RELATIVITY
By Lieutenant H. K. Lyle. U. S. N.
In the past year considerable publicity has been given to the discoveries of Dr. Albert Einstein. His work has been of great importance in the scientific world but it is impossible to have an accurate understanding of his theories without a fair degree of mental application. This article is an attempt to set forth Dr. Einstein's work in non-technical language omitting the mathematical side as far as possible.
The Theory of Relativity is advanced as an underlying law of nature which covers all. phenomena whether astronomical, optical, dynamic or electro-magnetic. In other words, all of these phenomena possess some similar qualities and are related closely to each other.
Heretofore time has been considered as an abstract quantity but a development of this theory of relativity will show that it is intimately connected with natural phenomena so that it can no longer be considered as a thing apart but more in the nature of a fourth dimension or factor in locating any event.
At present our basis of classical mechanics lies in the laws of Newton and the geometry of Euclid. Both have proved eminently satisfactory for practical engineering and it seems that they must be so firmly established that their position is unassailable. However, it has been definitely known for many years that these laws do not account for every phenomenon which has been observed. Sir Isaac Newton himself was aware of some of the deficiencies and tried without success to surmount them.
The Theory of Relativity has come to the aid of modern physicists and explained some of these events, such as the advance of the perihelion of Mercury. As a result, it has more pr less stated the limitations of Newton's laws and geometry as we have used it.
The reasoning of Dr. Einstein is directed at first principles, and definitions will be necessary at this point so that we may have a complete understanding of later arguments.
The first point in any problem is the necessity of locating any event in space. We do so by means of a "system of coordinates" which may be referred to as a "sphere of motion" or "system."
The "Cartesian system of coordinates" consists of the length of three perpendiculars from the point in question to the three mutually perpendicular planes which compose the frame of reference. The word "length" in this case means that a measuring rod, "L," has been laid along these perpendiculars a certain number of times and this number expresses length in terms of the unit "L." It assumes that the rod "L" does not change during the complete operation.
A system which is sufficiently far removed from all outside influence so that "a body in motion continues that motion in a straight line" is called a "Galileian system."
A "Gaussian system" is one in which an event is located arbitrarily. This system will be fully explained later.
Let us now consider two Galileian systems K and K' where K' has uniform motion in a straight line without rotation with respect to K. The Special or Restricted Theory of Relativity is expressed thus: "Natural phenomena run their course with respect to K' and according to the same general laws as with respect to K."
Take as an example a train moving along a straight track at a constant speed. Natural events may be observed from both systems, that is, from the train and from the embankment alongside the tracks. Suppose the train is running in a vacuum and a stone is dropped from the window. The observer on the train sees it fall in a straight line while the other sees a parabolic curve. The stone only described one path but both observers are correct. Since they were in different frames of motion, this event appeared differently to each.
Notwithstanding this different appearance all equations having to do with the passage of the stone along its trajectory have the same form and the coordinates for one system can be obtained by applying certain equations to the corresponding coordinate of the other system. These equations are known as the Lorentz Transformation.
These transformations are the result of investigations to solve the apparent incompatibility of the constancy of the velocity of light in vacuo and the principle of relativity. The solution resulted in a new conception of time.
It was found that time could no longer be considered as an abstract quantity but that it has a very definite relation to natural phenomena and is directly affected by the state of motion of the clock. Briefly, the result is this. Let us consider two clocks of similar construction which have exactly the same rate. We place one of these clocks on the moving train and the other on the embankment. Their rates will now vary slightly. The clock in motion will be slower and the greater the speed of the train the greater the difference.
This discovery permits the retention of both the law of propagation of light and the principle of relativity.
Investigation along these lines also brought out new ideas of measurements. A measuring rod "L" may be used to determine the length of our train by applying to the rails between the points where the front and rear may be observed at any given instant. The same rod may also be applied to the train itself. According to the principle of relativity we shall not obtain the same answer because the measuring rod undergoes a slight contraction relative to the embankment as a result of the motion of the train.
These statements are evolved after extensive mathematical work which is outside the scope of this article. Therefore, I must ask the reader to accept them without thorough explanation.
