The Inclusion of Gravitation
With the idea of investigating the problem from the very bottom, Einstein now undertook a broader and more daring point of view. In the first place he said that there is no apparent reason in the great scheme of world events why any one special system of coordinates should be fundamental to the description of phenomena, just as in the special theory a ray of light would appear the same whether viewed from a fixed system or a system moving with constant velocity with respect to the ether. This makes the very broad assumption that no matter what system of coordinates we may use, the mathematical expressions for the laws of nature must be the same. In Einstein’s own words, then, the first principle of this more general theory of relativity must be the following:
“The general laws of nature are expressed through equations which hold for all systems of coordinates, that is, they are covariant with respect to arbitrary substitutions.”[3]
But this was not enough to include gravitation so Einstein next formulated what he was pleased to call his “equivalence hypothesis.” This is best illustrated by an example. Suppose that we are mounting in an elevator and wish to investigate the world of events from our moving platform. We mount more and more rapidly, that is with constant acceleration, and we appear to be in a strong gravitational field due to our own inertia. Suppose, on the other hand, that the elevator descends with an acceleration equal to that of gravity. We would now feel certain that we were in empty space because our own relative acceleration has entirely destroyed that of the earth’s gravitational field and all objects placed upon scales in an elevator would apparently be without weight.
Applying this idea, then, Einstein decided to do away with gravitation entirely by referring all events in a gravitational field to a new set of axes which should move with constant acceleration with respect to the first. In other words we are going to deal with a system moving with uniform acceleration with respect to the ether, just as we considered a system moving with uniform velocity in the special theory.
The next step in the construction of this complicated theory is to reduce these two hypotheses to the language of mathematics and this was accomplished by Einstein with the help of M. Grossmann by means of the theory of tensors.
On account of the very great intricacy of the details, we must content ourselves with the mere statement that this really involved the generalization of the famous expressions known as Laplace’s and Poisson’s equations, on the explicit assumption that these two equations would still describe the gravitational field when we are content to use a first approximation to the truth. The set of ten differential equations which Einstein got as a result of his generalization he called his field equations of gravitation.[4]
[1] Dr. Davis went rather fully into the algebra of the Michelson-Morley experiment. But Dr. Russell has covered the same ground in a form somewhat more advantageous from the typographical viewpoint, and the point is not one which it is profitable to discuss twice; so we eliminate this part of Dr. Davis’ text.—Editor. [↑]
[2] This statement is objectionable, as explained in Chapter IV.—Editor. [↑]
[3] A. Einstein: Die Grundlage der allgemeinen Relativitätstheorie. Ann. d. Physik. 4, vol. 49, page 776. [↑]
[4] At this point we have again used the blue pencil on Dr. Davis’ text, his discussion of the three observational tests of the General Theory adding nothing to Dr. Pickering’s.—The Editor. [↑]
XIV
NEW CONCEPTS FOR OLD
What the World Looks Like After Einstein Has Had His Way With It
BY JOHN G. McHARDY, COMMANDER R.N.,
LONDON
“The new-created world, which fame in heaven
Long had foretold, a fabric wonderful,
Of absolute perfection.”
Einstein’s Theory of Relativity has led to determining a key law of nature—the law of gravitation—which is also the basic law of mechanics. Thus it embraces a whole realm of physics, and promises, through the researches of Professor Weyl, to embrace another realm—electro-dynamics. Its limitations are not yet reached, for Einstein has already postulated therefrom a theory of a finite, yet unbounded, universe. This essay, however, is mainly concerned with mechanics, and electrical forces are not considered.
To have synthesised Newton’s two great principles—his law of motion and law of gravitation—interpreting in the process the empirical law of equality of gravitational and inertial mass, is alone an immense achievement; but Einstein’s researches have opened up a new world to the physicist and philosopher which is of greater importance. He has given us a vision of the immaterial world, a geometrical or mathematical vision, which is more satisfying than the “ether” conceptions hitherto presented. The fabric of his vision is not baseless. It is this fabric we shall consider, touching on certain aspects of the Einstein theory in the endeavor to present an image in miniature of his edifice of thought and to show the firmness of its foundations. That they are well and truly laid was demonstrated by the verification, from observations made during the solar eclipse in 1919, of Einstein’s prediction of the displacement of a wave of light in a gravitational field, showing light to have the property of weight.
The physical world is shown by Einstein to be a world of “relations.” Underlying it there is an absolute world of which physical phenomena are the manifestation. “Give me matter and motion,” says Descartes, “and I will construct the world.” “Give me a world in which there are ordered relations,” says the Relativist, “and I will show you the behavior of matter therein” (mechanics). We first view this underlying world as an abstraction, abstracting energy (“bound” as in matter and electrons, “free” as in light), and its attribute force. This abstraction we will call the “World-Frame.” Later, we will study the underlying world in connection with energy, and will call this absolute world the “World-Fabric.” The connection between the geometrical character of the World-Frame and the geometrical characters of the World-Fabric is the key to the law of gravitation.