Part 4
Anything that exists, that is to say, persists, is moving along the time dimension at what appears to be a uniform rate. Of course you can, if you like, conceive of time itself as a stream flowing through things. Since all motion is relative, that way of looking at it is just as “true” as the other. But it is simpler and more sensible to think of things moving through a stationary time just as we think of them moving through a stationary space. A material point that is at rest, such as the dot of an _i_ on this page, (we continue to disregard the motion of the earth) is not moving about in space but is moving forward in time. Its track then is a straight line along the time dimension. That is, a material point is a line in the fourth dimension. If you move the page to the right the forward movement of the dot of the _i_ in the time dimension is combined with the sideways motion in a single slanting line. If you move the page simultaneously upward, rightward and backward the track of the point is a line combining the movement in all four dimensions. Such a track of a point moving through space and time is called its “world-line.” It is a continuity of one dimension. Any event is the point of intersection of one or more such world lines and we can never observe anything except such intersections. That is to say, everything happens somewhere and sometime.
A picture flashed on a cinema screen has three dimensions. It is, say, 10 feet long and 6 feet high and lasts ¹⁄₁₆ of a second, but it has no thickness. A man necessarily has four dimensions. He may measure from 24 to 72 inches in one dimension, from 8 to 18 inches in the second, from 4 to 9 inches in the third and 70 years in the fourth.
After all, the idea of the relativity of time ought to be easier to accept than that of space for it is in accord with experience instead of contrary to it. We drop off to sleep and wake the next instant if we credit our personal perceptions. Why should we believe the sun and the clock in preference to ourselves?
Bergson bases his whole philosophy upon the distinction between _duration_ as it is felt by the individual while he is living through it and _time_ as it is employed by the physicist in his calculations. The latter conception, physical time, is, as Bergson says, a mere invention of man and virtually a fourth dimension of space, so he concludes:
To sum up; every demand for explanation in regard to freedom comes back, without our suspecting it, to the following question: “Can time be adequately represented by space?” To which we answer: Yes, if you are dealing with time flown; No, if you speak of time flowing.[1]
Past and future are alike to the physicist, differing only in direction, like east and west. But to the living person they are altogether different things. For man rolls up his past, as a tourist his rug, and carries it with him wherever he goes. That is why Wells’s “Time Machine” and the reversed reels of the movies are so funny. There is nothing absurd about running a wheel backward but there is about running a man backward.[2] The physicist feels no reluctance about turning the stream of time backward for all physical phenomena are reversible under the proper conditions. If we interpret the universe as merely matter in motion and imagine at a certain instant that every individual particle reverses its motion and goes in just the opposite direction at the same speed, then the whole history of the world would be reënacted in the opposite order and the earth would return to its primeval nebulæ.
In Wells’s story, “The New Accelerator,” a professor invents an elixir that speeds up the rate of living a thousandfold. A person taking a dose of it sees people as wax figures apparently motionless in the midst of violent action. Falling objects seem to stand still in the air. The music of a band is reduced to “a low-pitched, wheezy rattle” or “the slow muffled ticking of some monstrous clock.” But in compensation for this the accelerated drug-fiend could watch at leisure the slow flapping of a bee’s wings.
But even Wells with his seven-league-boots imagination finds it difficult to keep ahead of the march of science. What he then saw only with his mind’s eye we can actually observe. By moving the accelerating lever on your phonograph toward the S end of the scale you can slow up the tune and lower its pitch until it becomes inaudible as music. The new Pathé ultra-rapid camera can take pictures at the rate of 160 to the second. When these are projected on the screen at the usual rate of 16 to the second all movement takes place ten times slower than in actual life. This gives opportunity for the study in detail of the action of a ballplayer pitching a curve or of the wing motion of a humming bird or of the splash of a marble falling into water or of the flight of a bullet. We can magnify motion or minify it as much as we will. The cinematograph owes its origin the desire of Senator Leland Stanford to study the movement of a horse’s legs so as to find out why one racer went faster than another.
