diff options
Diffstat (limited to '78274-0.txt')
| -rw-r--r-- | 78274-0.txt | 1545 |
1 files changed, 1545 insertions, 0 deletions
diff --git a/78274-0.txt b/78274-0.txt new file mode 100644 index 0000000..52a18fa --- /dev/null +++ b/78274-0.txt @@ -0,0 +1,1545 @@ +*** START OF THE PROJECT GUTENBERG EBOOK 78274 *** + + + + + TEN CENT POCKET SERIES NO. 408 + Edited by E. Haldeman-Julius + + Introduction to + Einstein + + William F. Hudgings + + Copyright, 1923, by the author + Copyright, 1923, Haldeman-Julius Company + + HALDEMAN-JULIUS COMPANY + GIRARD, KANSAS + + +[Illustration: DR. ALBERT EINSTEIN] + + + + + _A Popular Explanation of the Einstein Theory That Approaches an + Actual Visualization of the Subject._ + + +The preparation of this essay on the Principle of Relativity has been +undertaken after a careful study of practically all the literature +hitherto published on the subject purporting to be of a popular or +semi-popular nature. With due appreciation of the merit of many of +these works, a perusal of them has made it increasingly apparent that +the writers have not succeeded in eliminating technical phraseology +sufficiently to bring the subject down to the level of the untrained +reader, and that a _really_ popular exposition of both the Special +and the General Theory is needed. + +Notwithstanding the laudable efforts of some to meet this need it +is manifest that the public continues to be in the dark and many +erroneously suppose that “only twelve men in the world” can really +comprehend this new theory of the universe. Even book reviews ofttimes +betray a surprising lack of appreciation of the fundamentals of +Einstein’s noble contribution to science. + +It is the purpose of this booklet therefore, to bring within the +scope of visualization, as nearly as possible, the very essence of +relativity, and to state its principles in terms devoid of unfamiliar +phrases and illustrations. The popular mind is not satisfied with +vague statements, mathematical equations and formulæ. It calls for an +explanation in words of every day usage, and where a technical term +_must_ be employed then a definition of that term in language +simple and lucid. To meet these requirements without sacrificing +scientific accuracy has been the earnest endeavor of the author. + +Acknowledgment is hereby made to Dr. S. I. Bailey and Dr. G. D. +Birkhoff of Harvard, Dr. Edward Kasner of Columbia, Prof. Victor +Schmidt of Cincinnati, and Dean J. A. Robison of the Oakland Technical +High School (Calif.), all of whom have kindly criticized the manuscript +and made valuable suggestions for improvement of the text.--W. F. H. + +New York, Dec., 1922. + + + + + Introduction to Einstein and Universal Relativity + + + What Is Einstein’s Contribution? + +Albert Einstein, orthodox Jew of German birth and ardent Zionist, is +the most discussed figure in intellectual circles in the world today. +It is not as a Jew nor as a Zionist, however, that we find him so +much in the limelight of public opinion, although as such he is a +distinguished figure and has only recently toured Europe and America +in the interest of the Zionist movement. What has lately made the name +of Einstein a byword upon the lips of scientists, philosophers and the +multitudes of more common mortals is his revolutionary treatment of +time, space and gravitation which materially affects various branches +of accepted science and thereby threatens to throw certain portions of +our school books and other scholarly works into the discard. + +The ideas which led to the development of Einstein’s theory are not +new. Newton appreciated the principle of mechanical relativity and +presented it in much detail. But to Einstein belongs the credit of +completely disentangling the idea of relativity of motion from the +ancient conception of absoluteness of velocity and carrying out the +relativistic viewpoint to its logical conclusion. In so doing, +however, he found it necessary to discard or revamp many popular +theories and “laws of nature.” But consistency is a jewel wherever +found, and Einstein seems to have placed the gem into its proper +setting. + +This Jewish scientist, born in 1878 of humble parentage, began +delving into the mysteries of the universe at an early age. When only +sixteen, it is said, he contributed a paper, modest in its brevity, +which was read before an assemblage of German scientists with marked +appreciation of its merit, and soon thereafter his active mind began +to traverse new and unbeaten paths of scientific thought. During the +years that followed his ideas took more definite form, and in 1905, +while occupying a chair in a Swiss university, Einstein published +his first “Principle of Relativity,” now called the “Special Theory +of Relativity,” which treats the physical laws relating to uniform, +rectilinear motion, i. e., motion in its simplest form. + +Einstein’s hypothesis, unlike most theories advanced, did not originate +with any physical “discovery” or observation. In a sense it grew out +of the philosophic notion of the relativity of all knowledge. He +laid hold of the idea that everything is measured by or considered +_relative_ to something else; that our concepts of absolute +position, absolute motion and absolute time are groundless for the +reason that we have no immovable and unchangeable standard anywhere +in the universe as a starting point. While there is nothing new or +revolutionary in this fact, yet it actually produced certain scientific +“revolutions” when carried out to a consistent end. + +Prior to Einstein, the principle of relativity had been recognized +and philosophized upon to a certain point and then dropped when it +ran counter to hitherto unquestioned conceptions. Einstein, on the +contrary, contended that if the principle is admitted at all it must +be admitted at every point irrespective of what it contradicts. After, +therefore, his theory was fully developed it was then subjected +to certain physical tests, with the result that it has been quite +fully confirmed. This mode of theoretical development is called _a +priori_ as against the more usual _a posteriori_ method, +wherein the theory _follows_ rather than precedes certain +particular phenomena. + +It was in 1915, during the stress of the great war, that Professor +Einstein, working unconcernedly in his laboratory in Berlin, completed +and published what is now called his “General Principle of Relativity.” +Due to hostilities, the details of his developed theory did not reach +the outside world until after the armistice in 1918; but such meagre +accounts of it as did cross the German border were sufficient to awaken +Eddington and other British scientists to the extent that they hastily +equipped two research expeditions, dispatching one to the Island of +Principe off the coast of Africa and the other to Sobral, Brazil, to +test Einstein’s hypothesis about the deflection of light rays in a +gravitational field, by taking photographs of stars near the sun +during the solar eclipse of May 29, 1919, which was total in the +regions mentioned. + +[Illustration: WILLIAM F. HUDGINGS] + +It was the announced results of these expeditions, confirming as they +did this new theory of gravitation, that has made Albert Einstein +famous. + +There are, of course, opponents of the Einstein theory in the circle of +scientists in both Europe and America, though they are comparatively +few in proportion to the number who see merit in his postulates if not, +indeed, a complete solution of gravitational and other phenomena which +have baffled the sages of every century. Sir Oliver Lodge is a leading +opponent of the Einstein theory among a certain few British scientists, +but his opposition is easily accounted for when it is remembered that +Einstein, by ignoring the ether and denying other unproven hypotheses, +has materially upset Sir Oliver’s spiritistic playground. + +Einstein, to say the least, has stimulated the scientific thought +of the world to an extent unprecedented by anyone since the days of +Newton. If the Principle of Relativity is not universal, then any +theory concerning it cannot long stand; it would predict results +which actual experiment would disprove. It must be admitted, however, +that experiments to date tend to confirm the Theory of Relativity as +propounded by Einstein. + + + =Concepts Versus Reality= + +Relativity is a term appropriately applicable to that school of +thought standing midway between mathematics, physics and astronomy, +which concerns itself with determining the relationship between +observers and their objects. Two or more observers may view a certain +phenomenon or set of phenomena and reach divergent conclusions as +to distance, position, size or other physical quantities commonly +involved. It is not for anyone to arbitrarily conclude that one +particular observation is correct and that all the others are +consequently in error. + +It is the task of the relativist to consider the findings of the +various observers, note the conditions attending each, eliminate the +peculiarities of the varying viewpoints and the qualities which have +been unmindfully superimposed by each observer upon the external +object. He must then set down a general rule or law which will embody +the inter-relations of the data before him and which will also hold +good for all other observers regardless of distance, position or motion +of their frames of reference. + +_A frame of reference_ is simply an arbitrary set of mutually +perpendicular lines or planes from which measurements may be made for +purposes of describing the location of points within a given area +or region. If a _two_-dimensional area (i. e., a _plane_ +surface) is being dealt with, then our frame of reference would consist +merely of two _lines_ drawn perpendicular to each other. Thence, +any point within the right-angle formed by these two lines may be +described by specifying its distance from each line. + +If we are to locate a given point in a _three_-dimensional region +(i.e., where length, breadth and also _height_ are involved), then +our frame of reference must consist of _planes_ instead of lines, +three in number, each perpendicular to the other two, like the floor +and two adjacent walls of a room. If any point within a room is to be +located we would first select as a frame of reference, let us say, the +east and south walls together with the floor. The location of the point +may then be described with accuracy by simply specifying its distance +from each of the two walls together with its height above the floor. + +The perpendiculars of a frame of reference are called its _axes_, +and the distances from a given point to the axes are called its +_co-ordinates_. A little reflection will suffice to show that it +is quite impossible to record the exact location of anything, either in +space or in a plane, without making use of a reference frame and two or +more co-ordinates. + + + =Unlimited Choice of Reference Frames= + +We are not limited in our selection or choice of reference frames. +They may be close by or relatively remote from the point or object +whose location we wish to describe. The axes may be physical or purely +imaginary. They may be in any direction or position whatsoever, so +long as they are mutually perpendicular and embrace the area in which +the point or object is located. They may be at rest relative to the +object and in motion relative to the observer, or vice versa, or either +at rest or in motion relative to both. If the relative velocity is +known, the co-ordinates may be determined as certainly as though both +the object and the observer, together with his reference frame, were +relatively at rest. + +An observer usually prefers to select a frame of reference which +partakes of his own motion, for the sake of convenience and simplicity. +Hence for purposes of either local or astronomical observation we may +choose an imaginary axis perpendicular to the surface of the earth. But +we must admit that as the earth rotates our perpendicular rotates with +it. Consequently it is really changing its direction from moment to +moment. Such rotation, however, is not detected so long as we confine +our observations to terrestrial things, because our frame of reference +is at rest relative to the earth and to all things fixed upon the +earth; but for astronomical observations the matter becomes somewhat +complicated due to the various relative motions involved. + +Astronomers often choose as a reference frame an imaginary line +reaching from the earth’s center to the center of the sun, together +with another line perpendicular thereto extending from the earth to +some other planet or point in space. But neither the earth or the sun, +or in fact any other body of matter, is at rest; nor do they move in a +perfectly straight line or plane. The axes of such a frame of reference +continuously undergo a change of direction. Indeed, a reference frame +attached to any physical body in the universe must of necessity move, +because all matter of which we have any knowledge is in motion; and a +frame of reference attached to nothing would be meaningless. + +By reason of the motion of all objects and of frames of reference it +is necessary that the latter be supplemented by clocks for determining +the times of occurrences of physical events encountered in our +measurements. We may therefore say that no frame of reference, when +used for physical purposes, is complete without a set of accurately +synchronized timepieces. The element of time is an essential factor +in any calculation of physical quantities because motion is forever +involved. + +Velocity (rate of motion) means distance traveled within a given +time. Only when distance and time are combined can we determine +our co-ordinates with universal accuracy. This will be more fully +understood when we come to consider the four-dimensional aspect of +the universe. But for all practical purposes the time element cannot +be divorced from the distance measurement of moving bodies even when +viewed in the ordinary three-dimensional sense. + +Suppose, for sake of illustration, we are assigned the task of charting +or recording the precise course of a ball shot from a cannon. In order +to specifically state the history of its flight it will be necessary +to indicate the position of the cannon ball during each moment of +its journey. We will say this is a possibility without attempting +an explanation of the method of measurement used. When our task is +completed we will have a set of four figures for each successive +second, as follows: + +(1) distance from cannon. +(2) distance from ground. +(3) distance to right or left. +(4) duration of time. + +When these four sets of figures are carefully set down they may be +transmitted on paper or by telegraph and the recipient will be able to +determine therefrom the history of the fired ball with as much accuracy +as though he had been personally present and made the observations +himself. This, however, would be impossible were the time factor +omitted. + +The foregoing illustration is given merely to show that time is fully +as important as distance measurement in specifying the location of +any moving body (and everything in the universe is moving). It is in +no sense intended to show how time is really a fourth dimension. This +fact will be brought out in subsequent pages; but thus far we have +followed, rather, our ordinary concepts of time as an independent +element, detached from any unit of space measurement whatsoever. Such a +conception, however, comes from a lifelong training which is based upon +a limited scope of vision. + + + =Knowledge Broadens Many Concepts= + +Concepts do not necessarily represent the reality merely because they +are easy to believe. They may be simply following in a groove caused +by centuries of “thoughtless reasoning.” Prior to Galileo, human +concept taught that “up” and “down” were absolute directions which +never vary. It was easy to so believe, and nobody even thought to call +such concept in question. But eventually it became known that two plumb +lines which point “down” to earth do not hang parallel to each other. +They each point to a common center like the spokes of a wheel. + +Now if two plumb lines happen to be about six thousand miles +(one-fourth the circumference of the earth) apart, they then hang +practically perpendicular to each other. When this erstwhile paradox +was established as a certain fact mankind were obliged to change their +former concepts of the absoluteness of the “up” and “down” directions. +Einstein now asks us to revise our concepts of the absoluteness and +independence of time and space and gives us his reasons therefor. + +In the light of Einstein’s treatment of the principle of relativity it +is seen that no particular reference frame possesses any advantage over +any other for mathematical accuracy irrespective of the physical laws +involved. It would not be difficult, of course, to grasp this fact in +a mechanical sense provided units of length and of time were absolute +quantities which are unaffected by the motion of the reference frame. +But as will be subsequently seen, such units when viewed independently, +do vary constantly, and rigidity becomes in reality a meaningless term. + +In the face of this apparently insurmountable difficulty, however, +Einstein shows in his Special Theory that any unaccelerated (i.e., +uniformly moving) frame of reference is as suitable as any other for +the mathematical expression of physical laws. This is accomplished by +regarding the universe and all objects therein as existing in four +dimensions, viz., length, breadth, thickness and _time_--the +latter altering the length unit according to the relative velocities of +the reference frames. + +It is naturally impossible to mentally visualize or graphically portray +more than three dimensions, but they can be mathematically conceived. +We shall endeavor presently to