Chapter VI: completes the preliminary course in the fundamentals of (8)

Thus, for an observer moving past our earth with a velocity which is nine-tenths that of light, a meter stick on the earth would be 44 centimeters as measured by him, while a second on our clocks would be about two and a half seconds as marked by his clock. Similarly, what he calls a meter length would, for us, be only 44 centimeters and he would appear to us to be living about two and a half times slower than we are. Each observer is perfectly consistent in his measurements of time and space as long as he confines his observations to his own system, but when he tries to make observations on another system moving past his, he finds that the results which he obtains do not agree with those obtained by the other observer.

It is not surprising that in accordance with this conclusion it also follows that the mass of a body must increase with its velocity. For low velocities the increase is so small that we cannot ever hope to measure it, but as the velocity of light is approached the difference becomes more and more appreciable and a body having the velocity of light would possess infinite mass, which simply means that such a velocity cannot be attained by any material object. This conclusion has been experimentally confirmed by observations on the mass of the extremely small negatively charged particles which are emitted by radioactive elements. Some of these particles are ejected with velocities which are over nine-tenths that of light, and measurements show that the increase in mass is in accord with this theory.

The relativity theory also throws new light on the nature of mass itself. According to this view, mass and energy are equivalent. The absolute destruction of 1 gram of any substance, if possible, would yield an amount of energy which is one hundred million times as much as that obtained by burning the same mass of coal. Conversely, energy changes are accompanied by changes in mass. The latter are ordinarily so inappreciably small as to escape our most refined methods of measurements, but in the case of the radioactive elements we actually observe this phenomenon. From this standpoint, also, the laws of conservation of energy and of mass are shown to be intimately related.

UNIVERSAL RELATIVITY

So far we have dealt with what has been designated as the special theory of relativity. This, as we have seen, applies to uniform motion only. In extending the theory to include non-uniform or accelerated motion, Einstein has at the same time deduced a law of gravitation which is much more general than that of Newton.

A body falling towards the earth increases in velocity as it falls. The motion is said to be accelerated. We ascribe this increase in velocity to a gravitational force exerted by the earth on all objects. As shown by Newton, this force acts between all particles of matter in the universe, and varies inversely as the square of the distance, and directly as the product of the masses.

Of course, we have had a number of theories of gravitation, and none of them have proven successful. Einstein, however, was the first one to suggest a conception of gravitation which has proven extremely significant. He points out that a gravitational force is non-existent for a person falling freely with the acceleration due to gravity. For this person there is no sensation of weight, and if he were in a closed box which is also falling with the same acceleration, he would be unable to decide as to whether his system were falling or situated in interplanetary space where there is no gravitational field. Furthermore, if he were to carry out any optical or electrical experiments in this box he would observe the same results as an experimenter on the earth. A ray of light would travel in a straight line so far as this observer can perceive, while an external observer would, of course, judge differently.

Einstein shows that this is equally true for all kinds of acceleration including that due to rotation. In the case of a rotating body there exists a centrifugal force which tends to make objects on the surface fly outwards, but for an external observer this force does not exist any more than gravity exists for the observer falling freely.

Thus we can draw the general conclusion that a gravitational field or any other field of force may be eliminated by choosing an observer moving with the proper acceleration. For this observer, however, the laws of optics and electricity must be just as valid as for an observer on the earth.

In postulating this equivalence hypothesis Einstein merely makes use of the very familiar observation that, independently of the nature of the material, all bodies possess the same acceleration in a given field of force.

The problem which Einstein now sets out to solve is that of determining the law which shall describe the motion of any system in a field of force in such a general manner as to leave unaltered the fundamental relations of electricity and optics.

In connection with the solution of this problem he finds it necessary to discard the limitations placed on us by ordinary or Euclidean geometry. In this manner geometrical concepts as well as those of force are completely robbed of all notions of absoluteness, and the goal of a general theory of relativity is attained.

THE GEOMETRY OF GRAVITATION

Let us consider a circular disc rotating with a uniform peripheral speed. According to the deductions from the "special theory" of relativity, an observer situated near the edge of this disc, but not rotating with it, will observe that units of length measured along the circumference of the disc are contracted. On the other hand, measurements along the diameter, which is at right angles to the direction of motion of the circumference, will show no contraction whatever, and, consequently the observer will find that the ratio of circumference to diameter has not the well known value 3.14159 ... but exceeds this value, the difference being greater and greater as the peripheral speed approaches that of light. That is, the laws of ordinary geometry no longer hold true.

However, we know other cases in which the ordinary or Euclidean geometry is not applicable. Thus suppose that on the surface of a sphere we describe a series of concentric circles. Since the surface is curved, we are not surprised at finding that the circumference of any one of these circles is less than 3.14159 ... times the distance across the circle as measured on the surface of the sphere. What this means, therefore, is that we cannot use Euclidean geometry to describe measurements on the surface of a sphere, and every schoolboy knows this from comparing Mercator's projection of the earth's surface with the actual representation on a globe.

