth is at the center of a finite sphere, this suggests that we present each simultaneity-slice as a finite line segment with the earth-event in the middle. The other events that are interior as well as those on the surface are similarly representable. Thus we get:

Figure 9-5

The events at the end points of the finite line segments are events occurring at the boundary of the sphere. If we connect all the earth events together we get a straight line through the center of each simultaneity- slice which is parallel to the T-axis.

Figure 9-6

Such a line which represents the history of an object is called a world-line. Since the earth is not a point but is spread out in space, the collection of earth events or any simultaneity-slice will be a 'fat' event. WL is better represented, then, as a world-tube. It is convenient if we adopt the convention that the history of objects which remain at rest throughout their careers be represented by straight lines which are parallel to the T-axis. Using this convention, the representation of motion becomes very simple. An object is moving just at these times which its world line deviates from being parallel to the T-axis.

It still remains to order the other events on each simultaneity-slice. The diagram naturally suggests that we order the other events in such a way that the separation of a given event from the earth event at a given time is a reflection of how far from the earth the given event occurs. Since space has three dimensions, we must specify three coordinate positions for each "place" in the sphere:

(l) distance from the earth (r)

(2) angular displacement (~)

(3) azimuthal displacement (p) For the sake of simplification, we have suppressed the azimuthal displacement. In principle, this presents no problem since the direction to the pole star can serve as a fixed direction from which azimuthal displacement can be measured.

Aristotle suggests that heavy bodies travel faster as they approach the center of the earth. This provides the neo-Aristotelian view with an operational method for determining distances from the earth. Suppose that a small body (made of earth, of course) is dropped by a neo-Aristotelian from an initial position near the boundary of the sphere. Then its speed will steadily increase as it falls in a straight line to the earth. The velocity of such a test particle at any given moment is an operational measure of distance from the center of the earth. Below we represent this in a space-time diagram. Notice that the slope of the world line continuously increases. This represents the increase of velocity.

Figure 9-7

It only remains to specify a preferred direction for the measurement of horizontal angular displacements. Here again, Aristotle suggests that right and left can be absolutely defined in relation to the rising and setting of the sun. For a first approximation, we may take as baseline the direction defined by the center of Athens and the point where the sun rises on the horizon on March 21. With these coordinates each event can be given a precise spatial location. Time must be measured from an arbitrary starting point, e.g., the 1st Olympiad. Ignoring, for the sake of simplicity, the actual variations in the motion of the sun and stars, the rotation of the heavenly sphere serves as a clock for elapsed time. Each event can then be uniquely determined by a set of 4 numbers el = (tl, r ~ 1) which give the coordinates of the event with respect to the frame of reference described above. (l) It is an empirical fact about the universe in which we live that 4 numbers suffice to locate any event with respect to any other event.

(2) The neo-Aristotelian model assumes that unique determinations of spatial distance, temporal separation, and betweenness can be defined for any two events with respect to a frame of reference which, allowing for disagreements over where to put the axes, all observers can agree upon as unique.

(3) There is an absolute time function.

(4) There is an obvious method for defining absolute places, i.e., positions in the finite universe whose relative position with respect to other positions all observers will agree upon. Suppressing the third dimension, our neo-Aristotelian space time diagram looks like this:

Figure 9-8

Each simultaneity-slice is a circle. WLe is the world line of the fixed earth. WLt is the world line of an Areo-Aristotelian test particle which falls from the boundary of the sphere to the center of the earth (the fact that it is curved indicates that it moves faster as it approaches the earth). WLh is the world line of a heavenly body moving in uniform circular motion around the edge of the heavenly sphere.

III. Newtonian Spacetime

We turn now to a formulation of Newtonian conception of space and time in terms of the "event" point of view. The construction proceeds in two stages. In stage 1, we develop a modification of Aristotelian space-time which takes into account a principle of relativity discovered by Galileo. The resulting spacetime is called "Galilean spacetime." In stage 2, certain absolute structures representing Absolute space and Absolute time are imposed upon Galilean spacetime. The resulting spacetime is called "Newtonian (or Neo-Newtonian) spacetime."