Such playful flights of the scientific imagination as Wells and Flammarion indulge in and such freaks of projection as the camera man amuses us with are of use to those of us who find difficulty in translating a mathematical formula into terms of everyday life. There is no better place to study metaphysics than in the world of the flickering screen, for there man has complete control of time and space. He can enlarge and reduce any object. He can hasten, retard or reverse any action. He can throw upon the screen at the same time events happening months and miles apart. Therefore to those of us who have had the advantage of an education in the movies, Einstein’s ideas of the relativity of time and space do not seem startling or inconceivable.
Kant not only conceived the possibility of more than three dimensions but believed in the probability of it. His argument is based on greater insight into the intentions of the Almighty than we of this day would claim:
“If it is _possible_ that there be developments of other dimensions in space, it is also very _probable_ that God has somewhere produced them. For His works have all the grandeur and variety that can possibly be conceived.”
In this temporal, spatial and material world of ours reality requires that the four dimensions should hang together. But at an infinite distance from all matter this fourfold combination would be dissolved into a three-dimensional space and a one-dimensional time. In that extra-mundane realm time ceases to flow, gravitation no longer drags downward, matter is non-existent, light is immovable and change is impossible.[3] Thus the new mathematics leads to a state curiously like the conventional conception of heaven.
We talk as our forefathers did about “the ends of the earth” but we know that one might start from his home and walk forever in any direction without coming to an end of it. But though the earth’s surface is infinite in the sense of endless, yet one never can get more than 8,000 miles away from home where’er he may roam. If a man stood on the top of the highest mountain on earth and aimed a level gun in any direction, the bullet, if it could be given sufficient velocity to counteract the influence of gravity, would go around the world and hit him in the back of the head. Or if light were sufficiently deflected by gravitation to follow a level line around the earth--another absurd assumption--the man looking through a level telescope in any direction could see how his hair was combed in the back. Such happenings, though impossible, are not inconceivable but are logical consequences of our knowledge that the world is round and that what we call straight or level lines as measured on plain or sea are really great circles around a center four thousand miles below.
Now is it not also conceivable that the lines we call straight in astronomical space may also have an imperceptible curvature in some unknown fourth dimension? If this curve is closed like the circumferences of the earth a ray of light pursuing a straight course in a certain direction might eventually return upon its track, even though not refracted or reflected by the matter it passes through or by. A telescope of unlimited power pointed into space at a tangent might then show the observer his own back, if light were transmitted instantaneously, but, since it is not and since the curvature of space, if there be any, is exceedingly minute, what the observer would see, assuming that the earth had come back to its former position, might be the scenes of some geological age millions of years ago.
NON-EUCLIDEAN GEOMETRY
The idea that space itself may be curved and that the axioms and assumptions on which our geometry since the time of Euclid have been based, may not be absolutely and exactly and eternally and universally true has been diligently studied during the last fifty years. The Russian Lobatchewsky, the Hungarian Bolyai and the German Riemann have developed systems of geometry by starting from premises the opposite of those of Euclid and these systems are just as logical and consistent with themselves as the ordinary or Euclidean geometry. These non-Euclidean geometries were at first commonly regarded as mere freaks of the mathematical imagination, but they have already proved valuable in leading to a reconsideration of the fundamental principles of our thinking and, if Einstein is right, they may be necessary to explain physical phenomena. It is hard for the mathematician to discover anything useless. A distinguished American mathematician in announcing a new theorem exclaimed: “And thank Heaven, no possible use can ever be found for it.” But, whatever it was, he made a rash boast for nowadays the mechanic treads on the heels of the mathematician and uses imaginary quantities, actual only in the fourth dimension, like √-1, in figuring out the winding of his dynamo.
Readers whose mathematical faculty is weak or undeveloped and who like something concrete with “human interest” in it will find what they want in “Flatland by A Square,” a book published in Boston in 1891. The author, who turned out to be the Reverend Edwin Abbott, tells of a land in only two dimensions. The ruling class consisted of polygons, the bourgeoisie of squares and equilateral triangles, the lower class of isosceles triangles of narrow base, while the criminals had more irregular forms and the women were mere needles. Since all were confined to a surface, four lines set in a square made a tight prison. The inhabitants of Flatland, even the aristocratic and intellectual individuals who had so many sides as to be almost circular, could not conceive of a third dimension from which a person like ourselves could look down and see at a glance the insides of their houses, their safes and their bodies just as a being in the fourth dimension could see the insides of ours. The narrator, that is, A Square of Flatland, visits as a missionary the land of two dimensions where all the people lie in a line and refuse to believe in anything outside it and finally he encounters and endeavors to convert a solitary point of no dimensions but finds him, as we should expect, an incorrigible solipsist.