show how time takes its place alongside +the ordinary three dimensions in the true geometry of the universe, +possessing the value of a fourth co-ordinate or dimension. In reality +it supersedes in importance the other three in the sense that it +possesses the illusive quality of automatically correcting or adjusting +physical values which otherwise would be inconsistently altered by the +velocities of our frames of reference. + +Irrespective of whether we can visualize the matter or not, it will be +necessary for the reader to divest himself of all previous conceptions +of time and of space as universally absolute, separate and unvarying +in their unit length if he would comprehend the Einstein Theory of +Relativity. He must guard against the notion that time and space are +independent elements that should be measured separately. He must +adjust himself to the relativistic viewpoint that time and space are so +interlinked that either, when taken alone, becomes meaningless except +by analogy. When, therefore, we measure the distance between bodies or +the dimensions of the bodies themselves, we are not calculating the +miles or units of _space_ merely, but of _space-time_. + +We will not attempt but will purposely avoid, in a work of this scope, +the setting out of algebraic equations, believing they would not tend +to make the essay _popularly_ readable however much they may +appeal to the mathematical student. Therefore it must suffice here to +state that the numerical value of time required for a light ray to +traverse a given distance, together with the relative velocity between +the frame of reference and the object, become considerable physical +factors in calculation where great distances and enormous velocities +are being dealt with. These items, however, are infinitesimal when +merely earthly distances and ordinary velocities are involved. + +Einstein’s equations, therefore, may be said to have no practical +bearing upon the ordinary things with which we have to do in daily +experience--we may continue to use our yardstick and our pocket +timepiece exactly as before. But this in no wise diminishes the +fundamental importance of the matter. Scientific interest rests not in +the _amount_ of variance from accustomed laws, but rather in the +fact that a variance exists and why. + + + =Universe a Four-Dimensional Continuum= + +We shall at this point merely touch upon the space-time character of +the universe as Einstein sees it, leaving the subject for treatment in +appropriate order later on. Einstein did not originate the geometry +which he uses; he has simply made a masterful application of the work +of Riemann, Minkowski and others in the outworking of his theory. +Various geometers, notably Minkowski, departing from the beaten path of +Euclid, had come to view the universe as a four-dimensional continuum +in which space and time are inseparably interlinked, and Einstein saw +in this a solution of several phenomenal problems which had arisen in +recent years to which the laws of Newton appeared inapplicable. + +The term _continuum_ denotes a continuity of units. The geometer +speaks of a straight line as a one-dimensional continuum, because it +consists of a continuity of _points_ extending in one direction. A +plane, likewise, is termed a two-dimensional continuum for the reason +that it is a continuity of _lines_ laid side by side producing an +area of length and also breadth. Accordingly the whole spatial universe +has been long regarded as a three-dimensional continuum, i. e., a +continuity of _planes_ piled one on top of another, extending from +infinity to infinity. + +As for _time_, it has been commonly looked upon as something +entirely separate and apart from space. Humanity has habitually +regarded it as an independent one-dimensional continuum, i. e., a +continuity of _instants_, each being of universally definite +duration, beginning where its predecessor ended and ending where its +successor begins, and thus flowing on forever regardless of motion, +location or any physical condition. + + + =Time and Space Inseparable= + +But what ground have we for regarding space and time as independent and +unrelated continua? Does not _time_ thrust itself upon us at every +turn, wherever we undertake a measurement in space? This might not be +so if we could regard space as motionless and then really measure it +from point to point. But absolute space cannot be measured. The best +that can be done is to measure from one body of matter to another +body of matter--and all matter is in motion and continuously changing +position, hence the entrance of time into all physical calculation. + +Everything in the universe is somewhere at some time and somewhere else +at some other time. Thus it is seen that time must intersect space at +every point wherever moving matter is involved. And if for any reason +whatsoever the time units are shortened or lengthened then the points +which they intersect are in reality distorted irrespective of what our +conception of such a state of affairs might be. + +Instead, therefore, of regarding the universe as a continuity of +immovable points it is in reality a continuity of _events_ in +each one of which _time_ is an essentially governing factor. +Consequently the term _event_ in a four-dimensional continuum is +analogous to the term _point_ in a three-dimensional continuum. +Something does not necessarily need to _happen_ at each +_event_ in the continuum in order to constitute them “events.” +On the contrary, the continuum of events exists as a background for +phenomena, and when happenings occur in any region whatsoever, the +events (time and space points combined) are there, ready to give forth +their testimony to the mathematician when he calls for his location +data. + +The fact that we cannot diagram such a continuum in no sense detracts +from its reality. A combined space and time is no less real than +the conceptual independent space and time which it supplants. There +are various other continua in the world about us which cannot be +represented by lines and angles or physical models. There is, for +instance, the continuum of color, reaching from ultra violet to ultra +red with its infinite number of graduating hues in between. Then there +is the continuum of sound which cannot be visualized in any degree. +The scale of musical notes is a perfect continuity, extending from +the lowest to the highest audible sound and beyond, yet we cannot +_see_ a continuum of this character any more than we can visualize +the four-dimensional continuum of space-time. + +From the foregoing it will be seen that the difficulty of making clear +in few words the substance of the Einstein theory is due to its radical +departure from our ordinary concepts of things. Many pages might be +utilized to fully define such apparently simple terms as time, space, +distance, straight line, etc., and after these are seen from the +viewpoint of the relativist, then the reader is equipped to proceed +with his study of relativity, but not before. + +The aforementioned handicap has been apparent to every writer who has +attempted to make the theory of relativity popularly readable. The +following pages will discuss these terms in an applied fashion and +will attempt to give the necessary foundation knowledge without which +it would be impossible to appreciate the theory as expounded later in +this essay and in other works on the same subject. The reader will +undoubtedly find it profitable to review the foregoing paragraphs after +he has perused the arguments which follow. + + + =Relativity and Its Effects= + +Relativity, as applied to motion of all matter and systems of matter in +the universe, stands opposed to the idea of _absolute velocity_. +If nothing is stationary in the entire universe from which we may +determine the _actual_ rate of motion of bodies, then the best we +can do to describe the velocity of anything is to say that it moves +at such and such a speed _relative_ to something else which is +also moving at some unknown rate except as it is related to some other +moving body. + +We may say that the earth is traveling at the rate of over eighteen +miles per second in its annual trip around the sun. But this does not +represent the actual velocity of our earth. It describes merely our +motion relative to the sun; but who shall say how rapidly our sun is +moving through space[1] and carrying us with it, just as we carry +the moon with us as we revolve around the sun. We know the sun is +apparently approaching a distant star cluster, but we cannot determine +whether our system is moving toward it or whether the cluster is moving +toward us, or both. We may be, in fact, chasing it through the heavens +as a dog chases a rabbit, and gaining on it a trifle each century; or +it may be really chasing us. All we know about it is that the distance +between the two systems is growing gradually less. + +Then, again, who knows but that the entire stellar universe, including +not only our solar system but all other systems as well, may be +revolving about one common center located in the remote regions +of space? And if so, in what general direction does it revolve? +These reflections immediately convince us that all motion is purely +relative; that no velocity can be looked upon as being absolute. Hence +our eighteen-mile-per-second velocity around the sun is probably +infinitesimal in comparison to our _actual_ speed through space, +if such could really be determined by some stationary standard. + +Einstein did not originate the doctrine of relativity; it has been +a much discussed philosophic subject for centuries and particularly +of the nineteenth century. What he did, however, was to formulate a +particular theory concerning it which co-ordinates and satisfies the +observed laws of nature and accounts for discrepancies which have +long troubled mathematicians and scientists who have based their +calculations on the theories of the past, notably Newton’s laws. + + + =Mercury’s Perihelion= + +A striking example of such a discrepancy which Einstein has accounted +for, is the unusual yearly advancement of the perihelion of Mercury’s +orbit. Due to gravity, all planets revolve about the sun in ellipses +rather than in perfect circles, with the sun a trifle to one side of +the center of such ellipse. This brings the planet nearer to the sun +at one end of the ellipse than at the other. The near end of the orbit +is called the _perihelion_, while the distant end is called the +_aphelion_. See Fig. 1. + +Newton’s law would indicate that if our spherical sun had but one +planet revolving around it, the orbit of that planet would never change +its position unless disturbed by some outside cause, its perihelion +and aphelion being fixed. But where there are more than one planet in +a system, a slow annual advance of the perihelion would be produced. +The amount of such advance is easily calculated; hence it has been +an astonishment to astronomers to find that the perihelion of Mercury +actually advances 42 minutes (that is, seven-tenths of a degree) per +century more than Newton’s law allows for. Einstein, however simply +points out that at perihelion a planet is moving with greater velocity +than at aphelion because of its relative nearness to the sun, and +that its velocity (the _time_ element) must be reckoned with in +addition to the Newtonian gravitational advance. He computed that this +should increase the advance of Mercury’s perihelion by 43 minutes per +century, which most fully accounts for the observed discrepancy. + +[Illustration: FIG. 1.] + +Opponents of Einstein have attempted to account for the aforementioned +discrepancy on the ground that the sun is not a perfect sphere, and +that its equatorial diameter exceeds its polar diameter sufficiently +to add the required amount to the attraction at perihelion. But this +involves other difficulties, as for instance, a change of 3 minutes +per century in the inclination of the orbit, which manifestly does not +exist. The orbits of the other planets in our solar system are not +sufficiently eccentric to reveal any marked difference between Newton’s +and Einstein’s calculated results. But Einstein’s success in connection +with Mercury has placed his theory upon a very satisfactory foundation. + +We have seen that all motion is relative. The same is true of +_time_ because motion and time are inseparable. But even if this +were not so, where would we find an absolute standard or universal unit +of time any more than an absolute rate of motion of matter? We on earth +count time according to the rotation of the earth on its axis, and we +call the period of rotation a day, but the other planets in our solar +system have days of very different length from ours, some shorter and +some longer. + +All heavenly bodies possess their respective time standards, all +different from ours and different from each other. Which shall be +taken as the _absolute_ standard? There is no universal standard. +Time is not an absolute quantity; it is relative even as motion is +relative. A “perfect timekeeper” if suddenly transferred from earth to +Jupiter would immediately be seen to keep a different time due to the +differences in velocity of the two planets. + + + =The Universal Unit= + +We have already emphasized that conceptual time, as an independent +one-dimensional continuum, is fictitious. It does not really exist as +such, but is a component part of space-time. The question naturally +arises in our minds: if time does not exist in and of itself, and if +there is no universal time unit, then how is it possible for Einstein +or anybody else to make a calculation in which time is involved and +arrive at any definite conclusion? The answer is that there is a +universal unit, but not a universal _time_ unit. This true unit +is the _separation-interval_ between events in the space-time +continuum. It is a combination of distance and time. + +Such a combination unit may be partially illustrated by a crude +analogy. Suppose we are calculating the distance between two points +in a plane. We would first describe a triangle and let the hypotenuse +of the triangle connect the two points in question. But somebody else +might erect a different triangle from ours which would describe the +distance between the two points equally well. The triangles would have +the same hypotenuse, but their respective bases and altitudes would be +dissimilar, as shown in Figure 2. + +Let us think of the bases as representing time and the altitudes +as representing space, while the hypotenuse stands for our +separation-interval. Even as we may have many base-altitude +combinations for our common hypotenuse, so we may have numerous +combinations of space and time for the same separation-interval. In +certain combinations the space element is greater than in others, +while the time element is correspondingly smaller, and vice versa, +although the separation-interval, like our hypotenuse, remains a +constant. + +[Illustration: FIG. 2.] + +It may be somewhat easier for the reader to appreciate the +non-existence of a universal unit of length than it is for him to +comprehend the unreality of independent time. No unit of length can +be taken as a universal standard of measure, because measurements +are relative and dependent upon the motion of the observer and his +reference frame, or upon the velocity of the object relative thereto. +This has been scientifically established by experiments made with +particles emitted by radioactive substances whose velocities range +from 20,000 to 170,000 miles per second. + +Lorentz and Fitzgerald, previous to Einstein, had suggested that all +moving matter suffers a physical contraction in the direction of its +motion, but their theory is not particularly convincing. To Einstein +belongs the credit of postulating upon this subject in a manner that +agrees with experiment and satisfactorily answers several phenomenal +questions. He points out that there is an _apparent_ contraction +which is proportional to the relative velocity between object and +observer, but that this “contraction” does not exist if the observer +happens to be moving along with the object which he is measuring. + +To illustrate: If an ocean liner measures 1,000 feet in length while +lying at the pier, theoretically it would be a trifle less than 1,000 +feet while under way if viewed by an observer on shore. If, however, +the measurement were taken by an observer aboard the moving ship, +using the same yardstick that was used at the pier, the result would +still be 1,000 feet. Einstein’s contention is that the ship undergoes +no physical shortening such as Lorentz and Fitzgerald supposed, but +it is simply the victim of a phenomenon of observation. The apparent +contraction, however, holds good for all object or bodies of matter in +exact ratio to their velocity relative to the observer as specified +in Einstein’s mathematical formula, and is just as real in practical +calculation as though it were a physical factor. + + + =Motion and Contraction= + +It is within the realm of possibility, of course, that some degree of +physical contraction does result from motion, on the theory that the +electro-magnetic forces operating between the atoms and molecules of +matter undergo a change due to velocity. If, therefore, the yardstick +and everything else aboard the moving ship suffered a physical +contraction exactly proportional to the length of the ship itself, then +there would be no way of detecting it by any measurement taken aboard +the vessel. + +It hardly appears reasonable, however, that materials of different +density would undergo the same proportional contraction, as for +instance a wooden yardstick and the steel sides of a ship, inasmuch +as they are of entirely different molecular composition. The Einstein +theory therefore proposes an observational variation rather than a +physical contraction of the object, and shows that it equally