Chapter VI: completes the preliminary course in the fundamentals of (6)

If we set up a surface geometry on a sphere, we find that the elimination of one dimension leaves us with a line-geometry that is still non-Euclidean since it pertains to the great circles of the sphere rather than to Euclidean straight lines. In shaking Minkowski's continuum down into a three-dimensional one by eliminating any one of his coordinates, if we eliminate either the $x$, the $y$ or the $z$, we have left a three-dimensional geometry in which the disturbing minus sign still occurs in the distance-formula, and which is therefore still non-Euclidean. If we omit the $t$, this does not occur. We see, then, that the time dimension is the disturbing factor, the one which gives to space-time its non-Euclidean character so far as the possession of curvature is concerned. And we see that this curvature is not the same in all directions, and in one direction is actually zero--whence the attempted analogy with a cylinder instead of with a sphere.

Many writers on relativity try to give the space-time continuum an appeal to our reason and a character of inevitableness by insisting on the lack of any fundamental distinction between space and time. The very expression for the space-time invariant denies this. Time is distinguishable from space. The three dimensions of space are quite indistinguishable--we can interchange them without affecting the formula, we can drop one out and never know which is gone. But the very formula singles out time as distinct from space, as inherently different in some way. It is not so inherently different as we have always supposed; it is not sufficiently different to offer any obstacle to our thinking in terms of the four-dimensional continuum. But while we can group space and time together in this way, [this does not mean at all that space and time cease to differ. A cook may combine meat with potatoes and call the product hash, but meat and potatoes do not thereby become identical.]223

THE QUESTION OF VISUALIZATION

To the layman there is a great temptation to say that while, mathematically speaking, the space-time continuum may be a great simplification, it does not really represent the external world. To be sure, you can't see the space-time continuum in precisely the same way that you can the three-dimensional space continuum, but this is only because Einstein finds the time dimension to be not quite freely interchangeable with the space dimension. Yet you do perceive this space-time continuum, in the manner appropriate for its perception; and it would be just as sensible to throw out the space continuum itself on the ground that perception of the two is not of exactly the same sort, as to throw out the space-time continuum on this ground. With appropriate conventions, either may stand as the mental picture of the external world; it is for us to choose which is the more convenient and useful image. Einstein tells us that his image is the better, and tells us why.

Before we look into this, we must let him tell us something more about the geometry of his continuum. What he tells us is, in its essentials, just this. The observer in a pure space continuum of three dimensions finds that as he changes his position, his right-and-left, his backward-and-forward, and his up-and-down are not fixed directions inherent in nature, but are fully interchangeable. The observers, in the adjoined sketch, whose verticals are as indicated by the arrows, find very different vertical and horizontal components for the distance between the points $O$ and $P$; a similar situation would prevail if we used all three space directions. The statement analogous to this for Einstein's four-dimensional continuum of space and time combined is that, as observers change their relative motion, their time axes take slightly different directions, so that what is purely space or purely time for the one becomes space with a small component in the time direction, or time with a small component in the space direction, for the other. This it will be seen explains fully why observers in relative motion can differ about space and time measurements. We should not be at all surprised if the two observers of the figure reported different values for horizontals and verticals; we should realize that what was vertical for one had become partly horizontal for the other. It is just so, says Einstein, with his observers of time and space who are in relative motion to one another; what one sees as space the other sees as partly time, because their time axes do not run quite parallel.

The natural question here, of course, is "Well, where are their time axes?" If you know what to look for, of course, you ought to be able to perceive them in just the way you perceive ordinary time intervals--with the reservation that they are imaginary, after all, just like your space axes, and that you must only expect to see them in imagination. If you look for a fourth axis in Euclidean three-space to represent your time axis, you will of course not find it. But you will by all means agree with me that your time runs in a definite direction; and this it is that defines your time axis. Einstein adds that if you and I are in relative motion, my time does not run in quite the same direction as yours.

How shall we prove it? Well, how would we prove it if he told us that our space axes did not run in precisely the same direction? Of course we could not proceed through direct measures upon the axes themselves; we know these are imaginary. What we should do would be to strike out, each of us, a very long line indeed in what seemed the true horizontal direction; and we should hope that if we made them long enough, and measured them accurately enough, we should be able to detect any divergence that might exist. This is precisely what we must do with our time axes if we wish to verify Einstein's statement that they are not precisely parallel; and what better evidence could we demand of the truth of this statement than the evidence already presented--that when we measure our respective time components between two events, we get different results?

