Einstein 1920: Minkowski’s Four-Dimensional Space

Reference: Einstein’s 1920 Book

This paper presents Part 1, Chapter 17 from the book RELATIVITY: THE SPECIAL AND GENERAL THEORY by A. EINSTEIN. The contents are from the original publication of this book by Henry Holt and Company, New York (1920).

The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding.  Feedback on these comments is appreciated.

The heading below is linked to the original materials.


Minkowski’s Four-Dimensional Space

The non-mathematician is seized by a mysterious shuddering when he hears of “four-dimensional” things, by a feeling not unlike that awakened by thoughts of the occult. And yet there is no more common-place statement than that the world in which we live is a four-dimensional space-time continuum.

Time (duration) is the inertial dimension, and the other three are spatial dimensions of substance.

Space is a three-dimensional continuum. By this we mean that it is possible to describe the position of a point (at rest) by means of three numbers (co-ordinates) x, y, z, and that there is an indefinite number of points in the neighbourhood of this one, the position of which can be described by co-ordinates such as x1, y1, z1, which may be as near as we choose to the respective values of the co-ordinates x, y, z of the first point. In virtue of the latter property we speak of a “continuum,” and owing to the fact that there are three co-ordinates we speak of it as being “three-dimensional.”

Space is a three-dimensional continuum of field substance. A “point” is a bit of field substance that is moving at the speed of light. Its duration at any one location is extremely small. The locations are abstract and arbitrary.

Similarly, the world of physical phenomena which was briefly called “world” by Minkowski is naturally four-dimensional in the space-time sense. For it is composed of individual events, each of which is described by four numbers, namely, three space co-ordinates x, y, z and a time co-ordinate, the time-value t. The “world” is in this sense also a continuum; for to every event there are as many “neighbouring” events (realised or at least thinkable) as we care to choose, the co-ordinates x1, y1, z1, t1 of which differ by an indefinitely small amount from those of the event x, y, z, t originally considered. That we have not been accustomed to regard the world in this sense as a four-dimensional continuum is due to the fact that in physics, before the advent of the theory of relativity, time played a different and more independent rôle, as compared with the space co-ordinates. It is for this reason that we have been in the habit of treating time as an independent continuum. As a matter of fact, according to classical mechanics, time is absolute, i.e. it is independent of the position and the condition of motion of the system of co-ordinates. We see this expressed in the last equation of the Galileian transformation (t’ = t).

In material domain, time (duration), or inertia, is very large; so the substance at any one location is hardly moving. The time (duration, inertia) is treated as constant at all locations.

The four-dimensional mode of consideration of the “world” is natural on the theory of relativity, since according to this theory time is robbed of its independence. This is shown by the fourth equation of the Lorentz transformation:

This equation basically states how the duration (inertia) of substance is decreasing with velocity (in the material domain only). For v = c/2, t’ = 0.866t. That means inertia (the “density” of point mass) has decreased by a factor of 0.866, or that “point mass” has smeared over a larger area.

Moreover, according to this equation the time difference Δt’ of two events with respect to K’ does not in general vanish, even when the time difference Δt of the same events with reference to K vanishes. Pure “space-distance” of two events with respect to K results in “time-distance” of the same events with respect to K’. But the discovery of Minkowski, which was of importance for the formal development of the theory of relativity, does not lie here. It is to be found rather in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space. (Note: confer the somewhat more detailed discussion in Appendix II.) In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude √(-1) × ct proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time co-ordinate plays exactly the same rôle as the three space coordinates. Formally, these four co-ordinates correspond exactly to the three space co-ordinates in Euclidean geometry. It must be clear even to the non-mathematician that, as a consequence of this purely formal addition to our knowledge, the theory perforce gained clearness in no mean measure.

Math is fine but it is not interpreted clearly in special relativity. The time coordinate is the dimension of inertia, which varies with changes in spatial dimensions (velocity).

These inadequate remarks can give the reader only a vague notion of the important idea contributed by Minkowski. Without it the general theory of relativity, of which the fundamental ideas are developed in the following pages, would perhaps have got no farther than its long clothes. Minkowski’s work is doubtless difficult of access to anyone inexperienced in mathematics, but since it is not necessary to have a very exact grasp of this work in order to understand the fundamental ideas of either the special or the general theory of relativity, I shall at present leave it here, and shall revert to it only towards the end of Part II.



Minkowski’s four-dimensional space essentially relates the inertia of substance to its velocity.


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