When we come to think of it, the reason we realize all this is because our sense of three dimensions enables us to differentiate flat surfaces from those that are curved. Let us, however, imagine a two-dimensional being living on the surface of a large sphere. So long as his measurements are confined to relatively small areas he will find it possible to describe all his measurements in terms of Euclidean geometry. As, however, his area of operation increases he will begin to observe greater and greater discrepancies. Being unfamiliar with the existence of such a three-dimensional object as a sphere, and therefore not realizing that he is on the surface of one, our intelligent two-dimensional being will conclude that the disturbance in his geometry is due to the action of a force, and by means of plausible assumptions on the "law" of this force he will reconcile his observations with the laws of plane geometry.

Now since an acceleration in a gravitational field is identical with that due to centrifugal force produced by rotation, we concluded that the geometry in a gravitational field must also be non-Euclidean. That is, space in the neighborhood of matter is distorted or curved. The curvature of space bears the same relation to three dimensions that the curvature of a spherical surface bears to two dimensions, and that is why we do not perceive it, any more than the intelligent two-dimensional being would be aware of the distortion of his space (or surface). Furthermore, like this being, we have assumed the existence of a gravitational force to account for discrepancies in our geometrical measurements.

The identification in this manner of gravitational effects with geometrical curvature of space enables Einstein to derive a general law for the path of any particle in a gravitational field, with respect both to space and to time. Furthermore, the law expresses this motion in terms which are independent of the relative motion and position of the observer, and satisfies the condition that the fundamental laws of physics be equally valid for all observers. The solution of the problem involved the use of a new kind of higher calculus, elaborated by two Italian mathematicians, Ricci and Levi-Civita. The result is a law of motion which is extremely general in its validity.

For low velocities it approximates to Newton's solution, and in the absence of a gravitational field it leads to the same conclusions as the special theory of relativity. There are three deductions from this law which have aroused a great deal of interest, and the confirmation of two of these by actual observation must be regarded as striking proof of Einstein's theory.

XIII

AN INTRODUCTION TO RELATIVITY

A Treatment in Which the Mathematical Connections of Einstein's Work are Brought Out More Strongly and More Successfully Than Usual in a Popular Explanation

BY HAROLD T. DAVIS, UNIVERSITY OF WISCONSIN, Madison, Wis.

One of the first questions which appears in philosophy is this: What is the great reality that underlies space and time and the phenomena of the physical universe? Kant, the philosopher, dismissed it as a subjective problem, affirming that space and time are "a priori" concepts beyond which we can say no more.

Then the world came upon some startling facts. In 1905 a paper appeared by Professor Albert Einstein which asserted that the explanation of certain remarkable discoveries in physics gave us a new conception of this strange four-dimensional manifold in which we live. Thus, the great difference between the space and time of philosophy and the new knowledge is the objective reality of the latter. It rests upon an amazing sequence of physical facts, and the generalized theory, which appeared several years later, founded as it is upon the abstruse differential calculus of Riemann, Christoffel, Ricci and Levi-Civita, emerges from its maze of formulas with the prediction of real phenomena to be sought for the in the world of facts.

We shall, therefore, approach the subject from this objective point of view. Let us go to the realm of actual physical events and see how the ideas of relativity gradually unfolded themselves from the first crude wonderings of science to the stately researches that first discovered the great ocean of ether and then penetrated in such a marvelous manner into some of its most mysterious properties.

THE ELECTROMAGNETIC THEORY OF LIGHT

Suppose that we go out on a summer night and look into the dark depths of the sky. A thousand bright specks are flashing there, blue, red, yellow against the dark velvet of space. And as we look we must all be impressed by the fact that such remote objects as the stars can be known to us at all. How is it that light, that curious thing which falls upon the optic nerve and transmits its pictures to the brain, can ever reach us through the black regions of interstellar space? That is the question which has for its answer the electromagnetic theory of light.

The first theory to be advanced was Newton's "corpuscular" theory which supposed that the stars are sending off into space little pellets of matter so infinitesimally small that they can move at the rate of 186,000 miles a second without injuring even so delicate a thing as the eye when they strike against it.

But in 1801, when Thomas Young made the very important discovery of interference, this had to give way to the wave theory, first proposed by Huyghens in the 17th century. The first great deduction from this, of course, was the "luminiferous ether," because a wave without some medium for its propagation was quite unthinkable. Certain peculiar properties of the ether were at once evident, since we deduce that it must fill all space and at the same time be so extremely tenuous that it will not retard to any noticeable degree the motion through it of material bodies like the planets.

But how light was propagated through the ether still remained a perplexing problem and various theories were proposed, most prominent among them being the "elastic solid" theory which tried to ascribe to ether the properties of an elastic body. This theory, however, laid itself open to serious objection on the ground that no longitudinal waves had been detected in the ether, so that it began to appear that further insight into the nature of light had to be sought for in another direction.

