Einstein's Theories of Relativity and Gravitation by Albert Einstein, is part of the HackerNoon Books Series. You can jump to any chapter in this book here. Einstein's Theory of Gravitation
EINSTEIN’S THEORY OF GRAVITATION
The Discussion of the General Theory and Its Most Important Application, from the Essay by
PROF. W. F. G. SWANN, UNIVERSITY OF MINNESOTA, MINNEAPOLIS
Newton’s great discovery regarding the motion of the planets consisted in his showing that these could all be summed up in the following statement: consider any planet in its relation to all particles in the universe. Write down, for the planet, in the line joining it to any particle, an acceleration proportional to the mass of the particle and to the inverse square of its distance from the planet. Then calculate the planet’s resultant acceleration by combining all the accelerations thus obtained.
We have here purposely avoided the use of the word “force,” for Newton’s law is complete as a practical statement of fact without it; and this word adds nothing to the law by way of enhancing its power in actual use. Nevertheless, the fact that the acceleration is made up as it were of non-interfering contributions from each particle in the line joining it to the planet strongly suggests to the mind something of the nature of an elastic pull for which the [328]particle is responsible, and to which the planet’s departure from a straight-line motion is due. The mind likes to think of the elastic; ever since the time of Newton people have sought to devise some mechanism by which these pulls might be visualized as responsible for the phenomena in the same way as one pictures an elastic thread as controlling the motion of a stone which swings around at its end.
This search has been always without success; and now Einstein has found a rather different law which fits the facts better than Newton’s law. It is of such a type that it does not lend itself conveniently to expression in terms of force; the mind would gain nothing by trying to picture such forces as are necessary. It compensates for this, however, in being capable of visualization in terms of what is ultimately a much simpler concept.
In order to appreciate the fundamental ideas involved, suppose for a moment that gravitation could be annihilated, completely, and suppose I find myself upon this earth in empty space. You shall be seated at some point in space and shall watch my doings. If I am in the condition of mind of the people of the reign of King Henry VIII, I shall believe that the earth does not rotate. If I let go a stone, there being no gravity, I shall find that it flies away from me with an acceleration. You will know, however, that the stone really moves in a straight line with constant velocity, and that the apparent acceleration which I perceive is due to the earth’s rotation. If I have argued that acceleration is due to force, I shall say that the earth repels the stone, and shall try to find the law governing the [329]variation of this force with distance. I may go farther, and try to imagine some reason for the force, some pushing action transmitted from the earth to the stone through a surrounding medium; and, you will pity me for all this wasted labor, and particularly for my attempt to find a mechanism to account for the force, since you know that if I would only accept your measurements all would appear so simple.
Let us probe this matter a little farther, however, from the stand-point of myself. I must believe in the reality of the force, since I have to be tied to my chair to prevent my departure from the earth. I might wonder how this field of force would affect the propagation of light, chemical action and so forth. For, even though I had discovered that, by using your measures, I could transform away the apparent effects of my field of force as far as concerned its power to hurl stones about, I could still regard this as a mathematical accident, and believe that the force was really there. Although I might suspect that the same transformation of view-point that would annul the field’s effect as regards the stones would also annul its effect as regards light, etc., I should not be sure of this, as you would be; and my conscience would hardly allow me to do more than look upon the assumption of complete equivalence between the apparent field and a change in the system of measurement as a hypothesis. I should be strongly tempted to make the hypothesis, however.
Now the question raised by Einstein is whether the force of gravity, which we experience as a very [330]real thing, may be put upon a footing which is in some way analogous to that of the obviously fictitious centrifugal force cited above: whether gravitation may be regarded as a figment of our imagination engendered by the way in which we measure things. He found that it could be so regarded. He went still farther, and in his Principle of Equivalence, he postulated that the apparent effects of gravitation in all phenomena could be attributed to the same change in the system of our measurements that would account for the ordinary phenomena of gravitation. On the basis of this hypothesis he was able to deduce for subsequent experimental verification, the effects of gravitation on light. He did not limit himself to such simple changes in our measurements as were sufficient to serve the purpose of the problem of centrifugal force cited above; but, emboldened by the assumptions, in the older theory of relativity, of change in standards of length and time on account of motion, he went even farther than this, and considered the possibility of change of our measures due to mere proximity to matter.
His problem amounted to an attempt to find some way in which it was possible to conceive our scales and clocks as altered, relatively to some more fundamental set, so as to allow of the planetary motions being uniform and rectilinear with respect to these fundamental measures, although they appear as they do to us. If we allow our imaginations perfect freedom as to how the scales may be altered, we shall not balk at assuming alterations varying in any way we please, Einstein does, however, introduce [331]restrictions for reasons which we will now discern.
If we imagine our whole universe, with its observers, planetary orbits, instruments, and everything else, embedded in a jelly, and then distort the jelly and contents in any way, the numbers at which our planetary orbits (or rather their telescopic images) intersect our scales will be unaltered. Moreover, we could vary, in any manner, the times at which all objects (including the clock hands) occupied their distorted positions, and the hand of some clock near the point where the planetary image crossed the scale would record for this occurrence the same dial reading as before. An inhabitant of this distorted universe would be absolutely unconscious of the change. Now the General Theory of Relativity which expresses itself in slightly varied forms, amounts to satisfying a certain philosophical craving of the mind, by asserting that the laws of nature which control our universe ought to be such that another universe like the above, whose inhabitants would be unconscious of their change, would also satisfy these laws, not merely from the standpoint of its own inhabitants, but also from the standpoint of our measurements. In other words, this second universe ought to appear possible to us as well as to its inhabitants.
Einstein decides to make his theory conform to this philosophical desire, and this greatly limits the modifications of clocks and scales which he permits himself for the purpose of representing gravitation. Further, if we express the alterations of the measures as functions of proximity to matter, velocity [332]and so forth, our expressions for these alterations will include, as a particular case, that where matter is absent, although the scales and observer may still remain. Our alteration of the scales and clocks with velocity must thus revert, for this case, to that corresponding to the older theory of relativity, in order to avoid predicting that two observers, in uniform motion relative to each other in empty space, will measure different values for the velocity of light. In this way, the velocity of light comes to play a part in expressing the alterations of the measures.
Even with these restrictions, Einstein was able to do the equivalent of finding an alteration of scales and clocks in the presence of matter which would account for our finding that the planetary motions take place very nearly in accordance with Newton’s law. The new law has accounted with surprising accuracy for certain astronomical irregularities for which Newton’s law failed to account, and has predicted at least one previously unknown phenomenon which was immediately verified.
In conclusion, it may be of interest to state how the new law describes the motion of a particle in the vicinity of a body like the earth. The law amounts to stating that, if we measure a short distance, radially as regards the earth’s center, we must allow for the peculiarity of our units by dividing by
where r is the distance from the earth’s center, m [333]the mass of the earth, c the velocity of light, and G the Newtonian gravitational constant. Tangential measurements require no correction, but intervals of time as measured by our clocks must be multiplied, for each particular place, by the above factor. Then, in terms of the corrected measures so obtained, the particle will be found to describe a straight line with constant velocity although, in terms of our actual measures, it appears to fall with an acceleration.
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This book is part of the public domain. Albert Einstein (2020). Einstein's Theories of Relativity and Gravitation. Urbana, Illinois: Project Gutenberg. Retrieved October 2022.
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