exists +whether we regard ourselves as at rest and the object as moving away +from us, or whether we consider the object as stationary and ourselves +as speeding past it. + +Let us suppose our observational instruments are lifted from their +fixed position on shore and placed aboard a railway train and carried +in the same direction and at the same velocity as the coastwise vessel. +In this event the ship would measure full 1,000 feet in length just as +it did when we measured it at the pier, because the observer under the +conditions stated would be at rest relative to the moving ship. But if +our train carried us faster than the ship, then the ship would again +begin to measure short because our _relative_ motion would be the +same as though we were stationary and the ship were moving away from us +in the opposite direction. + +The foregoing illustration is merely theoretical, however, because the +variation is too small for observation in cases of small distances +and low velocities. Nevertheless, when the velocity approximates that +of light rays the apparent contraction becomes plainly visible. Some +of the particles emitted by radioactive substances possess a velocity +of about nine-tenths that of light and in such cases the amount of +apparent shortening which they undergo can be computed because it is +very great. And should the velocity become equal to that of light rays +(i. e., 186,300 miles a second), then, says Einstein, the observed +length of the particle would be reduced to zero. + +Einstein does not claim that relative velocities greater than 186,300 +miles a second cannot be attained, but he does contend that velocities +greater than that relative to an observer cannot be observed. Thus if +a body of matter were moving away from an observer at only half the +velocity of light, and the observer himself should suddenly become +accelerated in the opposite direction until the relative velocity +between the two became equal to or greater than that of light, then the +observed _length_ of the body in the direction of motion would be +zero, although its width would be unaffected. + + + =“Absolute Length” Fictitious= + +All this leads us ultimately to the conclusion that there is no such +thing as determining the _absolute_ dimensions of anything, +because relativity of motion and the time element are undeniable +factors in all measurements. The assertion that these are infinitesimal +so far as the quantities we ordinarily have to deal with are concerned +does not alter the fact that “_absolute_ length” is a fictitious +phrase. For this reason we cannot reckon the _absolute distance_ +between any two conceptual points in the universe; we must calculate in +units of _space and time_ combined or else accept the fact that +our conclusions are simply of local and not of universal significance. + +We may, for instance, make a measurement of the distance between +Neptune and the sun according to Euclidean geometry, obtaining a +certain result. But if we were on another planet and there chose a +reference frame which has a wholly different relative velocity to the +frame of reference which we used here on earth, and with this new +reference frame we measured the identically same space between Neptune +and the sun we would obtain a very different result. Which measurement +would be correct? Neither would be correct if by that term we mean the +_absolute_ distance if the universe were motionless and unwarped +by matter. + +Hence in measuring great distances involving enormous velocities +we cannot ignore the principle of relativity and hope to obtain +universally accurate results. It is because of this oversight that +Euclid’s geometry is found to be inadequate in such cases. But as +already suggested, there is a geometry that is universal in its +application, in which _time_ enters as a fourth dimension. The +measurements taken according to such a geometry do not, therefore, +represent distance merely, but a blending of distance and time. This is +the geometry which Einstein employs. It will be discussed in greater +detail further on. + + + =The Special Theory of Relativity= + +We will now consider Einstein’s original theory of 1905, which has +subsequently been called the “Special” or “Restricted” Theory of +Relativity, before attempting an examination of his “General Principle” +which he announced ten years later. By so doing we will find it a +stepping stone of much worth, leading us naturally to the General +Theory which will be discussed later. + +The Special Theory is summarized in two postulates (propositions), +one relating to uniform, straight-ahead motion, and the other to the +velocity of light rays. It is the combination of these two propositions +that necessitates the interrelation of time and space. If both are +true, and the weight of evidence seems to be on Einstein’s side, then +the space-time geometry already alluded to must be correct. In any +event our new conception of space and time as heretofore mentioned has +doubtless prepared our minds to appreciate the postulates which we will +now paraphrase in non-technical terms as follows: + + (1) EVERY LAW OF NATURE MAY BE STATED IN A FORM WHICH WILL HOLD + GOOD BETWEEN ALL OBSERVERS AND OBJECTS PROVIDED THE OBSERVERS AND + OBJECTS ARE EACH MOVING IN A STRAIGHT LINE AND WITH UNIFORM VELOCITY; + AND UNDER THESE CONDITIONS NO OBSERVER COULD POSSIBLY DETECT HIS OWN + MOTION BY ANY LOCAL EXPERIMENT WHATSOEVER, UNLESS HE MAKES REFERENCE + TO OBJECTS OUTSIDE HIS OWN MOVING SYSTEM. + + (2) LIGHT RAYS IF UNOBSTRUCTED HAVE AN OBSERVED CONSTANT VELOCITY + IRRESPECTIVE OF THE RELATIVE VELOCITY BETWEEN THE OBSERVER AND THE + SOURCE OF LIGHT. + +On the surface there is nothing unusual about the first postulate; +it appears to be simply another way of setting forth the mechanical +principle of relativity announced by Newton. But it really involves +more than that, because so long as we regard all bodies and measured +lengths and times as rigid and unchanged by the motion of our reference +frames, then it is not true that every law of nature holds good between +observer and objects as set forth in the postulate. Electro-magnetic +laws, for instance, are an exception, for they really do change their +form in proportion to the relative motion of our reference frames. + +This first postulate simply demands a restatement of the laws of nature +to make them harmonize with the principle of relativity _in toto_. +As already observed, many inconsistencies arise in regard to time, +distances and dimensions if we hold to our old conceptions that units +of time and of lengths are absolute quantities which cannot vary under +any circumstance or condition. We know by experiment that they do vary, +hence the need of a restatement of nature’s laws to account for the +facts resulting from motion. This postulate, however, concerns only +one kind of motion, viz., uniform and rectilinear motion. Rotating and +accelerated and generally irregular motions are dealt with under the +General Theory which will be considered later. + + + =Laws of Nature Not Unalterable= + +In this connection it is well to bear in mind that a “law of nature,” +as the term is commonly used, is not some God-given formula that cannot +be altered. It is but a human description of the operation of nature, +based on observed facts. As our powers of observation and knowledge +increase it sometimes occurs that our “laws of nature” are found to be +inadequate and need revision. That is precisely what Einstein calls +for. In mathematics we may stipulate that “things equal to the same +thing are equal to each other,” but physically speaking the phrase is +wholly ambiguous. Things observed to be equal while in uniform relative +motion lose their equality when their velocities vary. + +It will be noted that Einstein in his first postulate also specifies +that when an observer is moving straight ahead at uniform velocity he +will be unable to detect his own motion and will believe himself to be +at rest unless he performs an experiment on some outside object that +is not moving along with him. The truth of this is apparent to anyone +who has been aboard a smoothly running train while entering or leaving +a station, and has been unable to determine whether it is his train or +the train on the next track that is in motion, or both, until finally +his train ceases to move uniformly and he experiences a jolt as it +suddenly comes to a halt, or is thrown against the back of his seat as +it begins to speed up. Or perhaps while the train was moving uniformly +he performed an observational experiment on some outside object, such +as a house or a telegraph pole, and thereby arrived at the conclusion +that his train is moving rather than the one alongside it. + +Similarly we on earth are prone to consider ourselves as at rest and +the heavenly bodies as moving around us, and for untold centuries +mankind never stopped to question that conception. Even since we +have come to know that we are