The External World and its Geometry

Einstein's Theories of Relativity and Gravitation by Albert Einstein, is part of the HackerNoon Books Series. You can jump to any chapter in this book here. The External World and its Geometry

The External World and its Geometry

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 [209]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 [210]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.”[211]

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.

About HackerNoon Book Series: We bring you the most important technical, scientific, and insightful public domain books.

This book is part of the public domain. Albert Einstein (2020). Einstein's Theories of Relativity and Gravitation. Urbana, Illinois: Project Gutenberg. Retrieved October 2022.

This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org, located at https://www.gutenberg.org/cache/epub/63372/pg63372-images.html