Part 4

Riemann produced his general work along this line in 1854 which was far ahead of his time. He actually prophesied the connection of geometry with matter, and had he possessed a little more vision he would doubtless have worked out the details as well as the principles underlying gravitation in much the same manner as Einstein has done. Riemann’s efforts in the field of non-Euclidean geometry has materially aided Einstein in the development of the present theory. Minkowski’s work was utilized by Einstein to much profit in the outworking of the Special Theory, particularly his clarification of _time_ as a fourth dimension.

=Time as a Fourth Dimension=

It is natural for us to think of all matter as possessing but three dimensions--length, breadth and thickness--and we have been accustomed to making our measurements of matter and of space on that basis. Using the formula of Pythagoras we have ascertained the distance between any two points in a _plane_ (a two-dimensional area) by extracting the square root of the sum of the squares of the co-ordinate axes, i. e., the base and the altitude as in the accompanying diagram. See Figure 4.

If point A is 8 miles south and 6 miles west of point B then A and B are 10 miles apart, thus:

The square of 8 is 64 The square of 6 is 36 --- The square root of 100 is 10

Likewise, the distance between any two points in a three-dimensional region (as from an upper to the remotest lower corner of a room) is generally considered to be “the square root of the sum of the squares of the three sides” (Fig. 5).

Thus if the distance O to X is 12 feet and X to Y is also 12 feet, while Y to Z is 14 feet, then the straight diagonal distance from O directly through the room to point Z is 22 feet, because the sum of the squares of the three sides (144 + 144 + 196) yields a total of 484, and the square root of that number is 22. This simple formula will hold good for all ordinary measurements, but for great distances in space a slight correction is found necessary because of the little trick that light rays are prone to factor, i. e., the numerical value of the interval of time required for a light ray to traverse play upon us. We must subtract the _time_ from the distance. Hence if our cube were large enough to fill a goodly portion of the universe we would no longer say that the diagonal distance from O to Z is √(x^2 + y^2 + z^2) but rather √(x^2 + y^2 + z^2 - t^2), ----t, of course, representing _time_.

Now recall what we learned in the preceding pages, that the velocity of light always appears to be the same to all observers irrespective of the relative velocity between the observer and the source of light. It is manifest, therefore, that in making measurements the time factor (t) _really_ represents one quantity for one observer and a totally different quantity for another observer notwithstanding the fact that it appears to be a constant to all observers. Inasmuch as the velocity of light does appear to be constant to all observers its actual stretching or contracting of units is not manifest. Therefore the corrected equation as given above (the subtraction of the _time_ element) holds good for all observers irrespective of their motion.

The point of interest to the non-Euclidean geometer in connection with any measurement, be it remembered, is not the abstract _distance_ between points, because distance is not a constant and is not determinable unless we know the _absolute_ velocity of the observer and of the points being measured, which knowledge we do not possess. What we should look for, then, is the _distance and time combined_, or the _separation-interval_ as it is aptly called. The time factor automatically corrects the units for each observer, no matter what his motion may be, and thus the separation-interval appears a constant.

The foregoing illustrates how time takes its place alongside the ordinary three dimensions of space, and is in reality a fourth dimension, although it is not a thing that can be visualized as we can visualize the length or breadth or thickness of any object. In the following paragraphs we shall examine further into the geometry of the universe, particularly as it relates to the phenomenon of gravitation.

=Geometry with a Physical Meaning=

Certain news dispatches and book reviews have erroneously reported Professor Einstein as having said “only twelve men in the world can understand the Principle of Relativity.” The statement becomes absurd in view of the scores of volumes now in print, all of which set forth more or less clearly the details of the Einstein theory. What he alluded to in the remark so generally misquoted and misconstrued is his mathematical equations (calculus of tensors). He questioned if there are more than a dozen mathematicians in the world who are familiar with this abstruse differential calculus because it is not generally taught in the university text books.

This calculus is a veritable maze of formulæ, really invented by Riemann and Cristoffel, but systematized by the celebrated Italian mathematicians, Ricci and Levi-Cevita, and is impractical for any ordinary use. This is why so few mathematicians have familiarized themselves with it. Einstein, however, found it invaluable in dealing with such complex geometrical problems as his theory produced.

Briefly, the non-Euclidean geometer deals with _surfaces_ rather than planes, and his fundamental postulates are sufficiently broad to apply to all regular surfaces whether they be planes, spheres, cylinders, conicoids or even spheroids or ellipsoids. He considers a “straight” line as being the shortest distance between two points _on a surface_, hence if the surface is curved the “straight” line connecting any two points thereon will also be curved. This _shortest_ distance between points is called a _geodesic_. If the surface happens to be a _plane_ then the geodesics connecting points thereon are really straight lines in the Euclidean sense, but this would not be true for any other kind of a surface. Thus it is seen that Euclidean geometry is simply a limiting case of this more general geometry.

Geometers of the elliptic or spherical school, including Einstein, declare that in nature there is no such thing as a purely Euclidean straight line such as may be prolonged in opposite directions to infinity. On the contrary they hold that any “straight” line if prolonged sufficiently would return upon itself, because the universe is so constructed. In other words, what we ordinarily call a straight line is but an arc of a near infinite circle which possesses the _least possible curvature_. Magazine writers in an endeavor to make clear this portion of the theory of Relativity have strikingly declared that “according to Einstein a man might look through a telescope in any direction whatsoever and behold the back of his neck.” This jest, though omitting essential facts, is not without geometrical foundation. If we possessed a near infinite telescope and should live for a near infinite period of time to enable the rays of light to traverse this near infinite circle, then, if there were no obstructions along our line of sight, we might be rewarded with a round trip view of the rear portion of our body--though the simpler method would be to use two ordinary mirrors.

