THE GENERAL THEORY
Fragments of Particular Merit on This Phase of the Subject
BY VARIOUS CONTRIBUTORS
When Dorothy was carried by the cyclone from her home in Kansas to the land of Oz, together with her uncle’s house and her little dog Toto, she neglected to lower the trap door over the hole in the floor which formerly led to the cyclone cellar and Toto stepped through. Dorothy rushed to the opening expecting to see him dashed onto the rocks below but found him floating just below the floor. She drew him back into the room and closed the trap.
The author of the chronicle of Dorothy’s adventures explains that the same force which held up the house held up Toto but this explanation is not necessary. Dorothy was now floating through space and house and dog were subject to the same forces of gravitation which gave them identical motions. Dorothy must have pushed the dog down onto the floor and in doing so must herself have floated to the ceiling whence she might have pushed herself back to the floor. In fact gravitation was apparently suspended and Dorothy was in a position to have tried certain experiments which Einstein has never tried because he was never in Dorothy’s unique position.][188]
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The Principle of Equivalence, of which Einstein’s suspended cage experiment is the usual illustration, and upon which the generalized theory of relativity is built, is thus stated by Prof. Eddington: “A gravitational field of force is precisely equivalent to an artificial field of force, so that in any small region it is impossible, by any conceivable experiment, to distinguish between them. In other words, force is purely relative.”
This may be otherwise stated by going back to our idea of a four-dimensional world, the points of which represent the positions and times of events. If we mark in such a space-time the successive positions of an object we get a line, or curve, which represents the whole history of the object, inasmuch as it shows us the position of the object at every time. The reader may imagine that all events happen in one plane, so that only two perpendicular dimensions are needed to fix positions in space, with a third perpendicular dimension for time. He may then conceive, if he may not picture, an analogous process for four-dimensional space-time. These lines, “tracks of objects through space-time,” were called by Minkowski “world-lines.” We may now say that all the events we observe are the intersections of world-lines. The temperature at noon was 70°. This means that if I plot the world-line of the top of the mercury column and the world-line of a certain mark on the glass they intersect in a certain point of space-time. All that we know are intersections of these world-lines. Suppose now we have a large number of them drawn in our four-dimensional world, satisfying all known intersections, and let us suppose the whole embedded in a jelly. We may distort this jelly in any way, changing our coordinates as we please, but we shall neither destroy nor create intersections of world-lines. It may be proved that a change from one system of reference, to which observations are referred, to any other system, moving in any way with respect to the first system, may be pictured as a distortion of the four-dimensional jelly. The laws of nature, therefore, being laws that describe intersections, must be expressible in a form independent of the reference system chosen.
From these postulates, Einstein was able to show such a formulation possible. His law may be stated very simply:—All bodies move through space-time in the straightest possible tracks.
The fact that an easy non-mathematical explanation can not be given, of how this law is reached, or of just why the straightest track of Mercury through space-time will give us an ellipse in space after we have split space-time up into space and time, is no valid objection to the theory. Newton’s law that bodies attract with a force proportional to their masses and inversely proportional to the square of the distance is simple, but no one has ever given an easy non-mathematical proof of how that law requires the path of Mercury to be an ellipse, with the sun at a focus, instead of some other curve.][182]