Part 2

=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.