WHAT IT ALL LEADS TO

The preceding chapters have been compiled and written with a view to putting the reader in a state of mind and in a state of informedness which shall enable him to derive profit from the reading of the actual competing essays which make up the balance of the book. For this purpose it has been profitable to take up in detail the preliminaries of the Special Theory of Relativity, and to allow the General Theory to go by default, in spite of the fact that it is the latter which constitutes Einstein's contribution of importance to science. The reason for this is precisely the same as that for taking up Euclidean geometry and mastering it before proceeding to the study of Newtonian mechanics. The fundamental ideas of the two theories, while by no means identical, are in general terms the same; and the conditions surrounding their application to the Special Theory are so very much simpler than those which confront us when we apply them to the more general case, that this may be taken as the controlling factor in a popular presentation. We cannot omit the General Theory from consideration, of course; but we can omit it from our preliminary discussion, and leave its development to the complete essays which follow, and which in almost every case give it the larger half of their space which its larger content demands. In the process of the slow and difficult preparation of the lay mind for the assimilation of an altogether new set of fundamental ideas, it is altogether desirable to give the Special Theory, with its simpler applications of these ideas, a place out of proportion to its importance in Einstein's completed structure; and this we have therefore done.

The Special Theory, postulating the relativity of uniform motion and deducing the consequences of that relativity, is often referred to as a "special case" of the General Theory, in which this restriction of uniformity is removed. This is not strictly speaking correct. The General Theory, when we have formulated it, will call our attention to something which we really knew all the time, but to which we chose not to give heed--that in the regions of space to which we have access, uniform motion does not exist. All bodies in these regions are under the gravitational influence of the other bodies therein, and this influence leads to accelerated motion. Nothing in our universe can possibly travel at uniform velocity; the interference of the rest of the bodies in the universe prevents this.

Obviously, we ought not to apply the term "special case" to a case that never occurs. Nevertheless, this case is of extreme value to us in our mental processes. Many of the motions with which we are concerned are so nearly at constant velocity that we find it convenient to treat them as though they were uniform, either ignoring the resulting error or correcting for it at the end of our work. In many other cases we are able to learn what actually occurs under accelerated motion by considering what would have occurred under uniform motion were such a thing possible. Science is full of complications which we unravel in this fashion. The physicist deals with gas pressures by assuming temperatures to be constant, though he knows temperature never is constant; and in turn he deals with temperatures by assuming pressures to be constant. After this, he is able to predict what will happen when, as in nature, pressures and temperatures are varying simultaneously. By using as a channel of attack the artificially simple case that never occurs, we get a grip on the complex case that gives us a true picture of the phenomenon. And because in actual nature we can come as close as we please to this artificial case, by supposing the variable factor to approach constancy, so when we assume it to be absolutely constant we speak of the result as the limiting case. This situation does not occur, but is the limiting case for those that do occur.

When, in the matter of motion, we abandon the artificial, limiting case of uniform velocity and look into the general, natural one of unrestricted motion, we find that the structure which we have built up to deal with the limiting case provides us with many of the necessary ideas and viewpoints. This is what we expect--in it lies the value of the limiting case. We shall see that the relativity of time and space, established for the limiting case, holds good in the general one. We shall see that the idea of the four-dimensional space-time continuum as representing the external world persists, forming the whole background of the General Theory much more definitely than in the Special Theory. Incidentally we shall see that the greater generality of the case under consideration will demand a greater degree of generality in the geometry of this continuum, a non-Euclideanism of a much more whole-hearted type than that of the Special Theory. But all the revisions of fundamental concepts which we have been at such pains to make for the sake of the Special Theory will remain with us in the General. With this we may consider our preliminary background as established, and give our attention to the essayists, who will try to take us more deeply into the subject than we have yet gone, without losing us in its intricacies.

VII

RELATIVITY

The Winning Essay in the Contest for the Eugene Higgins $5,000 Prize

BY LYNDON BOLTON SENIOR EXAMINER, BRITISH PATENT OFFICE LONDON

The reader is probably acquainted with the method of specifying positions of points in a plane by their distances from two mutually perpendicular lines, or if the points are in space by their distances from three mutually perpendicular planes like adjacent sides of a flat-sided box. The method is in fact in common use for exhibiting relations between quantities by graphs or diagrams. These sets of axes, as they are called, together with any scales used for measuring, must be supposed rigid, otherwise the events or points which they are used to specify are indefinite. The lengths which locate any point with reference to a set of axes are called its coordinates.

When such systems are used for physical purposes, they must be supplemented by clocks to enable the times at which events occur to be determined. The clocks must be synchronized, and must go at the same rate, but it must suffice here to state that this is possible without indicating how these conditions can be attained. A system of axes with its clocks will hereinafter be called a Frame of Reference, and every observer will be supposed to be provided with such a frame partaking of his motion. All the objects which partake of an observer's motion will be called his system.

It is a question whether among all possible frames of reference any one frame or class of frames is more suited than another for the mathematical statement of physical laws. This is for experience to decide, and a Principle of Relativity is a statement embodying the answer.

THE MECHANICAL PRINCIPLE OF RELATIVITY

It has been ascertained that all such frames are equally suitable for the mathematical statement of general mechanical laws, provided that their motion is rectilinear and uniform and without rotation. This fact is comprehended in the general statement that all unaccelerated frames of reference are equivalent for the statement of the general laws of mechanics. This is the mechanical principle of relativity.

