Imagine a man in a room out of which he cannot see. He notices that when he releases anything it falls to the floor with a constant acceleration. Further he observes that all his objects, independently of their chemical and physical properties, are affected in precisely the same way. Now, he previously has experimented with magnets, and has remarked that they attract certain bodies in essentially the same way that the things which he drops are “attracted” to whatever is beneath the floor. Having explained magnetic attraction in terms of “forces,” he makes his first hypothesis: (A) He and his room are in a strong “field of force,” which he designates gravitational. This force pulls all things downward with a constant acceleration. Here he notes a singular distinction between magnetic and gravitational “forces”: magnets attract only a few kinds of matter, notably iron; the novel “force,” if indeed a force at all, acts similarly upon all kinds of matter. He makes another hypothesis: (B) His room and he are being accelerated upward.

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Either (A) or (B) describes the facts perfectly. By no experiment can he discriminate between them. So he takes the great step, and formulates the Equivalence Hypothesis:

A gravitational field of force is precisely equivalent in its effects to an artificial field of force introduced by accelerating the framework of reference, so that in any small region it is impossible to distinguish between them by any experiment whatever.

Next reconsidering his magnetic “forces,” he extends the equivalence hypothesis to cover all manifestations of force: The effects attributed to forces of any kind whatever can be described equally well by saying that our reference frameworks are accelerated; and moreover there is possible no experiment which will discriminate between the descriptions.

If the accelerations are null, the frameworks are at rest or in uniform motion relatively to one another. This special case is the “restricted” principle of relativity, which asserts that it is impossible experimentally to detect a uniform motion through the ether. Being thus superfluous for descriptions of natural phenomena, the ether may be abandoned, at least temporarily. The older physics sought this absolute ether framework to which all motions could be unambiguously referred, and failed to find it. The most exacting experiments, notably that of Michelson-Morley, revealed no trace of the earth’s supposed motion through the ether. Fitzgerald accounted for the failure by assuming that such motion would remain undetected if every moving body contracted by an amount depending upon its velocity in the direction of motion. The contraction for ordinary velocities is imperceptible. Only when as in the case of the beta particles, the velocity is an appreciable fraction of the velocity of light, is the contraction revealed. This contraction follows immediately from Einstein’s generalization constructed upon the equivalence hypothesis and the restricted relativity principle. We shall see that the contraction inevitably follows from the actual geometry of the universe.1

Let us return for a moment to the moving ball. Four measures, three of distances and one of time, are required in specifying its position with reference to some framework at each point and at each instant. All of these measures can be summed up in one compendious statement—the equations of motion showed how in changing from our room to his accelerated auto we found a new summary, “transformed equations,” which seemed to indicate that the ball had traversed a strong, variable field of force. Is there then in the chaos of observational disagreements anything which is independent of all observers? There is, but it is hidden at the very heart of nature.

To exhibit this, we must recall a familiar proposition of geometry: the square on the longest side of a right-angled triangle is equal to the sum of the squares on the other two sides. It has long been known that from this alone all the metrical properties of Euclidean space—the space in which for 2,000 years we have imagined we were living—can be deduced. Metrical properties are those depending upon measurement. Now, in the geometry of any space, Euclidean or not, there is a single proposition of a similar sort which tells us how to find the most direct distance between any two points that are very close together. This small distance is expressed in terms of the two sets of distance measurements by which the end-points are located, just as two neighboring positions of our ball were located by two sets of four measurements each. We say by analogy that two consecutive positions of the ball are separated by a small interval of time-space. From the formula for the very small interval of time-space we can calculate mathematically all the metrical properties of the time and space in which measurements for the ball’s motion must be made. So in any geometry mathematical analysis predicts infallibly the truth about all facts depending upon measurements from the simple formula of the interval between neighboring points. Thus, on a sphere the sum of the angles of any triangle formed by arcs of great circles exceeds 180°, and this follows from the formula for the shortest (“geodesic”) distance between neighboring points on the spherical surface.

We saw that it takes four measurements, one for time and three for distances, to fix an elementary event, viz., the position of the centre of our ball at any instant. A system of all possible such sets of four measurements each, constitutes what mathematicians call a four-dimensional space. The study of the four-dimensional time-space geometry, once its shortest-distance proposition is known, reveals all those relations in nature which can be ascertained by measurements, that is, experimentally. We have then to find this indispensable proposition.

Imagine the path taken by a particle moving solely under the influence of gravitation. This being the simplest possible motion of an actual particle in the real world, it is natural to guess that its path will be such that the particle moves from one point of time-space to another by the most direct route. This in fact is verified by forming the equations of the free particle’s motion, which turn out to be precisely those that specify a geodesic (most direct line) joining the two points. On the (two-dimensional) surface of a sphere such a line is the position taken by a string stretched between two points on the surface, and this is the shortest distance on the surface between them. But in the time-space geometry we find a remarkable distinction: the interval between any two points of the path taken is the longest possible, and between any two points there is only one longest path. Translated into ordinary space and time this merely asserts that the time taken between any two points on the natural path is the longest possible.