merely one of the millions of rotating +heavenly spheres which go to make up a universe, we are disinclined +to admit that ours is not in some sense a favored position, or that +an earthly reference frame is not somehow intrinsically better. But +reflection convinces us that this is not so; and since it is not so, +we would do well to put our science on a universal rather than upon a +purely local basis by altering whatever age-old conception is necessary +to make it agree with the principle of relativity. + + + =Paradoxical Behavior of Light= + +It is the second postulate rather than the first that is astounding, +because it substantially states that light rays from a given source +will reach an observer who is running away from the rays just as +quickly, i. e., at the same velocity, as they would reach another +observer who is moving toward the rays. If an explosion should take +place on the sun at this moment it would, of course, take the light +about eight minutes to reach the earth, traveling at a constant +velocity of 186,300 miles a second. Einstein’s declaration is that +if two observers are on opposite sides of the rotating earth, one +revolving away from the sun and the other revolving toward it, the +instruments of each observer will indicate that the rays from the flash +are traveling past him at exactly 186,300 miles a second regardless of +whether he is traveling toward or away from the sun. + +Ordinary concept would assume that in the one case the observer’s own +velocity toward the sun should shorten the time it takes the light +to reach him and thereby make it appear to him that the light is +traveling faster than 186,300 miles a second, while in the other case +we would suppose that the observer’s instruments would indicate that +the light is traveling slower than 186,300 miles a second due to his +own velocity away from the sun which should lengthen the period of +transit of the rays in overtaking him. But actual experiment appears to +contradict this and to corroborate Einstein’s postulate, paradoxical +though it seems to our accustomed concepts. Only by viewing the +universe as a four dimensional (not as a three dimensional) continuum +can the matter be understood. + +It is not, of course, any more strange that when _light_ waves +once start on their journey, the velocity of those waves would +thereafter be unaffected by the movement of the source from which they +originated, than that waves of _water_ would not be increased or +retarded by any forward or backward movement of the ship after the +waves have been started on their course across the lake. Newton knew +that the velocity of light past an observer is not increased by reason +of the source of light moving toward the observer. But what he did not +see was that _neither is the velocity of light increased by reason of +the observer moving toward the light_. + +In other words, after admitting the principle of relativity in a +mechanical sense, practically as stated in Einstein’s first postulate, +Newton then denied or at least overlooked that principle when it +touched the subject of the propagation of light. Take, for instance, +the case of light rays reaching us from certain distant stars. We +cannot surely know whether the source of light is traveling toward us, +or whether we are moving toward the source of light. Newton would say +that the velocity of light would be unaffected in the first instance, +but increased in the second. Einstein says it makes no real difference +which way it is, because the principle of relativity is universal, +all motion is relative, and the universe is so constructed that the +velocity of light always appears constant to all observers irrespective +of their motion or of the motion of the source of light relative to +them. + +If, however, we were traveling away from a given source of light at +a greater velocity than light itself can travel it would certainly +not be true that light, under such a condition, would register a +constant velocity. But such a rate of motion is manifestly impossible +of physical attainment, light being considered to possess the maximum +velocity greater than which no material body can travel. In other +words, 186,300 miles a second is the limiting velocity beyond which +physical phenomena does not reach, because all matter would evidently +suffer complete dissolution, being reduced to the state of free +electrons, by the time such a velocity would be attained. Light is +matter in the free electronic state, and the velocity of such a form of +matter is known to be 186,300 miles a second, as has been stated. + +Light, being a form of matter, is acted upon by gravitation. This was +first demonstrated during the solar eclipse of May 29, 1919, when +photographs were taken of various stars whose light at that season had +to pass very near the sun in order to reach us. Three months later +photographs were taken of these same stars after they had moved from +their former positions. Their normal relative motion being known, it +was easy to determine from the two sets of photographs whether their +displacement was entirely due to that motion. It was found that their +displacement was considerably less than it would have been under normal +conditions, thereby indicating that the light from these stars had been +deflected as it passed near the sun’s rim, making it appear to the +observer that these stars then occupied positions which they did not +occupy. See Fig. 3. + +[Illustration: FIG. 3.] + + + =Electronic Structure of Matter= + +Light and electricity both travel at the same velocity, and in the +final analysis they are the same form of matter. All ponderable matter +is made up of molecules, and molecules consist of atoms of varying +elements, except where the substance is elementary throughout, in which +case the molecules are made up of a certain number of atoms of the same +element. An atom is an aggregation of negatively charged particles of +electricity, called electrons, which revolve and vibrate at enormous +rates around a central nucleus of “protons” carrying a positive charge. +If we conceive of a handful of sand as whirling around and around at +such an enormous velocity as to appear as a globe several feet in +diameter we have a visualization of the ultra microscopic atom and its +comparative dimension to that of the individual electrons and protons +which compose it. + +Electrons and protons are manifestly identical in all form of matter, +but all atoms do not contain the same number of them. An atom of +hydrogen contains only one electron and one proton, while atoms of +heavier matter contain a vast number. It is the _number of electrons +and protons per atom_, together with their respective vibratory +rates, that constitute the difference between elements. Hence every +known form of matter, if reduced to the electronic state, would be +found to consist of the same original stuff. Light, therefore, may be +termed _free electrons_, i. e., electrons which are not bound into +atoms, and consequently glide off in every direction at the highest +possible velocity of which matter is capable, viz., 186,300 miles per +second. + +Thus it may be seen that all other forms of matter, since they consist +of electrons in the atomic or “bound” state, could not possess a +velocity equal to that of _free_ electrons. But electrons do not +become “free” simply by reason of matter undergoing a change of form. +Water, for instance, may be changed into a solid (ice) or into gas +(steam), but the electrons continue to revolve around their nucleus +in true atomic order, and the atoms continue to hold together in the +molecular state, the only change being in their rate of vibration which +affects the degree of elasticity between the molecules and between the +atoms. Hence whether water be in the solid, liquid or gaseous state, +the molecules thereof will be found intact, each consisting of two +atoms of hydrogen and one atom of oxygen (H₂O). + +When matter is being burned up, producing a flame or a glow of light, +this phenomenon indicates that some of the electrons have been freed +from their atomic condition and have started off on their journey in +the form of light rays, while another portion of them remain bound as +atoms but undergo a change of form, becoming either gas or ashes. + + + =Michelson-Morley Experiment= + +Returning now to Einstein’s second postulate, that the velocity of +unobstructed light rays appears to be constant to all observers +irrespective of the relative velocity between the observer and the +source of light. The evidence on which this theory is based was first +produced by Michelson and Morley in 1887 who at that time undertook an +experiment to ascertain if possible the velocity of the earth relative +to the ether. The experiment revealed that light registers a constant +velocity, whether it travels in the direction of the earth’s rotation, +or against the rotation, or at right angles thereto. The experiment +was repeated many times