This was soon forthcoming for in 1864 a new theory was proposed by James Clerk Maxwell which seemed to solve all of the difficulties. Maxwell had been working with the facts derived from a study of electrical and magnetic phenomena and had shown that electromagnetic disturbances were propagated through the ether at a velocity identical with that of light. This, of course, might have been merely a strange coincidence, but Maxwell went further and demonstrated the interesting fact that an oscillating electric charge should give rise to a wave that would behave in a manner identical with all of the known properties of a light wave. One particularly impressive assertion was that these waves, consisting of an alternating electric field accompanied by an alternating magnetic field at right angles to it, and hence called electromagnetic waves, would advance in a direction perpendicular to the alternating fields. This satisfied the first essential property of light rays, i.e., that they must be transverse waves, and the ease with which it explained all of the fundamental phenomena of optics and predicted a most striking interrelation between the electrical and optical properties of material bodies, gave it at once a prominent place among the various theories.

The electromagnetic theory, however, had to wait until 1888 for verification when Heinrich Hertz, in a series of brilliant experiments, succeeded in producing electromagnetic waves in the laboratory and in showing that they possessed all of the properties predicted by Maxwell. These waves moved with the velocity of light: they could be reflected, refracted, and polarized: they exhibited the phenomenon of interference and, in short, could not be distinguished from light waves except for their difference in wave length.

THE MICHELSON-MORLEY EXPERIMENT

With the final establishment of the electromagnetic theory of light as a fact of physics, we have at last endowed the ether with an actual substantiality. The "empty void" is no longer empty, but a great ocean of ether through which the planets and the suns turn without ever being aware that it is there.

In 1881 A. A. Michelson undertook an experiment, originally suggested by Maxwell, to determine the relative motion of our earth to the ether ocean and six years later he repeated it with the assistance of E. W. Morley. The experiment is now known as the Michelson-Morley experiment and since it is the great physical fact upon which the theory of relativity rests, it will be well for us to examine it in detail.

Since we can scarcely think that our earth is privileged in the universe and that it is at rest with respect to this great ether ocean that fills space, we propose to discover how fast we are actually moving. But the startling fact is that the experiment devised for this purpose failed to detect any motion whatever of the earth relative to the ether. [3]

The explanation of this very curious fact was given by both H. A. Lorentz and G. F. Fitzgerald in what is now widely known under the name of the "contraction hypothesis." It is nothing more nor less than this:

Every solid body undergoes a slight change in dimensions, of the order of ($v^2/c^2$), when it moves with a velocity $v$ through the ether.

The reason why the experiment failed, then, was not because the earth was not moving through the ether, but because the instruments with which the experiment was being conducted had shrunk just enough to negative the effect that was being looked for. [4]

THE LORENTZ TRANSFORMATION

We can not at this point forebear introducing a little mathematics to further emphasize the theory and the very logical nature of this contraction hypothesis.

Let us suppose that we were on a world that was absolutely motionless with respect to the ether and were looking at a ray of light. The magnetic and electric fields which form the ray can be described by means of four mathematical expressions which have come to bear the name of "Maxwell's field equations." Now suppose that we ask ourselves the question: How must these equations be changed so that they will apply to a ray of light which is being observed by people on a world that is moving with a velocity v through the ether?

The answer is immediate. From the Michelson-Morley experiment we know that we can not tell how fast or how slowly we are moving with respect to the ether. This means that no matter what world we may be upon, the form of the Maxwell field equations will always be the same, even though the second set of axes (or frame of reference) may be moving with high velocity with respect to the first.

Starting from this hypothesis (called in technical language the covariance of the equations with respect to a transformation of coordinates), Lorentz found that the transformation which leaves the field equations unchanged in form was the following:

$$x' = k(x - vt), y' = y, z' = z, t' = k(t - vx/c)$$

where $k$ is as on page 92.

And what, now, can be deduced from these very simple looking equations? In the first place we see that the space of $x'$, $y'$, $z'$, $t'$ is not our ordinary concept of space at all, but a space in which time is all tangled up with length. To put it more concretely, we may deduce from them the interesting fact that whenever an aviator moves with respect to our earth, his shape changes, and if he were to compare his watch with one on the earth, he would find that his time had changed also. A sphere would flatten into an ellipse, a meter stick would shorten up, a watch would slow down and all because, as H. Minkowski has shown us from these very equations, we are really living in a physical world quite different from the world of Euclid's geometry in which we are accustomed to think we live.

A variety of objections has very naturally been made to this rather radical hypothesis in an attempt to discredit the entire theory, but it is easily seen that any result obtained through the field equations must necessarily be in conformity with the theory of contraction, since this theory is only the physical interpretation of that transformation which leaves the field equations unaltered. Indeed, it is even possible to postulate the Lorentz transformation together with the assumption that each element of charge is a center of uniformly diverging tubes of strain and derive the Maxwell field equations from this, which shows from another point of view the truly fundamental nature of the transformation.

THE FIRST THEORY OF RELATIVITY

The whole question of the ether had arrived at this very interesting point when Professor Einstein in 1905 stated the theory of relativity. He had noticed that the equations of dynamics as formulated by Newton did not admit the Lorentz transformation, but only the simple Galilean transformation:

$$x' = x - vt, y' = y, z' = z, t' = t\,.$$

Here, indeed, was a curious situation. Two physical principles, that of dynamics and that of electromagnetism, were coexistent and yet each one admitted a different transformation when the system of reference was transferred to axes moving with constant velocity with respect to the ether.

