L E C T U R E S A T T H E U N I V E R S I T Y O F M A D R I D 8 8 3 It turns out that not only the gravitational field but also the metric are given by the g’s, which are functions of the coordinates. These functions are arbitrary, since the x’s are also arbitrary. If you have understood well what I have just said, which is the essential part of the theory, the further difficulties are only formal ones. The first question to face is that of finding a law of pure gravitation, that is, in a field where there is only gravitation. We will follow a method analogous to that of special rela- tivity, for we will look for laws that do not change their form when the system of reference is changed arbitrarily, as long as the transformation is continuous. This condition is very strict and only a few laws satisfy it. In order to obtain the law, one can put a limit on the order of derivatives that must figure in it. In classical mechanics and in Laplace’s equation, the maximum order of derivatives is the second and the equation is linear. We can admit the same condition for general relativ- ity thus, the equations are determined and one finds the law of gravitation. Next we need a law of motion analogous to Galileo’s, and this we find in the condition that a free point describes a geodesic, a direct translation of the law of Galileo [from the language of classical to relativistic physics]. It is frequently said that the theory of relativity is a revolution. This is not true. Rather it is, in a certain way, a translation: the language varies but the basic idea is the same. It is not a simple translation but it adds the condition of the covariance of natural laws. If there is matter in the field, the Laplace-Poisson equation does not have a zero second term, but the density of mass figures in it. In general relativity, the corresponding equation also has a second [third] term, which is also a density—not a scalar but a tensorial one, with ten components. The form of the equations is . To obtain it, the principle of the conservation of energy has been applied. The method for obtaining the equations is formal. We have used the postulate that the maximum order of derivatives figuring in them is the second. As a result, the method says nothing about physical meaning—even a priori it may be doubted whether it implies any- thing similar to the laws of Newton, because instead of one potential, now we have ten quantities. Therefore, one must see whether, as a first approximation, Newton’s theory can be obtained and, in effect, with such a precision that no phenomenon can be found with any divergence from it. This case shows how necessary speculative methods are for science. The equations are of such enormous complexity (as anyone who has dealt with any particular problem knows well) that it is impossible to obtain them by the inductive method, because a general prin- ciple is required as a deductive base. Naturally, we get all the general principles from expe- rience in a more or less direct way. The Consequences There are three theoretical consequences of the theory that lead to measurable quantities: First . The elliptical orbits of planets are not absolutely fixed but each rotates in its plane in the direction of the planet’s revolution. This movement is unobservable for the planets, Rik 1 2 --gikR - – fTik =