The preceding conclusions bring us to believe that Shakespeare was right when he wrote, "All that glitters is not gold." Phenomena have different appearances to observers placed in different frames of motion.
This statement will be readily understood by consideration of our moving train once more. Suppose two events occur simultaneously at two widely separated points along the track, A and B. Since our conception of time has been revised, we must define the word "simultaneous."
An observer situated at M, a point equidistant from A and B, is provided with mirrors so that he may observe both. Light with its constant velocity will take the same interval of time to traverse AM or BM. Therefore if the events which occurred at A and B are observed at M at the same instant, they will be "simultaneous."
But if another observer with similar mirrors is moving on the train in the direction AB the ray of light from A must overtake him while that from B is approaching him and obviously they will not reach him at the same instant. In short, the so-called "simultaneous" events do not appear so to the moving observer because of his motion with respect to the embankment.
All of these examples are formulated under the Special or Restricted Theory of Relativity where the relative motion of one system with respect to the other is limited to a uniform motion in a straight line. It is clear that all motion uniform or accelerated in a straight line or rotation has some effect which cannot be ignored. There are other influences also, such as gravitation, that may not be disregarded. It has been shown that one kind of motion causes a variation in the behavior of measuring rods and clocks. These variations render the Cartesian system of coordinates inadequate for the statement of the General Theory of Relativity.
To overcome this difficulty, Gauss invented a system of coordinates which bears his name. Suppose we have under consideration a surface or two dimensional continuum. This surface is covered with an infinitely large number of arbitrary curves so arranged that two and only two of these curves pass through every point of the continuum. A number is assigned to each curve so that any point may be located by giving the numbers of the curves which intersect there. For continua of more than two dimensions additional curves are used so that each point is the intersection of as many curves as the continuum has dimensions. This Gaussian system is a generalization of the Cartesian system and by virtue of its arbitrary construction effectually overcomes the disadvantages of the Cartesian coordinates.
The General Theory of Relativity may be stated as follows: "All Gaussian coordinate systems are equivalent for the formulation of general laws of nature."
The most interesting part of any study of the theory of relativity lies in the experimental work which supports its results. Because of the infinitesimal intervals of time and distance to be measured, there are only a few of these experiments possible.
According to the principle, it is impossible to detect any motion of the earth with respect to the ether which we regard as space. In other words, by no experiment, optical or dynamic, are we able to determine absolute motion.
Professors Michelson and Morley conducted a series of experiments which, disregarding relativity, should have shown the existence of this ether drift.
Their method may be seen from a consideration of the following example:
A is a source of light, B and C are mirrors 186,000 miles distant from A, AB is perpendicular to AC. All three points are rigidly connected and the entire system is moving along AC at half the velocity of light, 93,000 miles per second.
If the system were at rest, a ray of light released at A would require two seconds to reach B or C and return to A. But because of the indicated motion light will require two seconds to overtake C at D and 2/3 of a second more to return meeting A at some point E making a total of 2.667 seconds.
The ray traveling to B would reach it at some point M requiring 2/√3 seconds and returning in the same time would meet A at some point F, making a total of 4/√3 or 2.309 seconds. The difference is quite perceptible.
If now we fix our mirrors on the surface of the earth and one path of light be along the direction of the ether drift, we should be able to detect this drift or motion of the ether with respect to the earth by this different return of the rays.
In the actual experiments the mirrors were only a few feet apart and the time interval to be measured only about one quadrillionth of a second but the apparatus employed could detect this interval. But in every case the two rays returned in exactly the same length of time.
The Theory of Relativity explains the failure very easily. Our time intervals are all relative to that imaginary fixed point in space. The mirror system is shortened along AC with respect to a system of coordinates attached to this point in such a manner that the experiment results in failure. Since Relativity and this result agree, we have one argument in support of the theory.
Three positive confirmations are required for the retention of the Principle. They are:
- Light is affected by gravity.
- Advance of the perihelion of Mercury.
- Slight difference in the lines of the solar spectrum as compared to the laboratory spectrum.
According to the Theory of Relativity light rays have a curvilinear path in gravitational fields.