with different apparatus and under various +conditions, but always with the same result. + +It is interesting to note the details of this experiment. Light is +known to travel in waves, because rays coming against each other +from opposite directions can be made to “interfere” in precisely the +same manner as waves in water. If waves in the ocean come together +from different directions, one of two things will happen; they will +either unite and produce a larger wave, or else they will strike in +a manner to measurably kill off both waves. If the crests of the two +waves coincide they reinforce each other, but if they strike at right +angles, the destruction of the wave motion results. This is called +“interference.” + +Light rays conduct themselves in the same manner, thereby revealing +their wave-like nature. This is not contradictory to the idea that +light rays really consist of individual electrons, freed from atomic +attraction. Possibly each separate electron which goes to make up a +ray of light vibrates in a wave-like manner, possessing a wave motion +within itself. Waves of light are exceedingly minute and we do not +ordinarily witness any reinforcement or interference of light rays +about us, because there is no occasion for them to “pile up.” However, +in laboratory experiments, interference of light waves has been +produced, and to whatever extent the interference kills off the wave +motion, to that extent darkness ensues irrespective of the brightness +of the light at its source. + +Michelson and Morley worked on the theory that if they sent rays of +light from west to east (the direction of the earth’s rotation) and +then reflected them back over their course it should take longer +to make the eastward trip than the westward, because in the first +instance the earth is carrying the objective point away from the light +while in the latter instance it would be rushing to meet the oncoming +reflected rays. Under this condition there should be a noticeable +_interference_ of the light waves due to the difference in +distance and time involved in making the two halves of the round +trip. But to the amazement of all there was no interference whatever, +notwithstanding the fact that the apparatus was ten times larger than +it needed to be to reveal such interference of the waves had it really +occurred. + +The conclusion reached by Einstein as a result of this experiment is +that since light rays consist of matter in its basic or electronic +state, freed from atomic attraction, they therefore possess the +limiting velocity of which matter is capable. Hence they could not +travel more rapidly than 186,300 miles a second even if given a quick +send-off, nor would our traveling toward the light affect its apparent +velocity to us--unless it were possible for us to be traveling forward +more rapidly than light itself can travel. This would undoubtedly be +impossible, inasmuch as any physical body would necessarily consist +of electrons in the atomic or “bound” state and therefore could not +possess the mobility that free electrons would enjoy. The universe, +then, being a four-dimensional continuum, is so constructed that the +velocity of light always appears constant to all observers within it. + +This is what Einstein means when he postulates that light _in +vacuuo_ (i. e., unobstructed) possesses a constant velocity +irrespective of the relative velocity of observer and source of light. +That is, it is constant so far as the observer is concerned. Thus if +a flash should occur on any heavenly body and we were moving toward +the flash at say 40,000 miles a second and another observer were +moving away from it at say 60,000 miles a second, the experiment of +each observer would indicate that the light has reached him at exactly +186,300 miles a second, although according to Euclid’s conception of +space the light has been obliged to travel 100,000 miles a second +faster to reach the one observer than the other. But Euclid’s +conception is faulty, as will be seen shortly. + +How, then, would it be possible for the light rays to possess the same +apparent velocity per second for the two observers? It would not be +possible if “time” and “distance” are absolute quantities having the +same meaning for all observers. But if “seconds” and “miles” mean one +thing to observer “A” and a totally different thing to observer “B,” +then the apparent contradiction of facts becomes harmonious. This +is the essence of the doctrine of relativity. Observer “A” himself +does not use the terms “seconds” and “miles” consistently, i. e., as +unvarying quantities, nor does anyone. They mean one thing today and +something else tomorrow, depending upon what we are measuring and the +relative velocity between the observer and the object. The observer +is not aware of this inconsistency. To him there is no inconsistency +whatever. Nevertheless, only by acknowledging the varying quantities of +time and of space, and admitting the geometry which combines the two +into one unit, can the Michelson-Morley experiment and other similar +observations be understood and explained. + + + =Simultaneity a Meaningless Term= + +We have been taught that the true length of a moving body is “the +distance between _simultaneous_ positions of its end points”--a +very good definition, but impossible of application for the reason +that we cannot determine the _simultaneous_ positions of any two +points in the universe. Simultaneity is a meaningless term so long as +the absolute velocity of the observer and the absolute velocity of the +object being measured are unknown. We may know the _relative_ +velocity between them, but that is not sufficient. The two may be +relatively at rest--but for all we know the entire universe may be +speeding through space at thousands of miles a second in either one +direction or another. + +We may _see_ two events occur at the same instant, but that does +not prove that they actually _occurred_ simultaneously. Before we +could compute the exact time of the occurrence of either of the events +we must know the direction in which, and the velocity at which the +universe as a whole is moving, together with any and all velocities of +the observer at the moment. This knowledge we do not possess. Until the +_absolute_ velocity of bodies can be determined the question of +simultaneity must remain unsolved. + + + =The General Principle of Relativity= + +When in 1905 Einstein published the foregoing postulates which are +limited to uniform, rectilinear motion he may have considered that +it would be expecting too much to look for a general principle of +relativity such as would hold good for all kinds of rotating and +irregular motions and by which observers of different and variable +velocities might agree as to the reality of things under their +observation. Concluding, however, that the universe must surely be +constructed in a consistent manner he finally set out to find some rule +or principle by means of which an observer in one region would be seen +to possess no advantage over an observer in any other region of the +great expanse in arriving at accurate conclusions. + +Of course Einstein hardly expects to go to the Pleiades or to +Betelguese and from there take measurements and make calculations; he +is doubtless content to make all his observations from this earth. +But how may he be sure that observations made from a reference frame +located in this particular region of the universe will be true to the +reality since it is manifest that observers located elsewhere and using +different reference frames must necessarily reach conclusions different +from ours if they employed our accustomed laws? Maybe they would be +much nearer the reality than we! What right have _we_ to assume a +monopoly on truth! None whatever until we can formulate nature’s laws +in a manner that will hold good for every part of the universe alike. + +Until we are able to do this our science must be like the vain efforts +of the unskilled fisherman who harpoons for fish. Ignorant of the +trick that water plays on the line of sight he strikes directly at +the spot where he “sees” the fish and always misses his prey. The +skilled harpooner, on the contrary, understands the law of refraction +of light rays in water, and knows how to allow for this refraction; +hence he strikes a little this side of where the fish appears to be +and is rewarded with success. He is guided by a proven law and thereby +ascertains the true location of the fish, whereas the other man follows +“blind” observation which is quite frequently deceptive. + +Einstein’s “General Principle of Relativity” is not, in fact, a mere +generalization of the Special Theory in the sense that it simply +enlarges upon the two postulates which we have already considered. +On the contrary it handles the subject of Relativity from quite a +new standpoint, and therefore might be said to belong to an entirely +different school of thought. It does not lend itself to