To describe the phenomena and derive laws from them, we locate them in space and time. To do this we use geometry. Here it is that the part contributed by the observer comes in. There are an infinite number of geometries, and a priori there seems to be no reason to choose one rather than the other. Taking geometry of two dimensions as an example, we can draw figures on a piece of paper, and discuss their properties, and we can also do so on the shell of an egg. But we cannot draw the same figures on the egg as on the paper. The ones will be distorted as compared with the others: the two surfaces have a different geometry. Similarly it is not possible to draw an accurate map of the earth on a sheet of paper, because the earth is spherical and its representation on the flat paper is always more or less distorted. The earth requires spherical geometry, which differs from the flat, or Euclidean, geometry of the paper.

Up to a few years ago Euclidean (i.e. flat) geometry of three dimensions had been exclusively used in physical theories. Why? Because it is the true one, is the one answer generally given. Now a statement about facts can be true or false, but a mathematical discipline is neither true nor false; it can only be correct—i.e. consistent in itself—or incorrect, and of course it always is correct. The assertion that a certain geometry is the “true” one can thus only mean, that it is the geometry of “true” space, and this again, if it is to have any meaning at all, can only mean that it corresponds to the physical “reality.” Leaving aside the question whether this reality has any geometry at all, we are confronted with the more immediately practical consideration how we shall verify the asserted correspondence. There is no other way than by comparing the conclusions derived from the laws based upon our geometry, with observations. It thus appears that the only justification for the use of the Euclidean geometry is its success in enabling us to “draw an accurate map” of the world. As soon as any other geometry is found to be more successful, that other must be used in physical theories, and we may, if we like, call it the “true” one.

Accurate observations always consist of measures, determining the position of material bodies in space. But the positions change, and for a complete description we also require measures of time. An important remark must be made here. Nobody has ever measured a pure space-distance, nor a pure lapse of time. The only thing that can be measured is the distance from a body at a certain point of space and a certain moment of time, to a body (either the same or another) at another point and another time. We can even go further and say that time cannot be measured at all. We profess to measure it by clocks. But a clock really measures space, and we derive the time from its space-measures by a fixed rule. This rule depends on the laws of motion of the mechanism of the clock. Thus finally time is defined by these laws. This is so, whether as a “clock” we use an ordinary chronometer, or the rotating earth, or an atom emitting light-waves, or anything else that may be suggested. The physical laws, of course, must be so adjusted that all these devices give the same time. About the reality of time, if it has any, we know nothing. All we know about time is that we want it. We cannot adequately describe nature with the three space-coordinates alone, we require a fourth one, which we call time. We might thus say with some reason that the physical world has four dimensions. But so long as it was found possible adequately to describe all known phenomena by a space of three dimensions and an independent time, the statement did not convey any very important information. Only after it had been found out that the space-coordinates and the time are not independent, did it acquire a real meaning.

As is well known the observation by which this was found out is the famous experiment of Michelson and Morley. It led to the “special” theory of relativity, which is the one referred to by Minkowski in 1908. In it a geometry of four dimensions is used, not a mere combination of a three-dimensional space and a one-dimensional time, but a continuum of truly fourfold order. This time-space is not Euclidean, since the time-component and the three space-components are not on the same footing, but its fundamental formula has a great resemblance to that of Euclidean geometry. We may call it “pseudo-Euclidean.”

This theory, which we need not explain here, was very satisfactory so far as the laws of electromagnetism, and especially the propagation of light, were concerned, but it did not include gravitation, and mechanics generally. We then had this curious state of affairs, that physicists actually believed in two different “realities.” When they were thinking of light they believed in Minkowski’s time-space; when they were thinking of gravitation they believed in the old Euclidean space and independent time. This, of course, could not last. Attempts were made so to alter Newton’s law of gravitation that it would fit into the four-dimensional world of the special relativity-theory, but these only succeeded in making the law, which had been a model of simplicity, extremely complicated, and, what was worse, it became ambiguous.

It is Einstein’s great merit to have perceived that gravitation is of such fundamental importance, that it must not be fitted into a ready-made theory, but must be woven into the space-time geometry from the beginning. And that he not only saw the necessity of doing this, but actually did it.

To see the necessity we must go back to Newton’s system of mechanics. Newton did two things (amongst others). He canonised Galileo’s system of mechanics into his famous “laws of motion,” the most important of which is the law of inertia, which says that:

a body, that is not interfered with, moves in a straight line with constant velocity.

The velocity, of course, can be nil, and the body at rest. This is a perfectly general law, the same for all material bodies, whatever their physical or chemical status. Newton took good care exactly to define what he meant by uniform motion in a straight line, and for this purpose he introduced the absolute Euclidean space and absolute time as an essential part of his system of laws at the very beginning of his great work. The other thing Newton did was to formulate the law of gravitation. Gravitation was in his system considered as an interference with the free, or inertial, motion of bodies, and accordingly required a law of its own.

But gravitation has this in common with inertia, and in this it differs from all other interferences, that it is perfectly general. All material bodies are equally subjected to it, whatever their physical or chemical status may be. But there is more. Gravitation and inertia are actually indistinguishable from each other, and are measured by the same number: the “mass”. This was already remarked by Newton himself, and from his point of view it was a most wonderful accidental coincidence. If an apple falls from the tree, that which makes it fall is its weight, which is the gravitational attraction by the earth, diminished by the centrifugal force due to the earth’s rotation and the apple’s inertia. In Newton’s system the gravitational attraction is a “real” force, whereas the centrifugal force is only “fictitious”. But the one is as real as the other. The most refined experiments, already begun by Newton himself, have not succeeded in distinguishing between them. Their identity is actually one of the best established facts in experimental physics. From this identity of “fictitious,” or inertial, and “real,” or gravitational, forces it follows that locally a gravitational field can be artificially created or destroyed. Thus inside a closed room which is falling freely, say a lift of which the cable has been broken, bodies have no weight: a balance could be in equilibrium with different weights in the two scales.