Now the electromagnetic equations and their transformation had been shown to be in accord with experimental fact, whereas it had long been felt that Newton's equations were only a first approximation to the truth. For example, the elliptic orbit of a planet had been observed by Leverrier to exhibit a disquieting tendency to rotate in the direction of motion. This precession, which in the case of Mercury was as large as 43'' per century, could not be accounted for in any way by the ordinary Newtonian laws and was, consequently, a very celebrated case of discordance in gravitational astronomy.

With this example clearly before him, Einstein took the great step and said that the laws of dynamics and all other physical laws had to be remade so that they, also, admit the Lorentz transformation. That is to say,

The laws of physical phenomena, or rather the mathematical expressions for these laws, are covariant (unchanged in form) when we apply the Lorentz transformation to them.

The deductions from the Michelson-Morley experiment now seem to have reached their ultimate conclusion.

One discordant fact in this new theory remained, however. That same precession of the perihelion of Mercury which had first lead Einstein to his theory remained unsettled. When the new approximations were applied to the formula of orbital motion, a precession was, indeed, obtained, but the computed value fell considerably below that of the observed 43'' per century.

THE INCLUSION OF GRAVITATION

With the idea of investigating the problem from the very bottom, Einstein now undertook a broader and more daring point of view. In the first place he said that there is no apparent reason in the great scheme of world events why any one special system of coordinates should be fundamental to the description of phenomena, just as in the special theory a ray of light would appear the same whether viewed from a fixed system or a system moving with constant velocity with respect to the ether. This makes the very broad assumption that no matter what system of coordinates we may use, the mathematical expressions for the laws of nature must be the same. In Einstein's own words, then, the first principle of this more general theory of relativity must be the following:

"The general laws of nature are expressed through equations which hold for all systems of coordinates, that is, they are covariant with respect to arbitrary substitutions." [5]

But this was not enough to include gravitation so Einstein next formulated what he was pleased to call his "equivalence hypothesis." This is best illustrated by an example. Suppose that we are mounting in an elevator and wish to investigate the world of events from our moving platform. We mount more and more rapidly, that is with constant acceleration, and we appear to be in a strong gravitational field due to our own inertia. Suppose, on the other hand, that the elevator descends with an acceleration equal to that of gravity. We would now feel certain that we were in empty space because our own relative acceleration has entirely destroyed that of the earth's gravitational field and all objects placed upon scales in an elevator would apparently be without weight.

Applying this idea, then, Einstein decided to do away with gravitation entirely by referring all events in a gravitational field to a new set of axes which should move with constant acceleration with respect to the first. In other words we are going to deal with a system moving with uniform acceleration with respect to the ether, just as we considered a system moving with uniform velocity in the special theory.

The next step in the construction of this complicated theory is to reduce these two hypotheses to the language of mathematics and this was accomplished by Einstein with the help of M. Grossmann by means of the theory of tensors.

On account of the very great intricacy of the details, we must content ourselves with the mere statement that this really involved the generalization of the famous expressions known as Laplace's and Poisson's equations, on the explicit assumption that these two equations would still describe the gravitational field when we are content to use a first approximation to the truth. The set of ten differential equations which Einstein got as a result of his generalization he called his field equations of gravitation. [6]

XIV

NEW CONCEPTS FOR OLD

What the World Looks Like After Einstein Has Had His Way With It

BY JOHN G. McHARDY, COMMANDER R.N., LONDON

"The new-created world, which fame in heaven
Long had foretold, a fabric wonderful,
Of absolute perfection."

Einstein's Theory of Relativity has led to determining a key law of nature--the law of gravitation--which is also the basic law of mechanics. Thus it embraces a whole realm of physics, and promises, through the researches of Professor Weyl, to embrace another realm--electro-dynamics. Its limitations are not yet reached, for Einstein has already postulated therefrom a theory of a finite, yet unbounded, universe. This essay, however, is mainly concerned with mechanics, and electrical forces are not considered.

To have synthesised Newton's two great principles--his law of motion and law of gravitation--interpreting in the process the empirical law of equality of gravitational and inertial mass, is alone an immense achievement; but Einstein's researches have opened up a new world to the physicist and philosopher which is of greater importance. He has given us a vision of the immaterial world, a geometrical or mathematical vision, which is more satisfying than the "ether" conceptions hitherto presented. The fabric of his vision is not baseless. It is this fabric we shall consider, touching on certain aspects of the Einstein theory in the endeavor to present an image in miniature of his edifice of thought and to show the firmness of its foundations. That they are well and truly laid was demonstrated by the verification, from observations made during the solar eclipse in 1919, of Einstein's prediction of the displacement of a wave of light in a gravitational field, showing light to have the property of weight.

The physical world is shown by Einstein to be a world of "relations." Underlying it there is an absolute world of which physical phenomena are the manifestation. "Give me matter and motion," says Descartes, "and I will construct the world." "Give me a world in which there are ordered relations," says the Relativist, "and I will show you the behavior of matter therein" (mechanics). We first view this underlying world as an abstraction, abstracting energy ("bound" as in matter and electrons, "free" as in light), and its attribute force. This abstraction we will call the "World-Frame." Later, we will study the underlying world in connection with energy, and will call this absolute world the "World-Fabric." The connection between the geometrical character of the World-Frame and the geometrical characters of the World-Fabric is the key to the law of gravitation.

