If we have a four-dimensional manifold to begin with, we may equally shake out one of the four dimensions, one of the four coordinates, and consider the three-dimensional result of this process as a cross-section of the original four-dimensional continuum. And where, in cross-sectioning a three-dimensioned world, we have but three choices of a coordinate to eliminate, in cross-sectioning a world of four dimensions we have four choices. By dropping out either the x, or the y, or the z, or the t, we get a three-dimensioned cross-section.
Now our accustomed three-dimensional space is strictly Euclidean. When we cross-section it, we get a Euclidean plane no matter what the direction in which we make the cut. Likewise the Euclidean plane is wholly Euclidean, because when we cross-section it in any direction whatever we get a Euclidean line. A cylindrical surface, on the other hand, is neither wholly Euclidean nor wholly non-Euclidean in this matter of cross-sectioning. If we take a section in one direction we get a Euclidean line and if we take a section in the other direction we get a circle (if the cylindrical surface be a circular one). And of course if we take an oblique section of any sort, it is neither line nor circle, but a compromise between the two—the significant thing being that it is still not a Euclidean line.
The space-time continuum presents an analogous situation. When we cross-section it by dropping out any one of the three space dimensions, we get a three-dimensional complex in which the distance formula is still non-Euclidean, retaining the minus sign before the time-difference and therefore retaining the geometric character of its parent. But if we take our cross-section in such a way as to eliminate the time coordinate, this peculiarity disappears. The signs in the invariant expression are then all plus, and the cross-section is in fact our familiar Euclidean three-space.
If we set up a surface geometry on a sphere, we find that the elimination of one dimension leaves us with a line-geometry that is still non-Euclidean since it pertains to the great circles of the sphere rather than to Euclidean straight lines. In shaking Minkowski’s continuum down into a three-dimensional one by eliminating any one of his coordinates, if we eliminate either the x, the y or the z, we have left a three-dimensional geometry in which the disturbing minus sign still occurs in the distance-formula, and which is therefore still non-Euclidean. If we omit the t, this does not occur. We see, then, that the time dimension is the disturbing factor, the one which gives to space-time its non-Euclidean character so far as the possession of curvature is concerned. And we see that this curvature is not the same in all directions, and in one direction is actually zero—whence the attempted analogy with a cylinder instead of with a sphere.
Many writers on relativity try to give the space-time continuum an appeal to our reason and a character of inevitableness by insisting on the lack of any fundamental distinction between space and time. The very expression for the space-time invariant denies this. Time is distinguishable from space. The three dimensions of space are quite indistinguishable—we can interchange them without affecting the formula, we can drop one out and never know which is gone. But the very formula singles out time as distinct from space, as inherently different in some way. It is not so inherently different as we have always supposed; it is not sufficiently different to offer any obstacle to our thinking in terms of the four-dimensional continuum. But while we can group space and time together in this way, [this does not mean at all that space and time cease to differ. A cook may combine meat with potatoes and call the product hash, but meat and potatoes do not thereby become identical.][223]
The Question of Visualization
To the layman there is a great temptation to say that while, mathematically speaking, the space-time continuum may be a great simplification, it does not really represent the external world. To be sure, you can’t see the space-time continuum in precisely the same way that you can the three-dimensional space continuum, but this is only because Einstein finds the time dimension to be not quite freely interchangeable with the space dimension. Yet you do perceive this space-time continuum, in the manner appropriate for its perception; and it would be just as sensible to throw out the space continuum itself on the ground that perception of the two is not of exactly the same sort, as to throw out the space-time continuum on this ground. With appropriate conventions, either may stand as the mental picture of the external world; it is for us to choose which is the more convenient and useful image. Einstein tells us that his image is the better, and tells us why.
Before we look into this, we must let him tell us something more about the geometry of his continuum. What he tells us is, in its essentials, just this. The observer in a pure space continuum of three dimensions finds that as he changes his position, his right-and-left, his backward-and-forward, and his up-and-down are not fixed directions inherent in nature, but are fully interchangeable. The observers, in the adjoined sketch, whose verticals are as indicated by the arrows, find very different vertical and horizontal components for the distance between the points O and P; a similar situation would prevail if we used all three space directions. The statement analogous to this for Einstein’s four-dimensional continuum of space and time combined