Take for example a reference body undergoing an acceleration with respect to another. Any body moving in a straight line with respect to one will have a curvilinear path with respect to the other. Rays of light will come under this principle.
Now consider an observer in a large chest free from all gravitational influences. Newton's law of motion holds absolutely within the chest. But if a constant force be directed up from an outside source, the chest will undergo a constant acceleration.
At first the observer had nothing to hold him to the floor of his chest and the slightest pressure on the floor caused him to float upward. But as the chest is accelerated he notices a downward pressure which must be accounted for. If he knows of gravitational fields and is unaware of his acceleration, he will come to the conclusion that he is in a gravitational field and may carry out various experiments to substantiate his theory. These experiments will run their course in exactly the same manner as if his theory were correct.
Therefore we conclude that an acceleration affects natural phenomena in the same way as gravitation. It is impossible for this observer to tell the difference, because he cannot detect the absolute motion of his chest. Suppose yourself in a train with another standing on the next track. Very often you become aware of a relative motion but until you feel a jar or look at the ground, you cannot tell which train is moving.
To go back, if one reference body be undergoing an acceleration with respect to another, light rays which travel in a straight line with respect to first will travel in a curved line with respect to the second. If an observer on the reference body undergoing the acceleration is unable to distinguish between this acceleration and a possible gravitational field, we may conclude that the light rays will be affected by gravitational force just as they are by the acceleration. In short, light travels in a curvilinear path when under the influence of gravity.
This postulate of Relativity was proved conclusively by observation of light rays passing through the gravitational field of the sun.
During the total eclipse of the sun on May 29, 1919, British astronomers took photographs showing the stars which were visible nearly in line with the sun. Several months later more photographs were taken of the same stars only at this time the sun was in another part of the heavens.
As compared to the second observation the stars on the first plate appeared to be displaced outward from the center of the sun. The calculated curvature of these light rays was only 1.7 seconds of arc and the photographic results coincided very closely with the displacement which would be caused by this figure. The discrepancies in each case were in such a direction as to show more rather than less curvature so that this point is confirmed beyond any doubt.
It has been known for some years that the perihelion of Mercury has been advancing along its orbit at a rate of 43 seconds of arc per century. According to Newton's laws the perihelion of all the planets should remain stationary with respect to the sun.
Calculation under the Theory of Relativity shows that the perihelion of each planet should revolve slowly about the sun and that in the case of Mercury this rotation should be exactly 43 seconds of arc per century.
In the case of the other planets, this advance is too small to be observed with our present means. This difficulty is increased by She fact that Venus, which is next to Mercury in proximity to the sun, travels in an orbit which is nearly a perfect circle and its perihelion is correspondingly difficult to locate exactly.
The last point to be confirmed arises from the statement that the frequency of light waves is dependent on the intensity of the gravitational field in which the source is situated.
As the earth is smaller than the sun, the difference should be detected by a comparison of a spectrum from each source. It manifests itself by a deflection of the lines toward the red in the solar spectrum compared to a laboratory spectrum.
The difference in frequency only amounts to about two millionths of a wave length so it is difficult to measure. However, scientists in various parts of the globe have detected the difference so that it is almost beyond any question of doubt. There are some investigators who have failed to get this result and so this point is still the subject of some dispute.
Taken all in all the results of these experiments are sufficient to establish the Theory of Relativity as a law of nature. The laws of Newton become close approximations when the Principle of Relativity is accepted. His work, like that of Euclid, requires certain conditions which are realized. The reasoning of Euclid which seems so logical and unassailable to the school-boy is correct but as we cannot in practice be sure that a distance measured in one direction is equal to one measured at right angles to it, we are not justified in accepting the various hypotheses of geometry as practical facts from a truly scientific standard.
Geometry then is not an exact science of natural phenomena but as a close approximation it will suffice as a base for mathematics.
The Principle of Relativity gives us a more correct understanding of the laws governing the phenomena of the universe but as it differs only very minutely from our present understanding, our everyday mathematics will not be affected.
As far as popular opinion goes, the greatest result will be the acknowledgement that Dr. Einstein is the greatest physicist of the age, if not of all time.