THE WORLD-FRAME

This is our conception of a world, if such were possible, entirely free from the influence of energy. We may conceive of it as an amorphous immaterial something containing "point-events" (a point-event being an instant of time at a point in space--a conception, not a definition). These point-events have a fourfold order and definite relation in this Frame, i.e. they can be specified by four variables or coordinates in reference to some base called a reference system, with respect to which they are forward or backward, right or left, above or below, sooner or later. This shows the World-Frame to be four-dimensional. Thus an aggregate of point-events (or an "event," which implies limited extension in space and limited duration in time) [7] would have what we familiarly describe as length, breadth, height and time. To express these metrical properties most simply we must choose a four-dimensional reference system having a particular form--rectilinear axes (Cartesian coordinates), and a particular motion--uniform and rectilinear, i.e. unaccelerated, and non-rotating with respect to the path of a light ray. We call this an inertial system because Newton's Law of Inertia holds for such a system alone. This system indicates how observers partition the World-Frame into space and time. It restricts observers to uniform rectilinear motion, and observations to bodies and light-pulses in such motion. Thus gravitational and other forces are discounted, and we obtain World-Frame conditions notwithstanding the fact that observers are in the presence of energy.

Now the separation between point-events which have a definite relation to each other must be absolute. The separation between two points in a plane is defined by the unique distance between them (the straight line joining them). Between point-events the analogue of this unique distance, which we call the "separation-interval" (to indicate its time-like and space-like nature), is also unique. Its unique and absolute character give it great importance as thereby it is the same for all observers regardless of their reference system.

If, in place of the rather cumbersome expression $X-x$ to indicate the difference between the $x$-coordinates of two points, we employ the more compact expression $dx$; if for the benefit of readers who have a little algebra but no analysis we state explicitly that this expression is a single symbol for a single quantity, and has nothing to do with any product of two quantities $d$ and $x$; and if we extend this notation to all our coordinates: then it is clear from previous essays that the distance $S$ between two points in a plane referred to a rectilinear system $OX$, $OY$, is given by the simple equation $S^2 = (dx)^2 + (dy)^2$. Einstein and Minkowski show that the value for the separation interval $\Omega$, the analogue of $S$, referred to an inertial system is given by the equation

$$\Omega^2 = (dx)^2 + (dy)^2 + (dz)^2 - (dt)^2\,,$$

which is seen to be a modified extension to four dimensions of the equation for $S$. We must measure $t$ in the same units as $x$, $y$, $z$. By taking the constant velocity of light (300,000 kilometres per second) as unit velocity, we can measure in length or time indiscriminately. [8]

We will analyse briefly this equation as it epitomizes the Special Theory of Relativity. If the World-Frame had been Euclidean the equation would have been

$$\Omega^2 = (dx)^2 + (dy)^2 + (dz)^2 + (dt)^2$$

but this would not satisfy the "transformation equations" which resulted from the Special Theory. These transformation equations arose directly from a reconciliation between two observed facts; (a) the observed agreement of all natural phenomena with the "Restricted Principle of Relativity"--a principle which shows that absolute rectilinear motion cannot be established--(as regards mechanics this was recognized by Newton; the Michelson-Morley and other experiments showed this principle also applied to optical and electro-dynamical phenomena); and (b) the observed disagreement of optical and electro-dynamical phenomena (notably the constancy of light velocity) with the laws of dynamics as given by classical mechanics, e.g., in regard to the compounding of relative velocities. Einstein effected this reconciliation by detecting a flaw in classical mechanics. He showed that by regarding space and time measurements as relative to the observer--not absolute as Newton defined them--there was nothing incompatible between the Principle of Relativity and the laws of dynamics so modified. Newton's definitions were founded on conception. Einstein's recognition of the relativity of space and time is based on observation.

Equation (1) shows that the geometry of the World-Frame referred to an inertial system is semi-Euclidean (hyperbolic), and that space and time measurements are relative to the observer's inertial reference system. The equation shows that the World-Frame has a certain geometrical character which we distinguish as four-dimensional "flatness." It is everywhere alike (homaloidal). Its flat character is shown by the straight line nature of the separation-interval and of the system to which it is most simply referred.

Thus we have found two absolute features in the World-Frame--(1) Its geometrical character--"flatness"; (2) The separation-interval--which can be expressed in terms of measurable variables called space and time partitions, this partitioning being dependent on the observer's motion.

We are now in a position to explore the World-Fabric. Already we see that, studied under inertial conditions (free of force), it agrees with the World-Frame.

THE WORLD-FABRIC

The General Theory of relativity is largely concerned with the investigation of the World-Fabric. Consider the World-Frame to be disturbed. We may regard this disturbance, which manifests itself in physical phenomena, as energy, or more correctly "action."

When energy is thwarted in its natural flow, force is manifested, with which are associated non-uniform motions such as accelerations and rotations. This disturbed World-Frame we distinguish as the World-Fabric. It is found to have various non-Euclidean characters differing from the simple "flat" character of the World-Frame according to the degree of disturbance (action) in the region. Disturbance gives the fabric a geometrical character of "curvature"; the more considerable the disturbance, the greater the curvature. Thus an empty region (not containing energy, but under its influence) has less curvature than a region in which free energy abounds.