It is well recognized however that the laws of dynamics as hitherto stated involve the assumptions that the lengths of rigid bodies are unaffected by the motion of the frame of reference, and that measured times are likewise unaffected; that is to say that any length measured on his own system by either of two relatively moving observers appears the same to both observers, or that lengths of objects and rates of clocks do not alter whatever the motion relative to an observer. These assumptions seem so obvious that it is scarcely perceived that they are assumptions at all. Yet this is the case, and as a matter of fact they are both untrue.

THE SPECIAL PRINCIPLE OF RELATIVITY

Although all unaccelerated frames of reference are equivalent for the purposes of mechanical laws, this is not the case for physical laws generally as long as the above suppositions are adhered to. Electromagnetic laws do alter their form according to the motion of the frame of reference; that is to say, if these suppositions are true, electromagnetic agencies act in different ways according to the motion of the system in which they occur. There is nothing a priori impossible in this, but it does not agree with experiment. The motion of each locality on the earth is continually changing from hour to hour but no corresponding changes occur in electromagnetic actions. It has however been ascertained that on discarding these suppositions the difficulty disappears, and electromagnetic laws retain their form under all circumstances of unaccelerated motion. According to the theory of relativity, the correct view which replaces these suppositions is deducible from the following postulates:

(1) By no experiment conducted on his own system can an observer
detect the unaccelerated motion of his system.
(2) The measure of the velocity of light in vacuo is unaffected
by relative motion between the observer and the source of light.

Both these postulates are well established by experiment. The first may be illustrated by the familiar difficulty of determining whether a slowly moving train one happens to be sitting in, or an adjacent one, is in motion. The passenger has either to wait for bumps (that is, accelerations) or else he has to look out at some adjacent object which he knows to be fixed, such as a building (that is, he has to perform an experiment on something outside his system), before he can decide.

The second postulate is an obvious consequence of the wave theory of light. Just as waves in water, once started by a ship, travel through the water with a velocity independent of the ship, so waves in space travel onward with a speed bearing no relation to that of the body which originated them. The statement however is based on experiment, and can be proved independently of any theory of light.

It is not difficult to deduce from these postulates certain remarkable conclusions relating to the systems of two observers, A and B, in relative motion, among them the following:

(1) Objects on B's system appear to A to be shorter in the
direction of relative motion than they appear to B.
(2) This opinion is reciprocal. B thinks that A's measurements
on A's system are too great.
(3) Similarly for times: each observer thinks that the other's
clocks have a slower rate than his own, so that B's durations
of time appear shorter to B than to A, and conversely.
(4) Events which appear simultaneous to A do not in general appear
so to B, and conversely.
(5) Lengths at right angles to the direction of motion are
unaffected.
(6) These effects vary with the ratio of the relative velocity
to that of light. The greater the relative velocity, the
greater the effects. They vanish if there is no relative
velocity.
(7) For ordinary velocities the effects are so small as to escape
notice. The remarkable point however is their occurrence rather
than their magnitude.
(8) The observers similarly form different estimates of the
velocities of bodies on each other's systems. The velocity of
light however appears the same to all observers.

Taking into account these revised views of lengths and times the mechanical principle of relativity may be extended to physical laws generally as follows: All unaccelerated frames of reference are equivalent for the statement of the general laws of physics. In this form the statement is called the Special, or Restricted, Principle of Relativity, because it is restricted to unaccelerated frames of reference. Naturally the laws of classical mechanics now require some modification, since the suppositions of unalterable lengths and times no longer apply.

THE FOUR DIMENSIONAL CONTINUUM

Lengths and times therefore have not the absolute character formerly attributed to them. As they present themselves to us they are relations between the object and the observer which change as their motion relative to him changes. Time can no longer be regarded as something independent of position and motion, and the question is what is the reality? The only possible answer is that objects must be regarded as existing in four dimensions, three of these being the ordinary ones of length, breadth and thickness, and the fourth, time. The term "space" is applicable only by analogy to such a region; it has been called a "continuum," and the analogue of a point in ordinary three-dimensional space has been appropriately called an "event." By "dimension" must be understood merely one of four independent quantities which locate an event in this continuum. In the nature of the case any clear mental picture of such a continuum is impossible; mankind does not possess the requisite faculties. In this respect the mathematician enjoys a great advantage. Not that he can picture the thing mentally any better than other people, but his symbols enable him to abstract the relevant properties from it and to express them in a form suitable for exact treatment without the necessity of picturing anything, or troubling whether or not the properties are those on which others rely for their conceptions.

GRAVITATION AND ACCELERATION

The limitation of statements of general law to uniformly moving systems is hardly satisfactory. The very concept of general law is opposed to the notion of limitation. But the difficulties of formulating a law so that the statement of it shall hold good for all observers, whose systems may be moving with different and possibly variable accelerations, are very great. Accelerations imply forces which might be expected to upset the formulation of any general dynamical principles, and besides, the behavior of measuring rods and clocks would be so erratic as to render unmeaning such terms as rigidity and measured time, and therefore to preclude the use of rigid scales, or of a rigid frame of reference which is the basis of the foregoing investigation.

