THE LOGIC OF RELATIVITY
principle he has called the principle of covariance. In detailed and precise form, it may be stated somewhat as follows:
Principle of Covariance. The laws of nature can be (and are to be) expressed in such mathematical form in terms of the space-and-time coordinates x^, Xt, xt, Xi that they shall remain invariant under every transformation of the form
where the functions f are subject to the following conditions:
1) They are (apart from exceptional points or regions of fewer than four dimensions) finite and continuous and indefinitely differentiable;
2) They are such that the transformation is uniquely reversible, the inverse transformation having the properties demanded for the direct transformation;
3) They are such that in both the transformation and its inverse the fourth variable has the character of a time variable while the other three have the character of space variables.
This principle demands the attainment of an ideal which is admittedly mathematical in its character. By means of it alone one could not come to grips with phenomena. One needs some additional hypotheses. One of these is to the effect that the restricted theory of relativity is valid in free space, that is, in space free of a gravitational field. The other is the celebrated law of the equivalence of gravitational forces and the apparent forces due to acceleration. This may be set forth as follows:
Principle of Equivalence. For an indefinitely small region of the world (that is, a region so small that the variation of gravitation in it in both time and space is negligible) there exists a coordinate system So (Xi, X%, Xz, Xa) with respect to which gravitation has no influence either upon the motions of mass particles or upon any other physical phenomena whatsoever.
Such is the logical basis from which the general theory of relativity proceeds. We can not here follow it in its high enterprise of conquest over the laws of nature. The road (at present and perhaps for a long time to come) can be followed only by one who is willing to give serious and long-continued attention to the study of certain branches of mathematics. In the earlier parts of the argument the reasoning is rather technical and abstruse in character and the general steps are intelligible only to those who have a considerable acquaintance with a certain range of mathematical ideas. After a time the exposition comes down, if not to earth, at least to the solar system and cases begin to appear in which it is possible to find means for choosing between the theory of Newton and that of Einstein.
Three crucial phenomena have been brought to light by means of which to test between the two theories. We shall now speak briefly of each of these.
For a long time astronomers have known that there is a certain forward advance in the perihelion position of the planet Mercury which can not be accounted for on Newton's theory. It amounts to about 42 seconds of angular measure per century. This is well accounted for by Einstein's theory.
Einstein predicted, on the basis of his theory, that a ray of light from a star which is seen apparently close to the edge of the sun would be found to be