Recall now that when the line-formula for any kind of space is known all the metrical properties of that space are completely determined, and combine with this what we have just found, namely, the equations of motion of a particle subject only to gravitation are the same equations as those which fix the line-formula for the four-dimensional time-space. Since gravitation alone determines the motion of the particle, and since this motion is completely described by the very equations which fix all the metrical properties of time-space, it follows that the metrical (experimentally determinable) properties of time-space are equivalent to those of gravitation, in the sense that each set of properties implies the other.

We have found the thing in nature which is independent of all observers, and it turns out to be the very structure of time-space itself. The motion of the free particle obviously is a thing unconditioned by accidents of observation; the particle under the influence of gravitation alone must go a way of its own. And if some observer in an artificial field of force produced by the acceleration of his reference framework describes the path as knotted, he merely is foisting eccentricities of his own motion upon the direct path of the particle. The conclusion is rational, for we believe that time-space exists independently of any man’s way of perceiving it.

Incidentally note that this space is that of the physical world. For only by measurements of distances and times can we become aware of our extension in time and space. If beyond this time-space geometry of measurements there is some “absolute geometry,” science can have no concern with it, for never can it be revealed by the one exploring device we possess—measurement.

We have followed a single particle. Let us now form a picture of several. Any event can be analyzed into a multitude of coincidences in time-space. For consider two moving particles—say electrons. If they collide they both are in very approximately one place at the same time. We imagine the path of an electron through time-space plotted by a line (in four-dimensional space), which will deviate from a “most direct” (geodesic) path if the electron is subjected to forces. This is the “world-line” of the electron. If the world lines of several electrons intersect at one point in time-space, the intersection pictures the fact of their coincidence somewhere and somewhen; for all their world-lines having a time-space point in common, at some instant they must have been in collision. Each point of a world-line pictures the position at a certain place at a certain time; and it is the intersections of world-lines which correspond to physical events. Of what lies between the intersections we have no experimental knowledge.

Imagine the world-lines of all the electrons in the universe threading time-space like threads in a jelly. The intersections of the tangle are a complete history of all physical events. Now distort the jelly. Clearly the mutual order of the intersections will be unchanged, but the distances between them will be shortened or lengthened. To a distortion of the jelly corresponds a special choice (by some observer) of a reference framework for describing the order of events. He cannot change the natural sequence of events. Again we have found something which is independent of all observers.

We can now recapitulate our conclusions and state the principle of relativity in its most general form.

(1) Observers describe events by measures of times and distances made with regard to their frameworks of reference.

(2) The complete history of any event is summarized in a set of equations giving the positions of all the particles involved at every instant.

(3) Two possibilities arise. (A) Either these equations are the same in form for all space-time reference frameworks, persisting formally unchanged for all shifts of the reference scheme; or (B), they subsist only when some special framework is used, altering their form as they are referred to different frameworks. If (B) holds, we naturally assume that the equations, and the phenomena which they profess to represent, owe their existence to some peculiarity of the reference framework. They do not, therefore, describe anything which is inherent in the nature of things, but merely some idiosyncrasy of the observer’s way of regarding nature. If (A) holds, then obviously the equations describe some real relation in nature which is independent of all possible ways of observing and recording it.

(4) In its most general form the principle of relativity states that those relations, and those alone, which persist unchanged in form for all possible space-time reference frameworks are the inherent laws of nature.

To find such relations Einstein has applied a mathematical method of great power—the calculus of tensors—with extraordinary success. This calculus threshes out the laws of nature, separating the observer’s eccentricities from what is independent of him, with the superb efficiency of a modern harvester. The residue is a physical geometry—or geometrical physics—of time-space, in which it appears that the times and spaces contributed by the several observers’ reference frameworks are shadows of their own contrivings; while the real, enduring universe is a fourfold order of time and space indissolubly bound together. One observer separates this time-space into his own “time” and “space” in one way, determined by his path through the world of events; another, moving relatively to the first, separates it differently, and what for one is time shades into space for another.

This time-space geometry is non-Euclidean. It is “warped” (curved), the amount of warping at any place being determined by the intensity of the gravitational field there. Thus again gravitation is rooted in the nature of things. In this sense it is not a force, but a property of space. Wherever there is matter there is a gravitational field, and hence a warping of space. Conversely, as long ago imagined by Clifford, wherever there is a warping of space, there is matter; and matter is resolved ultimately into wrinkles in time-space.

To visualize a warped space, consider a simple analogy. A man walks away from a polished globe; his image recedes into the mirror-space, shortening and thinning as it goes, and thinning (in the direction of motion) faster than it shortens. Everything around him experiences a like effect. If he tries to discover this by a footrule it automatically shortens faster as he turns it into the horizontal position, so his purpose eludes him. The mirror-space is warped in the direction of the image’s motion. So is our own. For all bodies, as evidenced by the Fitzgerald contraction, shorten in the direction of motion. And just as the image can never penetrate the mirror-space a greater distance than half its radius, so probably time-space is curved in such a way that our universe, like the surface of a sphere, is finite in extent, but unbounded.