The following example taken from Einstein will make this clear, and also indicate a way out of the difficulty. A rotating system is chosen, but since rotation is only a particular case of acceleration it will serve as an example of the method of treating accelerated systems generally. Moreover, as it will be seen, the attribution of acceleration to the system is simply a piece of scaffolding which can be discarded when the general theory has been further developed.

Let us note the experiences of an observer on a rotating disk which is isolated so that the observer has no direct means of perceiving the rotation. He will therefore refer all the occurrences on the disk to a frame of reference fixed with respect to it, and partaking of its motion.

He will notice as he walks about on the disk that he himself and all the objects on it, whatever their constitution or state, are acted upon by a force directed away from a certain point upon it and increasing with the distance from that point. This point is actually the center of rotation, though the observer does not recognize it as such. The space on the disk in fact presents the characteristic properties of a gravitational field. The force differs from gravity as we know it by the fact that it is directed away from instead of toward a center, and it obeys a different law of distance; but this does not affect the characteristic properties that it acts on all bodies alike, and cannot be screened from one body by the interposition of another. An observer aware of the rotation of the disk would say that the force was centrifugal force; that is, the force due to inertia which a body always exerts when it is accelerated.

Next suppose the observer to stand at the point of the disk where he feels no force, and to watch someone else comparing, by repeated applications of a small measuring rod, the circumference of a circle having its center at that point, with its diameter. The measuring rod when laid along the circumference is moving lengthwise relatively to the observer, and is therefore subject to contraction by his reckoning. When laid radially to measure the diameter this contraction does not occur. The rod will therefore require a greater proportional number of applications to the circumference than to the diameter, and the number representing the ratio of the circumference of the circle to the diameter thus measured will therefore be greater than 3.14159+, which is its normal value. Moreover the relative velocity decreases as the center is approached, so that the contraction of the measuring rod is less when applied to a smaller circle; and the ratio of the circumference to the diameter, while still greater than the normal, will be nearer to it than before, and the smaller the circle the less the difference from the normal. For circles whose centers are not at the point of zero force the confusion is still greater, since the velocities relative to the observer of points on them now change from point to point. The whole scheme of geometry as we know it is thus disorganized. Rigidity becomes an unmeaning term since the standards by which alone rigidity can be tested are themselves subject to alteration. These facts are expressed by the statement that the observer's measured space is non-Euclidean; that is to say, in the region under consideration measurements do not conform to the system of Euclid.

The same confusion arises in regard to clocks. No two clocks will in general go at the same rate, and the same clock will alter its rate when moved about.

THE GENERAL PRINCIPLE OF RELATIVITY

The region therefore requires a space-time geometry of its own, and be it noted that with this special geometry is associated a definite gravitational field, and if the gravitational field ceases to exist, for example if the disk were brought to rest, all the irregularities of measurement disappear, and the geometry of the region becomes Euclidean. This particular case illustrates the following propositions which form the basis of this part of the theory of relativity:

(1) Associated with every gravitational field is a system of
geometry, that is, a structure of measured space peculiar to
that field.
(2) Inertial mass and gravitational mass are one and the same.
(3) Since in such regions ordinary methods of measurement fail,
owing to the indefiniteness of the standards, the systems of
geometry must be independent of any particular measurements.
(4) The geometry of space in which no gravitational field exists
in Euclidean. [1]

The connection between a gravitational field and its appropriate geometry suggested by a case in which acceleration was their common cause is thus assumed to exist from whatever cause the gravitational field arises. This of course is pure hypothesis, to be tested by experimental trial of the results derived therefrom.

Gravitational fields arise in the presence of matter. Matter is therefore presumed to be accompanied by a special geometry, as though it imparted some peculiar kink or twist to space which renders the methods of Euclid inapplicable, or rather we should say that the geometry of Euclid is the particular form which the more general geometry assumes when matter is either absent or so remote as to have no influence. The dropping of the notion of acceleration is after all not a very violent change in point of view, since under any circumstances the observer is supposed to be unaware of the acceleration. All that he is aware of is that a gravitational field and his geometry coexist.

The prospect of constructing a system of geometry which does not depend upon measurement may not at first sight seem hopeful. Nevertheless this has been done. The system consists in defining points not by their distances from lines or planes (for this would involve measurement) but by assigning to them arbitrary numbers which serve as labels bearing no relation to measured distances, very much as a house is located in a town by its number and street. If this labeling be done systematically, regard being had to the condition that the label-numbers of points which are close together should differ from one another by infinitesimal amounts only, it has been found that a system of geometry can actually be worked out. Perhaps this will appear less artificial when the fact is called to mind that even when standards of length are available no more can be done to render lengths of objects amenable to calculation than to assign numbers to them, and this is precisely what is done in the present case. This system of labeling goes by the name of "Gaussian coordinates" after the mathematician Gauss who proposed it.

