Gravitation and Space-Time
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.
XI
THE PRINCIPLE OF GENERAL RELATIVITY
How Einstein, to a Degree Never Before Equalled, Isolates the External Reality from the Observer’s Contribution
BY E. T. BELL
UNIVERSITY OF WASHINGTON
SEATTLE
Einstein’s general relativity is of such vast compass, being coextensive with the realm of physical events, that in any brief account a strict selection from its numerous aspects is prescribed. The old, restricted principle being contained in the general, we shall treat the latter, its close relations with gravitation, and the significance of both for our knowledge of space and time. The essence of Einstein’s generalization is its final disentanglement of that part of any physical event which is contributed by the observer from that which is inherent in the nature of things and independent of all observers.
The argument turns upon the fact that an observer must describe any event with reference to some framework from which he makes measurements of time and distance. Thus, suppose that at nine o’clock a ball is tossed across the room. At one second past nine the ball occupies a definite position which we can specify by giving the three distances from the centre of the ball to the north and west walls and the floor. In this way, refining our measurements, we can give a precise description of the entire motion of the ball. Our final description will consist of innumerable separate statements, each of which contains four numbers corresponding to four measurements, and of these one will be for time and three for distances at the time indicated.
Imagine now that a man in an automobile looks in and observes the moving ball. Suppose he records the motion. To do so, he must refer to a timepiece and some body of reference. Say he selects his wrist-watch, the floor of his auto and two sides meeting in a corner. Fancy that just as he begins his series of observations his auto starts bucking and the main-spring of his watch breaks, so that he must measure “seconds” by the crazy running-down of his watch, and distances with reference to the sides of his erratic auto. Despite these handicaps he completes a set of observations, each of which consists of a time measured by his mad watch and three distances reckoned from the sides of his bucking machine. Let us assume him to have been so absorbed in his experiment that he noticed neither the disorders of his watch nor the motion of his auto. He gives us his sets of measurements. We remark that his seconds are only small fractions of ours, also his norths and wests are badly mixed. If we interpret his sets in terms of our stationary walls and sober clock we find the curious paradox that the ball zigzagged across the room like an intoxicated bee. He obstinately argues that we know no more than he about how the ball actually moved. For we got a smooth description, he asserts, by choosing an artificially simple reference framework, having no necessary relations whatever to the ball. The crooked path plotted from his observations proves, he declares, that the ball was subject to varying forces of which we in the room suspected nothing. He contends that our room was being jarred by a system of forces which exactly compensated and smoothed out the real jaggedness of path observed by himself. But if we know all about his watch and auto we can easily apply necessary corrections to his measurements, and, fitting the corrected set to our reference-framework of walls and clock, recover our own smooth description.
For consistency we must carry our readjustments farther. The path mapped from our measurements is a curve. Perhaps the curvature was introduced by some peculiarity of our reference framework? Possibly our own room is being accelerated upward, so that it makes the ball’s true path—whatever that may be—appear curved downward, just as the autoist’s zigzags made the path he mapped appear jagged. Tradition attributes the downward curving to the tug of gravity. This force we say accelerates the ball downward, producing the curved path. Is this the only possible explanation? Let us see.