Our problem, after showing the relativity of force (especially gravitational force), is to determine the law underlying the fabric's geometrical character; to ascertain how the degree of curvature is related to the energy influencing a region, and how the curvature of one region is linked by differential equations to that of neighboring regions. Such a law will be seen to be the law of gravitation.

We study the World-Fabric by considering tracks on which material particles and light-pulses progress; we find such tracks regulated and defined by the Fabric's curvature, and not, as hitherto supposed, by attractive force inherent in matter. As a track is measurable by summing the separation-intervals between near-by point-events on it, all observers will agree which is the unique track between two distant point-events. Einstein postulates that freely progressing bodies will follow unique tracks, which are therefore called natural tracks (geodesics).

If material bodies are prevented from following natural tracks by contact with matter or other causes, the phenomenon of gravitational force is manifested relative to them. Whenever the natural flow of energy is interrupted force is born. For example, when the piston interrupts the flow of steam, or golf ball flow of club, force results--the interruption is mutual, and the force relative to both. Likewise when the earth interrupts the natural track of a particle (or observer) gravitational force is manifested relative to both.

So long as a body moves freely no force is appreciated by it. A falling aviator (neglecting air resistance) will not appreciate any gravitational force. He follows a natural track, thereby freeing himself from the force experienced in contact with matter. He acquires an accelerating motion with respect to an inertial system. By acquiring a particular accelerating motion an observer can annul any force experienced in any small region where the field of force can be considered constant.

Thus Einstein, interpreting the equality of gravitational and inertial mass, showed that the same quality manifests itself according to circumstances as "weight" or as inertia, and that all force is purely relative and may be treated as one phenomenon (an interruption in energy flow). This "Principle of Equivalence" shows that small portions of the World-Fabric, observed from a freely moving particle (free of force), could be treated as small portions of the World-Frame. [9]

If such observations were practicable, we could determine the Fabric curvature by referring point-event measurements to equation (1). We cannot observe from unique tracks but we can observe them from our restrained situation. Their importance is now apparent, because, by tracing them over a region, we are tracing something absolute in the Fabric--its geometrical character. We study this curvature by exploring separation-intervals on the tracks of freely moving bodies, relating these separation-intervals to actual measurements in terms of space and time components depending on the observer's reference system. The law of curvature must be the law of gravitation. To illustrate the lines on which Einstein proceeded to survey the World-Fabric from the earth we will consider a similar but more simple problem--the survey of the sea-surface curvature from an airship. We study this curvature by exploring small distances on the tracks of ships (which we must suppose can only move uniformly on unique tracks--arcs of great circles), relating such distances to actual measurements in terms of length and breadth components depending on the observer's reference system. This two-dimensional surface problem can be extended to the four-dimensional Fabric one.

We consider the surface to be covered by two arbitrarily drawn intersecting series of curves: curves in one series not intersecting each other, vide figure. This Gaussian system of coordinates is appropriate only when the smaller the surface considered, the more nearly it approximates to Euclidean conditions. It admits of defining any point on the surface by two numbers indicating the curves intersecting at that point. $P$ is defined by $x_1$, $x_2$. $P_1$ (very near $P$) is defined by $x_1 + dx_1$, $x_2 + dx_2$. The equation for the minute distance $s$ between two adjacent points in such a system is given by the general formula

$$s^2 = g_{11}d{x_1}^2 + g_{12}dx_1dx_2 + g_{22}d{x_2}^2\,.$$

The $g$'s may be constants or functions of $x_1$, $x_2$. Their value is dependent on the observer's reference system and on the geometrical character of the surface observed. The curves being arbitrary, the formula is appropriate for any reference system, or even if the observer does not know exactly what his reference system is. (The Fabric observer does not know what his space and time partitioning actually is because he is in a gravitational field). It is the $g$'s which disclose the geometry of an observer's partitions, and their values also contain a reflection of the character of the region observed.

We find $s$ by direct exploration with a moving ship ($\Omega$ is found by direct exploration with a freely moving particle); $dx_1$, $dx_2$ are the observed length and breadth measurement differences which we have to relate to $s$. By making sufficient observations in a small area and referring them to the general formula we can find the values of the $g$'s for the observer's particular reference system. Different values for $g$'s will be found if the observer changes his reference system, but there is a limitation to the values so obtainable owing to the part played by the surface itself, which is diffidently expressing its intrinsic geometrical character in the $g$'s in each observation.