Again, we quantify events by using a set of four real numbers with one set for each physical event that is found to exist in the universe. Throughout our discussion, it will be well to keep in mind the following basic principle. In deciding what structures are to be imposed on the set of all events one isolates a privileged set of events that are deemed of fundamental significance. So important are the privileged set of events that they are used as the very basis upon which viable descriptions of the structures possessed by the rest of the set of events are founded. The process of choosing fundamental events is by no means unique to the Newtonian thinker. Aristotle clearly focused on a few fundamental types of events (those associated with the earth, which he assumed sits with its center at the very center of the cosmos) when formulating his views of space and time. The point we wish to make is that a common thread that runs through all attempts to decide what space and time structures the world possesses is the preferential treatment of one or more classes of events.

Galilean Spacetime

Before we discuss the neo-Newtonian space-time picture let us consider the imposition of some additional structure on the neo-Aristotelian view we have just developed. As we have emphasized, for Aristotle the concept of unique place was central. In what follows we will develop a view of space-time structure in which just the uniqueness of place is discarded. This shall be done by introducing some additional phenomena into our set of considerations of most central concepts -- concepts whose codification into the fabric of space-time is essential. We consider an idealized world without any gravitational forces whatever. Now one of the dominant qualitative features of gravity is that objects placed in free motion in a permanent gravitational field accelerate, i.e., they speed up or slow down. Later in Chapter 15, we will reintroduce gravity and thus reassess the whole situation. If we ignore gravity, test particles at rest will remain at rest unless acted upon by some external force. Here we use the term "test particle" to refer to an object so small that its influence on the world is negligible. It can however be influenced by whatever forces exist in the world. Test particles which are moving with a constant velocity will continue to move with that velocity in a straight line unless they are acted upon by some external force. This is just one way of stating the principle of inertia. If we agree that a particle at rest has a constant velocity (v = 0~, then the test particles which are not being acted upon by external forces all have the property that they are moving uniformly (i.e., with constant relative velocity) with respect to each other. Call them inertial particles. Another important assumption that we make is that the universe is infinite. Since the universe is infinite, there is no privileged central position. The net effect is that the particles are effectively indistinguishable from one another. To see this consider a particle arbitrarily chosen to be at "rest". Choose some other particle moving with constant velocity V away from it.

Figure 9-9

Since the universe is infinite, A is in no special position with respect to the 'center' or 'boundary'. Consider now an observer on B. Since everyone likes to be at the center of things, he considers himself (B) to be at rest and A to be moving away from him with velocity -V.

Figure 9-10

An argument now ensues between observers on A and observers on B. Each claims to be at rest. Neither can persuade the other of his own case. Neither is a privileged observer with the other. The argument can be generalized for all observers on inertial particles. The net effect is that each can, with as good a claim as any other, proceed to map out the space which they all inhabit. But, since there are no fixed guideposts that all can agree on, when they come to assign positions to places in the universe, they will not agree with one another. Lack of agreement among the observers means the concept of absolute place has lost its significance (for the moment): Newton felt constrained to postulate the existence of absolute space on the basis of the bucket and globe experiments. However, recall that these experiments involve considerations of accelerated motion, which we are for the moment ignoring.

An important empirical fact turns out to be that the laws of mechanical motion turn out to be the same for all inertial observers (in a gravity free universe). Thus, in such a world uniform relative motion is simple and there is no reason to choose one uniformly moving observer as more fundamental than another. We hereby elevate this class of uniformly moving observers to a privileged position. Thus the new phenomenal ingredient is just that there exists a class of observers, all of whom can move relative to one another with uniform velocities, for which all the laws of nature are demonstrably the same. Stated another way, the introduction of this class of equivalent observers endows nature with a "relativity principle", the so-called Principle of Galilean Relativity, in which the laws of nature do not depend on the velocity of a given observer relative to some other observer equivalent in all respects except for his uniform relative motion. However, we do not throw out the presumed existence of a universal time function. Hence it makes perfectly good sense to speak of two extremely distantly related spatial points to be associated with a unique time of occurrence. We emphasize that the rational neo-Galilean thinker would be hard-pressed to come up with an empirical base for keeping the Aristotelian universal time function. Nevertheless we will leave this absolute structure untouched. We shall see later that this last aspect of absolute structure --the universal time function -- too will fall when Einstein focuses on the fundamental nature of light.