Having thus come to the conclusion that gravitation is not an interference, but is identical with inertia, we are tempted to restate the law of motion, so as to include both, thus:

Bodies which are not interfered with—do not move in straight lines, but—fall.

Now this is exactly what Einstein did. Only the “falling” of course requires a precise mathematical definition (like the uniform motion in a straight line), and the whole gist of his theory is the finding of that definition. In our earthly experience the falling never lasts long, very soon something—the floor of the room, or the earth itself—interferes. But in free space bodies go on falling forever. The motion of the planets is, in fact, adequately described as falling, since it consists in nothing else but obeying Newton’s law of gravitation together with his law of inertia. A body very far removed from all other matter is not subjected to gravitation, consequently it falls with constant velocity in a straight line according to the law of inertia. The problem was thus to find a mathematical definition of “falling,” which would embrace the uniform straight-line motion very far from all matter as well as the complex paths of the planets around the sun, and of an apple or a cannon-ball on earth.

For the definition of the uniform rectilinear motion of pure inertia Newton’s Euclidean space and independent time were sufficient. For the much more complicated falling under the influence of gravitation and inertia together, evidently a more complicated geometry would be needed. Minkowski’s pseudo-Euclidean time-space also was insufficient. Einstein accordingly introduced a general non-Euclidean four-dimensional time-space, and enunciated his law of motion thus:

Bodies which are not interfered with move in geodesics.

A geodesic in curved space is exactly the same thing as a straight line in flat space. We only call it by its technical name, because the name “straight line” would remind us too much of the old Euclidean space. If the curvature gets very small, or zero, the geodesic becomes very nearly, or exactly, a straight line.

The problem has now become to assign to time-space such curvatures that the geodesics will exactly represent the tracks of falling bodies. Space of two dimensions can just be flat, like a sheet of paper, or curved, like an egg. But in geometry of four dimensions there are several steps from perfect flatness, or “pseudo-flatness,” to complete curvature. Now the law governing the curvature of Einstein’s time-space, i.e., the law of gravitation, is simply that it can never, outside matter, be curved more than just one step beyond perfect (pseudo-)flatness.

Since I have promised not to use any mathematics I can hardly convey to the reader an adequate idea of the difficulty of the problem, nor do justice to the elegance and beauty of the solution. It is, in fact, little short of miraculous that this solution, which was only adopted by Einstein because it was the simplest he could find, does so exactly coincide in all its effects with Newton’s law. Thus the remarkably accurate experimental verification of this law can at once be transferred to the new law. In only one instance do the two laws differ so much that the difference can be observed, and in this case the observations confirm the new law exactly. This is the well known case of the motion of the perihelion of Mercury, whose disagreement with Newton’s law had puzzled astronomers for more than half a century.

Since Einstein’s time-space includes Minkowski’s as a particular case, it can do all that the other was designed to do for electro-magnetism and light. But it does more. The track of a pulse of light is also a geodesic, and time-space being curved in the neighborhood of matter, rays of light are no longer straight lines. A ray of light from a star, passing near the sun, will be bent round, and the star consequently will be seen in a different direction from where it would be seen if the sun had not been so nearly in the way. This has been verified by the observations of the eclipse of the sun of 1919 of May 29.

There is one other new phenomenon predicted by the theory, which falls within the reach of observation with our present means. Gravitation chiefly affects the time-component of the four-dimensional continuum, in such a way that natural clocks appear to run slower in a strong gravitational field than in a weak one. Thus, if we make the hypothesis—which, though extremely probable, is still a hypothesis—that an atom emitting or absorbing light-waves is a natural clock, and the further hypothesis—still very probable, though less so than the former—that there is nothing to interfere with its perfect running, then an atom on the sun will give off light-waves of smaller frequency than a similar atom in a terrestrial laboratory emits. Opinions as yet differ as to whether this is confirmed or contradicted by observations.

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The great strength and the charm of Einstein’s theory do however not lie in verified predictions, nor in the explanation of small outstanding discrepancies, but in the complete attainment of its original aim: the identification of gravitation and inertia, and in the wide range of formerly apparently unconnected subjects which it embraces, and the broad view of nature which it affords.

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Outside matter, as has been explained, the law of gravitation restricts the curvature of time-space. Inside continuous matter the curvature can be of any arbitrary kind or amount; the law of gravitation then connects this curvature with measurable properties of the matter, such as density, velocity, stress, etc. Thus these properties define the curvature, or, if preferred, the curvature defines the properties of matter, i.e. matter itself.

From these definitions the laws of conservation of energy, and of conservation of momentum, can be deduced by a purely mathematical process. Thus these laws, which at one time used to be considered as the most fundamental ones of mechanics, now appear as simple corollaries from the law of gravitation. It must be pointed out that such things as length, velocity, energy, momentum, are not absolute, but relative, i.e. they are not attributes of the physical reality, but relations between this reality and the observer. Consequently the laws of conservation are not laws of the real world, like the law of gravitation, but of the observed phenomena. There is, however one law which, already before the days of relativity, had come to be considered as the most fundamental of all, viz: the principle of least action. Now action is absolute. Accordingly this principle retains its central position in Einstein’s theory. It is even more fundamental than the law of gravitation, since both this law, and the law of motion, can be derived from it. The principle of least action, so far as we can see at present, appears to be the law of the real world.