It is in terms of Gaussian coordinates that physical laws must be formulated if they are to have their widest generality, and the general principle of relativity is that all Gaussian systems are equivalent for the statement of general physical laws. For this purpose the labeling process is applied not to ordinary space but to the four dimensional space-time continuum. The concept is somewhat difficult and it may easily be aggravated into impossibility by anyone who thinks that he is expected to visualize it. Fortunately this is not necessary; it is merely one of these irrelevancies to which those who are unaccustomed to think in symbols are liable.

It will now be seen that among physical laws the law of gravitation stands pre-eminent, for it is gravitating matter which determines the geometry, and the geometry determines the form of every other law. The connection between the geometry and gravitation is the law of gravitation. This law has been worked out, with the result that Newton's law of the inverse square is found to be approximate only, but so closely approximate as to account for nearly all the motions of the heavenly bodies within the limits of observation. It has already been seen that departure from the Euclidean system is intensified by rapidity of motion, and the movements of these bodies are usually too slow for this departure to be manifest. In the case of the planet Mercury the motion is sufficiently rapid, and an irregularity in its motion which long puzzled astronomers has been explained by the more general law.

Another deduction is that light is subject to gravitation. This has given rise to two predictions, one of which has been verified. The verification of the other is as yet uncertain, though the extreme difficulty of the necessary observations may account for this.

Since light is subject to gravitation it follows that the constancy of the velocity of light assumed in the earlier part of this paper does not obtain in a gravitational field. There is really no inconsistency. The velocity of light is constant in the absence of gravitation, a condition which unaccelerated motion implies. The special principle of relativity is therefore a limiting case of the general principle.

VIII

THE NEW CONCEPTS OF TIME AND SPACE

The Essay in Behalf of Which the Greatest Number of Dissenting Opinions Have Been Recorded

BY MONTGOMERY FRANCIS NEW YORK

We have all had experiences, on trains and boats, illustrating our inability to tell, without looking off to some external body, whether we are at rest or moving uniformly; and when we do so look, to tell, without reference to the ground or some other point external to both systems, whether ours or the other be the seat of motion. Uniform motion must be relative, because we find nowhere in the universe a body in the unique state of absolute rest from which alone absolute motion might be measured.

True, the wave theory of light with its homogeneous space-filling ether seemed to provide a reference standard for the concept of absolute motion, and for its measurement by experiment with light rays. But when Michelson and Morley looked for this absolute motion they found no trace of it. To the physicist, observational student of the external world, nothing exists save observationally; what he can never observe is not there. So: I. By no means whatever may we regard uniform straight-line motion as other than relative.

As a further direct consequence of the Michelson-Morley experiment we have: II. Light in a vacuum presents the same velocity, $C = 186,330$ miles per second, to all observers whatever their velocity of relative motion. In addition to being experimentally established, this is necessary to support I, for if light will distinguish between our velocities, its medium is necessarily a universal standard for absolute motion. But it is contrary to common sense to suppose that if I pass you at 100 miles per hour, the same light impulse can pass us both at the same speed, $C$. We feel, instinctively, that space and time are not so constituted as to make this possible. But the fact has been repeatedly demonstrated. And when common sense and fundamental concepts clash with facts, it is not the facts that must yield. We have survived such crises, notably one where we had to change the fundamental concept of up-and-down; if another one is here, says Einstein, let us meet it.

This the Special Theory of Relativity does. It accepts Postulates I and II above; their consequences it deduces and interprets. For extensive demonstration of these I lack space, and this has been satisfactorily done by others so it is not my chief duty; but clearly they will be startling. For the very ray of light which refuses to recognize our relative motion is the medium through which I must observe your system and you mine.

It turns out that I get different values for lengths and time intervals in your system than you get, and vice versa. And we are both right! For me to accept your "correction" were for me to admit that you are at absolute rest and I in absolute motion, that your measure of light velocity is right and mine wrong: admissions barred by the postulates. We have nothing to correct; we can only recognize the reason for the discrepancy; and knowing our relative velocity, each can calculate from his own results what the other's will be. We find, of course, that at ordinary velocities the discrepancy is many times too small for detection; but at relative velocities at all comparable with that of light it rises above the observational horizon.

To inquire the "true" length is meaningless. Chicago is east of Denver, west of Pittsburgh, south of Milwaukee; we do not consider this contradictory, or demand the "true" direction of Chicago. Einstein finds that the concept of length, between points in space or events in time, does not as we had supposed represent an intrinsic property of the points or the events. Like direction, it is merely a relation between these and the observer--a relation whose value changes with the observer's velocity relative to the object. If our ideas of the part played in the world by time and space do not permit us to believe this, we must alter these ideas. Let us see how we may do this.

A WORLD OF POINTS

To deal with points in a plane the mathematician draws two perpendicular lines, and locates any point, as $P$, by measuring its distances, $X$ and $Y$, from these "coordinate axes." The directions of his axes acquire for him a peculiar significance, standing out above other directions; he is apt to measure the distances $X-x$ and $Y-y$ between the points $P$ and $Q$ in these directions, instead of measuring the single distance $PQ$. We do the same thing when we say that the railroad station is five blocks north and two east.

