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 Space-time Continuum

THE SPACE-TIME CONTINUUM

Minkowski’s World of Events, and the Way It Fits Into Einstein’s Structure

BY THE EDITOR, EXCEPT AS NOTED

Seeking a basis for the secure formulation of his results, and especially a means for expressing mathematically the facts of the dependence which he had found to exist between time and space, Einstein fell back upon the prior work of Minkowski. It may be stated right here that the idea of time as a fourth dimension is not particularly a new one. It has been a topic of abstract speculation for the best part of a century, even on the part of those whose notions of the fourth dimension were pretty closely tied down to the idea of a fourth dimension of Euclidean point-space, which would be marked by a fourth real line, perpendicular to the other three, and visible to us if we were only able to see it. Moreover, every mathematician, whether or not he be inclined to this sort of mental exercise, knows well that whenever time enters his equations at all, it does so on an absolutely equal footing with each of his space coordinates, so that as far as his algebra is concerned he could never distinguish between them. When the variables x, y, z, t come to the mathematician in connection [142]with some physical investigation, he knows before he starts that the first three represent the dimensions of Euclidean three-space and that the last stands for time. But if the algebraic expressions of such a problem were handed to him independently of all physical tie-up, he would never be able to tell, from them alone, whether one of the four variables represented time, or if so, which one to pick out for this distinction.

It was Minkowski who first formulated all this in a form susceptible of use in connection with the theory of relativity. His starting point lies in the distinction between the point and the event. Mr. Francis has brought this out rather well in his essay, being the only competitor to present the Euclidean geometry as a real predecessor of Newtonian science, rather than as a mere part of the Newtonian system. I think his point here is very well taken. As he says, Euclid looked into the world about him and saw it composed of points. Ignoring all dynamic considerations, he built up in his mind a static world of points, and constructed his geometry as a scientific machine for dealing with this world in which motion played no part. It could to be sure be introduced by the observer for his own purposes, but when so introduced it was specifically postulated to be a matter of no moment at all to the points or lines or figures that were moved. It was purely an observational device, intended for the observer’s convenience, and in the bargain a mental device, calling for no physical action and the play of no force. So far as Euclid in his daily life was obliged to take cognizance of the fact that in the world of work-a-day [143]realities motion existed, he must, as a true Greek, have looked upon this as a most unfortunate deviation of the reality from his beautiful world of intellectual abstraction, and as something to be deplored and ignored. Even in their statuary the Greeks clung to this idea. A group of marvelous action, like the Laocoon, they held to be distinctly a second rate production, a prostitution of the noble art; their ideal was a figure like the majestic Zeus—not necessarily a mere bust, be it understood, but always a figure in repose without action. Their statuary stood for things, not for action, just as their geometry stood for points, not for events.

Galileo and Newton took a different viewpoint. They were interested in the world as it is, not as it ought to be; and if motion appears to be a fundamental part of that world, they were bound to include it in their scheme. This made it necessary for them to pay much more attention to the concept of time and its place in the world than did the Greeks. In the superposition process, and even when he allowed a curve to be generated by a moving point, the sole interest which Euclid had in the motion was the effect which was to be observed upon his static figures after its completion. In this effect the rate of the motion did not enter. So all questions of velocity and time are completely ignored, and we have in fact the curious spectacle of motion without time.

To Galileo and Newton, on the other hand, the time which it took a body to pass from one point of its path to another was of paramount importance. The motion itself was the object of their study, and they recognized the part played by velocity. But [144]Galileo and Newton were still sufficiently under the influence of Euclid to fit the observed phenomena of motion, so far as they could, upon Euclid’s static world of points. This they effected by falling in with the age-old procedure of regarding time and space as something entirely disassociated and distinct. The motion of an object—in theory, of a point—was to be recorded by observing its successive positions. With each of these positions a time was to be associated, marking the instant at which the point attained that position. But in the face of this association, space and time were to be maintained as entirely separate entities.

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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.

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