EINSTEIN'S RESULTS

Thus we approach the absolute character of the surface through the relative nature of the observer's reference system. There is a relationship common to all values of the $g$'s that belong to the same curvature. This relationship is expressed by a differential equation. It is this equation of curvature that the airship's observer must find. Einstein's problem was similar, but he was concerned with four dimensions, which entailed a general formula with ten $g$'s, and he had to find a set of differential equations of the second order to determine the law of Fabric curvature. He divided the Fabric into regions: I. World-Frame--beyond influence of energy. II. Empty region--free of energy, but under its influence. III. Region containing free energy only. Each region has a characteristic curvature. By means of an absolute differential calculus--a wonderful mathematical scaffolding erected by Riemann, Christoffel and others--involving the theory of tensors, he succeeded in finding such a set of equations. He kept the following points in view: (1) The equations must not only give the character of region II, but must satisfy the special case of region I; (2) They must be independent of any partitioning system, because the General Theory of Relativity demands that a law of nature be in a form appropriate for all observers whatever their position and motion; (3) They must be concerned with energy which is conserved, not mass which the Special Theory showed dependent on velocity. This set of differential equations which shows how the curvature of the Fabric at any point links to the curvature at neighboring points is the law of gravitation, a law which has been severely tested by the practical observation of the solar eclipse already referred to. At a first approximation these equations degenerate into Newton's Law. At a second approximation they account for the motion of the perihelion of Mercury, which had hitherto baffled astronomers. All the laws of mechanics are deducible from this law of World-Fabric curvature, i.e. conservation of energy (which includes conservation of mass since we re-define mass as energy) and conservation of momentum (re-defined by a relativist). It must be noted that this law and the General Theory show that the velocity of light is not absolutely constant, but, like everything else, a light-pulse is affected by the Fabric curvature in a gravitational field. In conclusion we will contrast some conspicuous differences in the old world view of classical mechanics and the new view presented by Einstein.

1. A three-dimensional ether medium with variously conceived properties which communicated the supposed inherent attractive force in matter in some unexplained way, and transmitted electromagnetic waves, has been replaced by a four-dimensional external World-Fabric, the geometrical character of which controls the motion of matter (energy) and accounts for all mechanical laws.

2. After separating the observer's subjective share in definitions from nature's share in the things defined, space, time, and force, hitherto regarded as absolute, have been shown to be purely relative and dependent on the observer's track. Mass has also proved to be relative to velocity unless re-defined as energy. As classical mechanics bases all definitions on space, time, and mass units, the relativity of such defined quantities is now apparent.

3. Newton's laws of motion, his law of gravitation, and the laws of conservation, hitherto regarded as unrelated, are now synthesised in a basic law of mechanics.

Einstein has not disturbed the electric theory of matter, and both the old and new physics have in common the "Principle of Least Action." We obtain a glimpse of this principle in the unique tracks pursued by freely moving bodies, which may be regarded as tracks of least effort, force only being manifested as an expression of the Fabric's resentment when bodies depart from these natural tracks. Einstein has approached nearer to the truth in regard to the laws underlying nature, and, as always, this means a simplification. His theory, which entails a readjustment of such fundamental conceptions as space and time, opens up fresh fields to scientific investigation and to philosophic thought. It reveals a bridge uniting the domains of physics and philosophy, and it heralds a new era in the history of science.

XV

THE NEW WORLD

A Universe in Which Geometry Takes the Place of Physics, and Curvature That of Force

BY GEORGE FREDERICK HEMENS, M.C., B.SC., LONDON

It is familiar knowledge that the line, the surface and ordinary Euclidean space are to be regarded as spaces of one, two and three dimensions respectively and readers of this journal are aware that a hypothetical space of four dimensions has been closely investigated. The most convenient space to study is the surface or two-space, since we can regard it as embedded in a three-space. If a surface is curved it is generally impossible to draw a straight line on it, for as we see clearly, the "straightest" line is changing its direction at every point. To describe this property accurately it is necessary to ascribe to each point a magnitude which expresses what happens to the direction of a short line in the region when displaced a short distance parallel to itself. This is called the direction-defining magnitude. Different sets of values of this magnitude relate to surfaces of different curvatures.

A second fundamental property has recently been pointed out. There is inherent in every part of a space a measure of length peculiar to that particular region and which in general varies from region to region. To describe this variation accurately it is necessary to ascribe to each point another magnitude called the length-defining magnitude, which expresses the change from each point to the next of the unit of length. These two magnitudes define the surface completely.

Similarly, a space of any number of dimensions is defined completely by a similar pair of magnitudes. A space is the "field" of such a magnitude-pair and the nature of these magnitudes defines the dimensions of the space. The four-space usually described is the Euclidean member of an infinity of four-spaces.

When we look into a mirror we see a space differing from ordinary space in that right and left are interchanged and this is described mathematically by saying that if we locate points as usual by specifying three distances $X_1$, $X_2$, $X_3$ of the point from three mutually perpendicular planes, then a point $X_1$, $X_2$, $X_3$, in actual space corresponds with a point $X_1$, $X_2$, $ -X_3$ in the mirrored space: in other words the mirrored space is derived from the real space by multiplying the $X_3$ coordinates by $-1$. If we were to multiply by $\sqrt{-1}$ instead of $-1$ we should derive a different space; in this case, however, we have no mirror to show us what it looks like. Such a space is said to have one negative dimension and it has the peculiar property that in the figure derived from the right triangle of ordinary space the square of the "hypotenuse" equals the difference and not the sum of the squares of the other two sides, so that the length of a line may sometimes have to be represented by the square-root of a negative number, a "complex" number.