In the diagram below we illustrate the Aristotelian and neo-Galilean space time models. The only straight lines that have absolute significance are the vertical ones in the Aristotelian view. In the neo-Galilean view the slanted ones are just as valid for fundamental observers.

Figure 9-11

Only for the standard, absolute Aristotelian space-time picture does P have unique significance. In the Galilean space-time view it is not meaningful to ask what is the "distance" between two general events P and Q. Only for those events which happen to lie in the same T = constant slice is distance defined. To illustrate this point we give below two pairs of events for which distance is and is not defined.

Figure 9-12

In order to see that the "spatial distance" between P and Q has no observer-independent significance we proceed as follows. Recall that the presumed equivalence of inertial observers covers any pair of uniformly moving observers. Suppose we choose an observer, Harry, for whom the events P and Q occur at the same spatial point but lie on different time slices. An example might be two of Harry's hiccups. The space-time diagram with Harry included is shown below.

Figure 9-13

Now we would surely give zero as the spatial distance Harry would assign between P and Q. The problem is that no other general inertial observer who is moving relative to Harry will assign the same numerical value. To see this let us draw the world line of another inertial observer, Larry, and ask what Larry would conclude.

Figure 9-14

Note that Harry and Larry are experiencing the same two space-time events. In the diagram above we have indicated with the wiggly lines the signals that went out (in all directions) when the point-events P and Q happened. It is important to note that it takes no elapsed time on the universal clock for P and Q to make themselves known to all observers, irrespective of whether they are far from or close to P and Q. Now between T(P) and T(Q) Larry moves relative to Harry. If Larry moves with a large relative velocity then the difference between the position observed for Q as compared with P is proportionally large. If Larry moves slowly the distance between P and Q is smaller. Since Larry is an inertial observer, he is as justified as Harry in taking himself to be at rest and to consider Harry as moving with respect to him. From Larry's point of view, Harry's hiccups do not occur at the same place. Thus the observed distance between P and Q will vary from frame to frame and will not represent some observer-independent property. Similarly, any other observer who has non-zero relative velocity with respect to Harry will, with equivalent justification, disagree about the spatial distance between P and Q. Consequently we see that the concept of spatial distance between two events P and Q, which occur on different time slices, does not have any invariant significance.

To emphasize that it is the fact that P and Q occur on different time slices that is crucial, let us now consider the pair of events P' and Q' in Figure 9.12.

Since P' and Q' occur at the same time with respect to some Galilean observer, they are simultaneous with respect to all Galilean observers (since Galilean space-time has a Universal Time function).

Figure 9-15

Recall that each simultaneity slice is a space with Euclidean structure. Thus, the distance between P' and Q' will be given by the ordinary distance formula

d^2(p'q') = (xp' -xp')^2 + (yp' - yq')^2 + (zp' -zq')^2

which is invariant with respect to a change of origin. Both Harry and Larry "see" the same space at the instant when P' and Q' occur.

Figure 9-16

Fig. 9-16 represents space at the instant tp' = tq'. At this moment, the difference between Harry and Larry is a difference between the origin of the coordinate system they use to measure space. Thus, Harry and Larry agree on the spatial distance between P' and Q'. But, Harry and Larry were arbitrary observers and so, we can conclude that the spatial distance between two events on the same simultaneity slice is observer independent.

We may summarize the neo-Galilean space-time as follows:

1. There exists a Universal time function.

2. The spacetime is an infinite stack of Euclidean (and therefore infinite) 3-dimensional slices.

3. There are an infinite number of equivalent straight line families of world lines. These ar~ the world lines of inertial particles.

Figure 9-17

Adding Absolute Space

The spacetime structure associated with Newtonian physics is equivalent in all essential respects to the Galilean spacetime structure, with the added provision, that Newton postulated the existence of Absolute Space. Postulating the existence of Absolute space amounts to singling out one of the infinite number of privileged frames of reference as somehow more privileged than the others. In Chapter 7, we discussed some of Newton's reasons for holding this view. In the light of our current discussion, it seems like an arbitrary and unnecessary move. However, in the context within which Newton was working, he thought, rightly or wrongly, that a general mechanical theory of motion required such absolute structures.

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