We should all of us have been familiar with the fourth dimension for years if Slade had not turned out a trickster. Slade was an American medium--the original of Browning’s “Mr. Sludge”--who fooled Professor Zöllner by giving him what purported to be experimental evidence of the fourth dimension. Zöllner was a distinguished German physicist, Professor of Astronomy in the University of Leipzig, old, near-sighted, pre-disposed to spiritualism, and unskilled in legerdemain. Any proofs that Zöllner asked for, Slade was usually able at the next séance to produce. All the things that one might do in four dimensions but could not do in three were forthcoming by the obliging spirits whom Slade had at call.
[Illustration: In space of one dimension (a straight line) there could be neither bend, loop nor knot in a string.]
[Illustration: In space of two dimensions (a flat surface) a double bend could be made in the string but no loop or knot could be made.]
[Illustration: But if we raise one string (into the third dimension) and lay it over the other like this:]
[Illustration: We get a loop but cannot form a knot without using the ends.]
[Illustration: A knot like this cannot be made in a string so long as the ends are held by the hands. But if we could use a fourth dimension we could tie such a knot as easily as we made a bend by the use of the second dimension and a loop by the use of the third. If such a knot could be tied in a string so held it would be experimental evidence of the existence of four-dimensional space.]
Zöllner tied the ends of a string together and sealed them on the table top, letting the loop hang down under the table out of sight. He then asked to have a single knot tied in the string and the spirits tied four. Zöllner also reports that the coins he put into a sealed box were taken out and writing produced inside sealed slates.
On the basis of these experiments Zöllner wrote a volume on “Transcendental Physics” to prove the existence of another world in the fourth dimension. But when Slade tried his tricks in London he was caught at them by Professor E. Ray Lankester. He was convicted of deception with intent to defraud in the Bow Street Police Court and sentenced to three months’ imprisonment with hard labor. Nowadays the apparatus for Slade’s famous slate-writing trick can be purchased at any conjurer’s shop.
It is vain to expect anything scientific to come out of the séance room where the alleged phenomena are not reproducible under specified conditions but appear only occasionally and under circumstances prescribed by the medium who always may be and often is proved to be a sleight-of-hand--or sleight-of-foot--performer. The fourth dimension which Einstein and other scientists are now considering is not conceived of as the abode of departed spirits, a spare room for ghostly visitants, but merely as a new factor in a mathematical formula. It offers us no hope of ever being able to take coin out of a closed safe or put coin into an unopened coconut but it does promise to explain certain optical phenomena which, though rare and minute, are yet open to the observation of anybody, be he skeptical or credulous.
SOME SIMPLE EXAMPLES
Lisbon lies nearly straight east of New York but when a ship captain wants to go to Lisbon he does not sail straight east but sets his course a little northward in the beginning and a little southward toward the end and so gets there quicker than if he had followed a line of latitude. Draw his course on a flat map and you would think he was taking a roundabout route, but trace it on a globe and you will see that he is following a great circle, the geodetic line, which is the shortest distance between any two points on the earth’s surface.
An airman looking down on a rocky, hilly, woody country sees it as a flat plain and if he watched a hunter returning home with his bag of game would wonder that he did not go straight instead of wandering around in such an irregular way. Yet the hunter, being tired, is taking what is for him the shortest way home as he dodges rocks and circumambulates the hills. The easiest way is the shortest way.
A river in its desire to reach the sea always takes the shortest possible way. Its meanderings are not meaningless but determined by a law as rigid as a law of geometry, that is, the law of gravitation which prevents the river from taking a short cut over the hill.
If you look at a landscape over a heated plain or bonfire or through uneven glass you will see that the image is distorted and confused because the rays of light are refracted and entangled as they pass through this unequal medium. Yet each ray is going just as straight as it can toward your eye.
Now to such familiar cases where a ray of light is bent out of its straight course by the uneven density of the air or glass through which it passes Einstein has added another and unsuspected effect, namely, that light is likewise deflected in passing through a strong gravitational field such as the vicinity of a large body like the sun.