The mathematician visualizes himself as an observer, located on his coordinate framework. For another observer on another framework, the horizontal and vertical distances $X'-x'$ and $Y'-y'$ between $P$ and $Q$ are different. But for both, the distance from $P$ direct to $Q$ is the same. In each case the right triangle tells us that:

$$PQ = \sqrt{(X-x)^2 + (Y-y)^2} = \sqrt{(X'-x')^2 + (Y'-y')^2}$$

Imagine an observer so dominated by his coordinate system that he knows no way of relating $P$ with $Q$ save by their horizontal and vertical separation. His whole scheme of things would be shattered by the suggestion that other observers on other reference frames find different horizontal and vertical components. We have to show him the line $PQ$. We have to convince him that this length is the absolute property enjoyed by his pair of points; that horizontals and verticals are merely relations between the points and the observer, result of the observer's having analyzed the distance $PQ$ into two components; that different observers effect this decomposition differently; that this seems not to make sense to him only because of his erroneous concept of a fundamental difference between verticals and horizontals.

THE FOUR-DIMENSIONAL WORLD OF EVENTS

We too have created a distinction in our minds corresponding to no sufficient reality. Our minds seize on time as inherently separable from space. We see the world made up of things in a continuum of three space dimensions; to make this dead world live there runs through it a one-dimensional time continuum, imposed from without, unrelated.

But did you ever observe anything suggesting the presence of time in the absence of space, or vice versa? No; these vessels of the universe always occur together. Association of the space dimensions into a manifold from which time is excluded is purely a phenomenon of the mind. The space continuum cannot begin to exist until the time dimension is supplied, nor can time exist without a place to exist in.

The external world that we observe is composed, not of points, but of events. If a point lacks position in time it does not exist; give it this position and it becomes an event. This world of events is four-dimensional--which means nothing more terrifying than that you must make four measures to locate an event. It does not mean, at all, that you must visualize four mutually perpendicular lines in your accustomed three-space or in a four-space analogous to it. If this world of four dimensions seems to lack reality you will be able to exhibit no better reality for your old ideas. Time belongs, without question; and not as an afterthought, but as part of the world of events.

To locate an event we use four measures: $X$, $Y$ and $Z$ for space, $T$ for time. Using the same reference frame for time and space, we locate a second event by the measures $x$, $y$, $z$, $t$. Minkowski showed that the quantity

$$\sqrt{(X-x)^2 + (Y-y)^2 + (Z-z)^2 - (CT-Ct)^2}$$

is the same for all observers, no matter how different their $x$'s, $y$'s, $z$'s and $t$'s; just as in the plane the quantity

$$\sqrt{(X-x)^2 + (Y-y)^2}$$

is the same for all observers, no matter how different their $x$'s and $y$'s.

Such a quantity, having the same value for all observers, is absolute. In the plane it represents the true, absolute distance between the points--their intrinsic property. In dealing with events it represents the true, absolute "interval," in time and space together between the events. It is not space, nor time, but a combination of the two. We have always broken it down into separate space and time components. In this we are as naive as the plane observer who could not visualize the distance $PQ$ until it was split into separate horizontals and verticals. He understood with difficulty that another observer, employing a different reference frame because in different position, would make the decomposition differently. We understand with difficulty that another observer, employing a different reference frame because in uniform motion relative to us, will decompose the "interval" between events into time and space components different from ours. Time and space are relative to the observer; only the interval representing space-time is absolute. So common sense stands reconciled to the Special Theory of Relativity.

SUCCESSIVE STEPS TOWARD GENERALITY

Is then our laboriously acquired geometry of points in a three-dimensional space to go into the discard? By no means. Jeans, investigating the equilibrium of gaseous masses, found the general case too difficult for direct attack. So he considered the case where the masses involved are homogeneous and incompressible. This never occurs; but it throws such light on the general case as to point the way toward attack on it.

Euclidean geometry excludes motion, save that engineered by the observer; and then the time is immaterial. Time does not enter at all; the three space dimensions suffice. This simple case never occurs where matter exists; but its conclusions are of value in dealing with more general cases.

When we look into a world alleged to be that of Euclid and find motion, we may retain the Euclidean concept of what constitutes the world and invent a machinery to account for the motion; or we may abandon the Euclidean world, as inadequate, in favor of a more general one. We have adopted the second alternative.

Newton's laws tells us that a body free to move will do so, proceeding in a straight line at uniform velocity until interfered with. We do not ask, nor does the theory tell us, whence comes the initial motion. There is no machinery to produce it; it is an inherent property of Newton's world--assured by the superposition of the time continuum upon Euclid's world to make Newton's, accepted without question along with that world itself.