All this, however, has an important bearing upon Einstein’s interpretation of gravitation. Not only does he contend for Lobatschewsky’s “curvature of space” but he also holds that surrounding every body of matter there is a _special_ space-curvature (four dimensional), the degree of which depends upon the body’s observed mass. This special curvature or “warp” of space constitutes the “gravitational field” surrounding all large bodies of matter and causes the acceleration of falling particles in that field. This distortion of space increases in proportion to the mass of the body causing it, and decreases with the distance from that body until ultimately it becomes _nil_ or practically so in a region remote from all matter.

Perhaps the nearest approach to a visualization of this space-curvature (which constitutes a gravitational field) is to consider the lines of force in a magnetic field. The reader is doubtless familiar with the age-old experiment of placing file dust on a thin sheet of cardboard or plate of glass and then holding a horseshoe magnet underneath with the two poles touching the sheet or plate. Immediately the filings arrange themselves into curved lines between the poles as shown in Fig. 6.

This experiment indicates that between the poles of a magnet are constant lines of force, invisible to sight but manifesting themselves when attractable particles are in or near their path. The earth, likewise, is a great magnet, having one magnetic pole in upper Canada above Hudson Bay, about 70° north latitude, and another pole in the Antarctic Ocean south of Australia. Between these two magnetic poles continually flow these invisible curved lines of force, just as with our horseshoe magnet in Figure 6. These lines of force are the cause of compasses pointing in a northerly and southerly direction. Other planets undoubtedly possess magnetic poles similar to those of the earth.

=The Background of Gravitation=

Now let us conceive of invisible geometric lines pervading the entire spatial universe somewhat analogous to these lines of force in a local magnetic field. To each point in these lines let us ascribe an electric and gravitational potential (remembering, of course, that we are dealing with _four_-dimensional space), and we have before us, in a nearly visualized, sense, the background of the new theory of gravitation.

Einstein was the first to present the subject of gravitation from this viewpoint. For centuries up to this time geometry and physics were considered as belonging to entirely different schools of thought, but under the master hand of Einstein the two sciences have been welded together into one. As Freundlich puts it, “quantities which hitherto had only a purely geometrical import, for the first time became animated with physical meaning.” Thus “empty space” is no longer empty, even though the existence of the ether be denied. When the study of free electrons has sufficiently advanced, it may be seen that these elementary particles of electricity, or energy-particles, freed from atomic or mass attraction, play an important role in gravitational phenomena.

Figure 7 represents in a crude fashion the special curvature of space in the region of a large body of matter, for instance our earth, with the points (events) situated at finite instead of infinite nearness to each other for sake of illustration. It will be readily seen that the distortion of the geometric lines would necessarily alter the relative positions of the point-potentials.

Any falling body moves in a geodesic, i. e., from one point to the _next nearest_ point in space-time.[2] In an undisturbed region, remote from matter, the points (events) may be considered as so arranging themselves that any four neighboring ones would constitute practically a square. It may then be seen that the easiest path for falling bodies would be to follow the _sides_ of the squares because by so doing they would be following the _geodesic_ or shortest distance between points. (See Fig. 7.) But in the region of a large body of matter the lines of points become so distorted that the _diagonal_ of any four neighbor points becomes the geodesic. Then the path of the falling particle will accordingly deviate. It will always follow the geodesic, or easiest path.

This causes the falling particle to take a direction which points toward the center of the gravitational field--but the center of gravity is exerting no drawing or attractive force as Newton supposed. Gravity is thus seen to be not an external _drawing_ power operating between bodies of matter, but an inherent order of nature in space. The acceleration as well as the direction of the falling particle is accounted for by this theory. As the separation-interval between points becomes shorter--due to the constantly accentuated distortion as the large body is approached--the falling particle would be correspondingly accelerated. The distortion being constantly increased the acceleration would likewise be constant.

Newton, in his law of inertia, postulated that any particle of matter at rest will forever remain at rest if not disturbed, but when once set in motion it will continue to move at uniform velocity in a straight line unless interfered with by outside force. Einstein, on the contrary, holds that any particle of matter if left to itself will _move_ (let us say _fall_, if you please) in the easiest direction (i. e., in a geodesic) at constant velocity unless it encounters a gravitational field (a distorted region), in which event it will become accelerated, and will also, if necessary, change its direction, in obedience to the principle of least action. In other words, it is natural for matter to possess energy, therefore natural for it to be in motion and unnatural for it to be at rest. And the contention has this much in its favor: every particle of matter in the universe, from the infinitesimal electron to the more gigantic sun and super-system of outer space, _is moving_, so far as our most modern observations extend. Nothing has yet been discovered to be at rest.

[Footnote 1: When either “space” or “time” is mentioned independently of the other in this treatise it may be understood that the terms are used in the ordinary conceptual sense for purposes of simplicity.]

[Footnote 2: The reader must bear in mind that four-dimensional, not the ordinary three-dimensional, space is here discussed. The author has endeavored, however, to treat the matter in such a manner as to approach a visualization of this otherwise quite complex subject.]

=TRANSCRIBER’S NOTES=

Simple typographical errors have been silently corrected; unbalanced quotation marks were remedied when the change was obvious, and otherwise left unbalanced.

Punctuation, hyphenation, and spelling were made consistent when a predominant preference was found in the original book; otherwise they were not changed.