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. Our World of Four Dimensions

Our World of Four Dimensions

It will be observed that we have now a much broader definition of non-Euclideanism than the one which served us for the investigation of Euclid’s parallel postulate. If we may at pleasure accept this postulate or replace it by another and different one, we may presumably do the same for any other or any others of Euclid’s postulates. The very statement that the distance between elements of the continuum shall possess significance, and shall be measurable by considering a path in the continuum which involves other elements, is an assumption. If we discard it altogether, or replace it by one postulating [156]that some other joint property of the elements than their distance be the center of interest, we get a non-Euclidean geometry. So for any other of Euclid’s postulates; they are all necessary for a Euclidean system, and in the absence of any one of them we get a non-Euclidean system.

Now the four-dimensional time-space continuum of Minkowski is plainly of a sort which ought to make susceptible of measurement the separation between two of its events. We can pass from one element to another in this continuum—from one event to another—by traversing a path involving “successive” events. Our very lives consist in doing just this: we pass from the initial event of our career to the final event by traversing a path leading us from event to event, changing our time and space coordinates continuously and simultaneously in the process. And while we have not been in the habit of measuring anything except the space interval between two events and the time interval between two events, separately, I think it is clear enough that, considered as events, as elements in the world of four dimensions, there is a less separation between two events that occur in my office on the same day than between two which occur in my office a year apart; or between two events occurring 10 minutes apart when both take place in my office than when one takes place there and one in London or on Betelgeuse.

It is not at all unreasonable, a priori, then, to seek a numerical measure for the separation, in space-time of four-dimensions, of two events. If we find it, we shall doubtless be asked just what its [157]subjective significance to us is. This must be answered with some circumspection. It will presumably be something which we cannot observe with the visual sense alone, or it would have forced itself upon our attention thousands of years ago. It ought, I should think, to be something that we would sense by employing at the same time the visual sense and the sense of time-passage. In fact, I might very plausibly insist that, by my very remarks about it in the above paragraph, I have sensed it.

Minkowski, however, was not worried about this phase of the matter. He had only to identify the invariant expression for distance; sensing it could wait. He found, of course, that this expression was not the Euclidean expression for a four-dimensional interval. He had discarded several of the Euclidean assumptions and could not expect that the postulate governing the metric properties of Euclid’s space would persist. Especially had he violated the Euclidean canons in discarding, with Einstein, the notion that nothing which may happen to a measuring rod in the way of uniform translation at high velocity can affect its measures. So he had to be prepared to find that his geometry was non-Euclidean; yet it is surprising to learn how slightly it deviates from that of Euclid. Without any extended discussion to support the statement, we may say that he found that when two observers measure the time- and the space-coordinates of two events, using the assumptions and therefore the methods of Einstein and hence subjecting themselves to the condition that their measures of the pure time-interval and of the pure space-interval between these [158]events will not necessarily be the same, they will discover that they both get the same value for the expression

If our acceptance of this as the numerical measure of the separation in space-time between the two events should lead to contradiction we could not so accept it. No contradiction arises however and we may therefore accept it. And at once the mathematician is ready with some interpretative remarks.

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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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Our World of Four Dimensions

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