visualization +as readily as does the Special Theory, and is consequently more +difficult of explanation and comprehension. However, what we have +already learned concerning Relativity will materially aid us in +understanding what follows, for the two theories are, after all, +dealing with the same general subject matter. We shall therefore +endeavor to link the two phases of the subject in a logical and +consistent manner. + +We know, as a matter of fact, that “uniform, straight-ahead motion” +which Einstein in his original theory assumed to exist, is an ideality +that does not appear in nature, because all motion with which we are +familiar is to some extent irregular, nor does any material object +move in a perfectly straight line. But realizing the necessity for a +standard from which to proceed, Einstein properly enough assumed a +standard of absolute perfection and absolute simplicity of motion, +even though it does not actually exist anywhere around us. In exactly +the same manner Euclidean geometry assumes and deals with theoretical +points, lines and planes which have no material existence in fact. + +As set forth in Einstein’s first postulate of the Special Theory, an +observer on a uniformly moving system could not possibly detect the +motion of his system without making reference to some outside object. +In the case of bodies or systems moving irregularly (i. e., with +acceleration) however, an observer thereon would detect “forces” acting +upon himself and upon all other objects on his system, due, of course, +to the acceleration. Recalling the illustration of the moving train: +so long as it is moving with perfect uniformity an observer thereon +would not know he is in motion at all until he made a comparison +with some outside object. But if the train suddenly slows down he is +thrown forward in his seat; if it speeds up he is thrown backward. +This force is called _inertia_. Now if we had never experienced +it before and were put aboard a noiseless and uniformly moving car +from which we could not see out we would be unable to interpret these +strange “forces” that we would feel as the motion of the car became +accelerated. We would probably attempt to explain them as some sort +of magnetic attraction, exactly as we are accustomed to explain the +“force” of gravity. + + + =Gravitation and Inertia= + +In the General Principle of Relativity Einstein deals with these forces +(inertial and gravitational) and attributes them to a common cause, +viz., acceleration of motion, and has put the matter upon a consistent +mathematical basis which at once accounts for certain discrepancies +long observed in Euclidean geometry and in Newton’s laws. It is obvious +enough that where there is no acceleration of motion there could be no +centrifugal or inertial force exhibited: but we have been accustomed +to looking upon gravitation as something entirely different--as a +mysterious drawing power or attractive force that is somehow inherent +in matter. But gravitation is non-existent if we fall with the proper +acceleration. To use Einstein’s own illustration: if we were in a +closed room poised somewhere in gravitational space, and began to +fall with the acceleration common to that field, there would be no +gravitational effects to be observed. Objects released by our hand +would not fall but would remain where they are, and we could raise +ourselves from the floor and stand midway between the floor and the +ceiling as easily as upon the floor itself. + +Again assume we are in a closed room poised in space, in a region +remote from any gravitational field whatsoever. Then suppose we began +to rise with a constant acceleration. Forthwith we would feel our feet +pressing against the floor. Objects released from our hand would strike +the floor by reason of the floor rising up to meet them, and in all +respects the effects would be identical with that of gravitation. In +other words we would have created an artificial gravitational field, +and it would be due to our accelerated motion. + +The characteristics of gravitation and inertia are identical. No +amount of insulation or screening will diminish the “pull” of gravity +on anything. Furthermore, gravity acts on every kind and quantity of +matter alike, so that if a feather weighing less than an ounce and +a pig of lead weighing a ton were held side by side at the top of a +great vacuum tube and allowed to drop at the same instant, the feather +would reach bottom within the same time as the lead, each falling at an +acceleration of approximately 32 feet per second. It is the resistance +of the air that retards the fall of light materials, such as a feather, +but in a vacuum there is no resistance and gravity is found to act on +all matter to the same degree under such conditions. The same is true +of inertia _in vacuuo_. + +When this relationship between the two forces is recognized we are +prepared to believe Einstein when he states that inertial force and +gravitational force are due to a common cause, viz., acceleration. This +does not mean that our earth, for instance, is being accelerated in all +directions at once, expanding out to meet “falling” objects such as in +the case of the artificial gravitational field mentioned in the above +paragraph. It does mean, however, that the falling objects themselves +are accelerated, but as will be presently seen this acceleration is not +due to any attractive force exerted by a “center of gravity” but rather +to a warped condition of space which surrounds all bodies of matter. + +Neither Newton nor Einstein have attempted to analyze the structure +of matter and on this basis explain the phenomenon of gravitation. +Newton evidently believed, however, that every particle of matter +exerts a drawing force upon every other particle of matter, hence he +formulated his law which specifies this attraction between bodies as +being directly proportional to the product of their mass and inversely +proportional to the square of the distance between them. But he did not +attempt to make clear what that “drawing force” is, or _why_ it is +inherent in all matter, nor did he explain how or through what medium +or mechanism it operates. + +Newton contented himself with merely dealing with the phenomenon +of gravitation in the abstract. So does Einstein, but with this +difference: the latter denies the existence of any mechanism whatever +in connection with gravitational force so far as any attractive power +from within is concerned, and accounts for it on purely geometrical +grounds. This is the most difficult phase of the Einstein theory +for the layman to grasp, for the reason that it involves the whole +structure of non-Euclidean geometry with which the public is generally +unfamiliar. + + + =Non-Euclidean Geometry= + +Euclid, the famous Greek mathematician, in the third century B. C. +published the first systematic treatise on geometry (the science of +space and its measurement), and his axioms and theorems are generally +taught in our high schools and colleges today. Euclid proceeded upon +the simple theory that all space consists of points, lines and planes. +He defined a _point_ as that which has position but not size; a +_line_ (continuity of points) as possessing length but no breadth +or thickness; and a _plane_ (continuity of lines) as having length +and breadth, but no thickness. They are simply abstract terms having +no physical existence in nature, except as they exist in our minds. +Nevertheless they have proved themselves convenient in measurement and +calculation. + +But when mathematicians, after centuries of earnest effort, were unable +to prove Euclid’s postulate concerning parallel lines, it occurred +to some of them that possibly the whole Euclidean system rests upon +a faulty foundation. Then it was that Saccheri in Italy, Legendre +in France, Gauss in Germany, Bolyai in Hungary and Lobatschewsky in +Russia, all masters of Euclidean geometry, conceived of other methods +of decomposing space than that proposed in Euclid’s _Elements_. + +Thus it was that early in the nineteenth century, almost simultaneously +in many countries, did many non-Euclidean geometric works come to be +published. These were of the same general character or form, commonly +called Hyperbolic geometry. Each of them is as consistent in itself +as is the geometry of Euclid. But to Riemann belongs the credit of +formulating a geometry which in the light of Einstein is seen to +approach much nearer to the reality of nature than does the Euclidean +or any other system. + +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= + +[Illustration: FIG. 4.] + +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_. + +[Illustration: FIG. 5.] + +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. + +[Illustration: FIG. 6.] + + + =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. + +[Illustration: FIG. 7.] + +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. + +*** END OF THE PROJECT GUTENBERG EBOOK 78274 *** |