In considering what at first sight may appear to be fantastic statements made by this theory, it must be borne in mind that all our knowledge of the external universe comes through our sense-impressions, and our most confident statements about external things are really of the nature of inferences from these sense-impressions and, being inferences, liable to be wrong. So that if the theory says that a stone lying on the ground is not a simple three-dimensional object, and that its substance is not the same as its substance a moment before, the matter is one for due consideration and not immediate disbelief.

The idea that the universe extends in time as well as in space is not new, and fiction-writers have familiarized us with wonderful machines in which travellers journey in time and are present at various stages of the world's history. This conception of the universe, to which the name "space-time" is usually applied, is adopted by the new theory and assigned the status of a physical reality.

THE WORLD GEOMETRY

The fundamental creed of the new theory is that the space-time universe constitutes a true four-dimensional space of one negative dimension, this dimension being time. The variations from point to point of the direction-defining and length-defining magnitudes generate the geometrical properties of curvature, etc., and these are cognised by the human mind as physical phenomena: our sense-impressions are nothing more nor less than perceptions of the geometry of a fourspace. So instead of inferring from our sense-impressions the existence of matter, motion and the like as we are accustomed to do, we should with equal justice infer the existence of a geometrical fourspace. Thus it becomes necessary to prepare a dictionary in which the familiar things of our world are identified with those geometrical properties of the four-space which really constitute them, and in so doing parts of our geometrical knowledge assume the guise of new physical knowledge.

Through the fourspace our consciousness travels, cognising a changing three-dimensional section of it as it goes and thus giving rise to time. It becomes aware that the fourspace is pleated or folded along lines all running roughly in the same direction, and possibly because this is the easiest direction to follow, it travels along the lines. The direction of this motion is the negative dimension. Thus consciousness is always aware of the nearly constant forms of the cross-sections of the pleats along which it travels. These unvarying forms constitute matter: matter is the form of a section through a uniform pleat of the fourspace--a three-dimensional aspect of a four-dimensional curvature; so that in strict accuracy we should say that a stone is the shape or form of a changing section of a four-dimensional object, the complete object being a long fold in the fourspace. The physical interpretation of this conservation of form of the cross-section is that matter is conserved. It is thus seen that the conscious mind, by following these pleats, has so determined time that the law of the conservation of matter must hold. The mathematical treatment of the subject makes it clear that practically all other physical laws similarly follow as a direct result of this choice of time. The type of order prevailing in the physical universe, the laws of gravitation, heat, motion and the rest are not directly imposed by some external power, but are apparently chosen by mind itself.

In the neighborhood of these pleats the fourspace is still curved, but to a smaller degree. This we cognise as energy or as a field of force. Thus energy is seen to be the same kind of thing as matter and would therefore be expected to have weight. This was experimentally demonstrated in 1919 when light was in effect actually weighed. Conversely, matter consists of energy; and it is calculated that one liter of water contains sufficient energy to develop a million horsepower for about four years. It is now believed that the sun's energy is derived from the disintegration of the matter of which it is made.

The method of establishing these identifications will be clear from the following: We already knew that matter is made up of electrons and that radiant energy is electromagnetic and before the advent of this theory it was regarded as certain that practically all observed physical phenomena except gravitation were manifestations of the electromagnetic field. The new theory has confirmed this belief. It is found that the gravitational and electromagnetic conditions of the universe are completely defined if to each point of space-time a gravitational and an electric potential are ascribed. These are magnitudes of the same nature as the direction-defining and length-defining magnitudes which must necessarily be associated with every point of space-time if it is a true "space," and they are therefore identified with these. By performing ordinary mathematical operations on these magnitudes statements of fact clothed in mathematical form are obtained, which are to be interpreted on the one hand as physical laws and on the other as geometrical properties of the fourspace. Nearly all our physical laws are derivable mathematically in this way, so that an extensive identification is effected which has been fruitful of results.

It has been mentioned that a slight curvature is sometimes cognised as force and as this identification appeared originally as a postulate its history is interesting.

THE GENESIS OF THE THEORY

An experiment by Michelson and Morley (1887), on which the whole theory is based, made it appear that if a man measures the velocity at which light passes him he will get the same result whether he is stationary, rushing to meet the light, or moving in the same direction as the light. The solution was provided by Einstein in 1905. He suggested that since we know the results of these determinations ought not to agree, something must have happened to the clocks and measuring-rods used in measuring the velocity so that the standards of length and time were not the same in the three cases, the alterations being exactly such as to make the velocity of light constant. This solution is universally accepted as true and is the fundamental postulate. Thus the length of a stick and the rate at which time passes will change as the velocity of the person observing these things changes. If a man measured the length of an aeroplane going past him at 161,000 miles per second it would measure only half the length observed when stationary. If the aeroplane were going with the velocity of light, its length would vanish though its breadth and height would be unaltered. Similarly, if of two twin brothers one were continually moving with reference to the other their ages would gradually diverge, for time would go at different rates for the two. If one moved with the velocity of light, time would stand still for him while for the other it would go on as usual. To get actually younger it would be necessary to move quicker than light which is believed to be impossible. The velocity of light is assumed to be the greatest velocity occurring in nature.