But Newton saw that his world of uniform motion, like Euclid's, was never realized. In the neighborhood of one particle a second is interfered with, forced to give up its uniform motion and acquire a constant acceleration. This Newton explained by employing the first of the alternatives mentioned above. He tells us that in connection with all matter there exists a force which acts on other matter in a certain way. He does not display the actual machinery through which this "force" works, because he could not discover any machinery; he had to stop with his brilliant generalization of the observed facts. And all his successors have failed to detect the slightest trace of a machinery of gravitation.

Einstein asks whether this is not because the machinery is absent--because gravitation, like position in Euclid's world and motion in Newton's, is a fundamental property of the world in which it occurs. His point of attack here lay in precise formulation of certain familiar facts that had never been adequately appreciated. These facts indicate that even accelerated motion is relative, in spite of its apparently real and absolute effects.

GRAVITATION AND ACCELERATION

An observer in a closed compartment, moving with constant acceleration through empty space, finds that the "bottom" of his cage catches up with objects that he releases; that it presses on his feet to give him the sensation of weight, etc. It displays all the effects that he would expect if it were at rest in a gravitational field. On the other hand, if it were falling freely under gravitational influence, its occupant would sense no weight, objects released would not leave his hand, the reaction from his every motion would change his every position in his cage, and he could equally well assume himself at rest in a region of space free from gravitational action. Accelerated motion may always be interpreted, by the observer on the system, as ordinary force effects on his moving system, or as gravitational effects on his system at rest.

An alternative statement of the Special Theory is that the observed phenomena of uniform motion may equally be accounted for by supposing the object in motion and the observer with his reference frame at rest, or vice versa. We may similarly state the General Theory: The observed phenomena of uniformly accelerated motion may in every case be explained on a basis of stationary observer and accelerated objective, or of stationary objective with the observer and his reference system in accelerated motion. Gravitation is one of these phenomena. It follows that if the observer enjoy properly accelerated axes (in time-space, of course), the absolute character of the world about him must be such as to present to him the phenomenon of gravitation. It remains only to identify the sort of world, of which gravitation as it is observed would be a fundamental characteristic.

Euclid's and Newton's systems stand as first and second approximations to that world. The Special Relativity Theory constitutes a correction of Newton, presumably because it is a third approximation. We must seek in it those features which we may most hopefully carry along, into the still more general case.

Newton's system retained the geometry of Euclid. But Minkowski's invariant expression tells us that Einstein has had to abandon this; for in Euclidean geometry of four dimensions the invariant takes the form:

$$\sqrt{(X-x)^2 + (Y-y)^2 + (Z-z)^2 + (T-t)^2}\,,$$

analogous to that of two and three dimensions. It is not the presence of the constant $C$ in Minkowski's formula that counts; this is merely an adjustment so that we may measure space in miles and time in the unit that corresponds to a mile. It is the minus sign where Euclidean geometry demands a plus that makes Minkowski's continuum non-Euclidean.

The editor has told us what this statement means. I think he has made it clear that when we speak of the geometry of the four-dimensional world, we must not read into this term the restrictions surrounding the kind of geometry we are best acquainted with--that of the three-dimensional Euclidean continuum. So I need only point out that if we are to make a fourth (and we hope, final) approximation to the reality, its geometry must preserve the generality attained by that of the third step, if it goes no further.

EINSTEIN'S TIME-SPACE WORLD

Einstein accordingly examined the possible non-Euclidean geometries of four dimensions, in search of one displaying fundamental characteristics which, interpreted in terms of space-time, would lead to the observed facts of gravitation. The mathematics of this investigation is that part of his work which, we are told, but twelve men can follow; so we may only outline his conclusions.

If we assume that in the neighborhood of matter the world of space-time is non-Euclidean, and that its curvature or distortion or non-Euclideanism is of a certain type already known to mathematicians; that the curvature of this world in the neighborhood of matter increases with the mass, and decreases as the distance from the matter increases; and that every particle of matter that is not interfered with travels through space-time in the most direct path possible in that continuum; then the observed facts of gravitation are accounted for as an inherent geometric property of this space-time world. We usually say that the presence of matter distorts this world, and that this distortion gives the track of particles through the region affected its non-uniform character.

Gravitation then is not a force at all; it is the fundamental nature of things. A body free to move through the world must follow some definite path. Euclid says it will stand still; Newton that it will traverse a straight line in three-space at uniform time-rate; Einstein that it will move in a "geodesic" through time-space--in every-day language, that it will fall.

The numerical consequences of Einstein's theory are, within the limits of observation, the same as those of Newton's for all bodies save one--Mercury. This planet shows a small deviation from the path predicted by Newton's law; Einstein's theory gives its motion exactly. Again, when modern research showed that light must be affected by gravitation, Einstein's theory, because of the extreme velocity of light, deviates from Newton's, where the speed is less a determining factor; and observations of starlight deflected by the sun during the eclipse were in much better accord with Einstein's theory than Newton's. Moreover, the Special Theory predicts that mass is an observational variable like length and duration. Radioactive emanations have a velocity high enough to give appreciable results here, and the prediction is verified, tending to support the general theory by supporting its limiting case.