It has long been known that the displacement of the earth in space and time (that is to say, its motion) causes an apparent displacement of the stars in space.
The astronomer does not point his telescope straight at a star. If he did, he would not see it, for, owing to the forward motion of the earth, the telescope moves out of range of the rays that otherwise would have reached it.
[Illustration: Everyone knows that a ray of light is bent out of its straight course as it passes from the air into a denser medium like water or glass, and that this deflection apparently shifts the position of the object from which the light comes. Einstein’s theory and the British eclipse observations prove, what was not known before, that a ray of light as it passes through the gravitational field of a large body like the sun, is also perceptibly bent out of its straight course and likewise makes an apparent shift in the position of its source, the star.--From Black & Davis’ “Practical Physics.” Published by The Macmillan Company.]
If you have ever tried to shoot a bird on the wing, or, better, a prairie-dog from a train you will get the idea. Or, if you have not had this experience, you have doubtless watched the raindrops running down a car window and have noticed that when the rain is falling straight down the drops strike the pane on a slant when the car is moving forward. The faster the car moves the greater the deviation from the perpendicular. If the train runs backward the rain-streaks slant in the opposite direction. If then you should be asked to point out the direction of the cloud from which the rain is coming you would--unless you knew and made allowance for the movement of the train--point in a line with the streaks on the pane, sometimes backward, sometimes forward, but not straight upward where the raincloud really is.
Now the astronomer is on a moving train, the earth, which is rushing around a ring about 186,000,000 miles across. Consequently every star appears to wabble around in a little ellipse and the astronomer has to aim his telescope, now on one side, then on the other, of the real position of the star in order to bring it on the cross-hairs of his object glass. This apparent displacement of the stars is known as “the aberration of light” was explained by Fresnel in 1818--to everybody’s satisfaction until recently--on the assumption that all space is filled with an immovable medium, the ether, which transmits the rays of light in straight lines in the form of wave motion, and that the earth moves through the ether without displacing it, somewhat as an airplane moves through still air. But the aviator knows how fast he is moving by the current of air streaming back in his face. Why then, since the ether is in perfect repose, could we not determine the absolute motion of the earth through space by measuring the drift of the ether as it streams through the pores of the earth? Light appears to afford us a means of measuring such a drift of the ether through matter, if there be such. Since light is conveyed by the ether we should naturally expect it to take less time to travel a certain distance if the receiving instrument is carried toward the source of the light by the earth motion than if it is being carried away from it. This question was put to the crucial test by two American physicists, Michelson and Morley, who devised an instrument so delicate that it could detect differences of one-25,000,000th of an inch in the path of a light ray. But although this delicacy was ten times greater than was necessary to detect the ether drift, if there were any, no evidence of such drift could be discovered.
THE ECLIPSE OBSERVATIONS
In the history of science the year 1919 is likely to be known, not as the year of the overthrow of the German Empire, but as the year of the overthrow of Newton’s law of gravitation. The British astronomers who went to Africa to observe the eclipse of the sun May 29, 1919, came back with the proof that a ray of light passing close by the sun is bent out of its straight course. The photographs taken during the six minutes when the sun was shadowed show the surrounding stars in different positions from where they are seen when the sun’s disk is not in their midst. This is the second time that Einstein has scored over Newton. The first was in regard to the orbit of Mercury. If the sun and Mercury were alone in the universe the planet, according to Newton’s law, would revolve forever around the sun in the same elliptical track. But the presence of the other planets makes Mercury deviate from this regular route so the ellipse it describes is never quite the same but slowly shifts around so that in the course of centuries its longer diameter would be pointing in a different direction. Calculating by Newton’s law, the influence exerted by the other planets astronomers found that it would shift the orbit of Mercury 532 seconds of arc in a century. But when they took observations on Mercury they found that its orbit was shifting at the rate of 574 seconds. The discrepancy between observation and theory, 42 seconds, is thirty times greater than could be accounted for by errors of instruments or observation. But according to Einstein’s theory, if the sun and Mercury were alone in space with no other planets interfering, the orbit of Mercury would not remain the same, but would advance at the rate of 43 seconds a century. This, as the reader will observe, is in substantial agreement with the discrepancy which has for two centuries puzzled astronomers, since it was inexplicable on the Newtonian theory.