We like always to unify our science; and seldom, after effecting a unification, are we forced to give it up. Einstein for the first time brings mechanical, electromagnetic and gravitational phenomena within one structure. This is one reason why physicists are so open minded toward his theory--they want it to be true.

THE LAYMAN'S LAST DOUBT

The final answer to any series of questions is inevitably "because the world is so constructed." The things we are content to leave on that basis are those to which we are accustomed, and which we therefore think we understand; those for which this explanation leaves us unsatisfied are those which are new and unfamiliar. Newton told us that the world of three-dimensional space with one-dimensional time superposed was so constructed that bodies left to themselves would go on forever in a straight line at constant speed. We think we understand this, but our understanding consists merely of the unspoken query, "Why, of course; what is there to prevent?" The Greeks, an intelligent people, looked at this differently; they would have met Newton with the unanimous demand "Why so; what is there to keep them going?" So if, in seeking an explanation of anything, we come sooner than we had expected to the finality "Because the world is so constructed," let us not feel that we have been cheated.

IX

THE PRINCIPLE OF RELATIVITY

A Statement of What it is All About, in Ideas of One Syllable

BY HUGH ELLIOT CHISLEHURST, KENT, ENGLAND

The invariance of the laws of nature was one of the most popular themes of nineteenth century philosophy. For it was not till last century that general acceptance was accorded to the doctrine of the "Uniformity of Law," adumbrated in ancient times by Epicurus and Lucretius. It is now a cardinal axiom of science that the same cause in the same conditions is always followed by the same effect. There exists in nature no indeterminate element; all things are governed by fixed laws, and the discovery of these laws is the main business of science.

It is necessary to guard against reading into this statement an
erroneous idea of the content of a "law of nature." Such a law
is of course not an enactment of any sort; and it is not even to
be thought of as an actual explanation of the how and why of the
phenomena with which it has to do. It really is nothing but an
expression of our belief in the pronouncement of the preceding
paragraph, that like conditions do produce like results. It is
a prediction based on past experience, and is of value merely in
that past experience leads us to credit its accuracy. The composite
essay beginning on page 19 discusses this question of the reality
of natural laws, and should be consulted in connection with the
present contribution.--Editor.

This great philosophic principle was derived of course from the study of natural science; i.e., from observations and experiments conducted upon the earth. Their comprehensiveness is therefore limited by the fact that the observer is always in a state of rest, or nearly so, as compared with the earth. All observers upon the earth are moving through space at the same velocity; and it was possible to argue that the uniformity of law might only hold good, when experiments were conducted at this velocity. An observer moving at very different velocity might discover that the laws of nature under these new conditions were somewhat different.

Such a view could indeed never be very plausible, for motion is only a relative conception. Imagine a universe consisting of infinite "empty" space, in which there is poised a single material body. How shall we determine whether this body is at rest, or whether it is moving at high or low velocity through space? It is never getting nearer to anything or farther from anything, since there is no other body for it to get nearer to or farther from. If we say it is moving at a uniform velocity of a thousand miles a second, our statement really has no significance. We have no more reason for affirming that it is in motion than we have for affirming that it is at rest. In short, there is no such thing as absolute motion; the conception of motion only arises when there are two or more bodies changing their position relatively to one another. This is what is meant by the relativity of motion. It seemed therefore improbable that the laws of nature would be different if the observer were moving at high velocity; for the movement of the observer is not an absolute quantity, but merely a statement of his relation to other bodies, and if there were no other bodies, the statement itself would be meaningless.

THE BEHAVIOR OF LIGHT

Now among the established laws of nature is that which specifies the velocity of light moving through a vacuum. If the laws of nature are invariable, this velocity will always be the same. But consider what would happen under the following circumstances: Suppose that we are at rest, and that an observer on another body flies past us at 150,000 miles a second. Suppose that at the moment he passes, a piece of flint projecting from him grazes a piece of steel projecting from us, giving rise to a spark; and that we both thereupon set about to measure the velocity of the light so produced. After one second, we should find that the light had traveled about 186,000 miles away, and since during this second the other observer had traveled 150,000 miles, we should infer that the light traveling in his direction was only about 36,000 miles ahead of him. We should also infer that he would find this out by his experiment, and that he would estimate the velocity of light as only 36,000 miles a second in his own direction, and 336,000 miles a second in the opposite direction. But if this is so, then that law of nature which specifies the velocity of light is quite different for him and for us: the laws of nature must be dependent upon the observer's motion--a conclusion which appears incompatible with the idea of the relativity of motion.

And it so happens that it is also contradictory to experimental conclusions. Experiments undertaken to settle the point show that each observer finds the same velocity for the light of the spark; and after one second, each observer finds that the light has traveled 186,000 miles from himself. But how is it possible that when it has traveled 186,000 miles in the same direction as the other observer who himself has moved 150,000 miles meanwhile, he should still think it 186,000 miles ahead of him? That is the initial paradox; and since there has been no room for error in the experiments, we are forced to conclude that there was something wrong in the assumptions and preconceptions with which we started.

SPACE AND TIME