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 7 Let us see what mathematical ideas have guided these attempts. The theory of general relativity is based on the metric of space, that is, on the invariance of the line element . We have seen that the laws of gravity were obtained by seeking tensorial quantities derived from the g’s. There have been attempts to simplify this theory which, in its mathematical part, was already given by Riemann, who calculated the tensorial quantities that could be derived from the g’s. Levi-Civita and Weyl managed to simplify substantially the way a sec- ond-order tensor can be produced, and at the same time they came up with a very practical concept: that of parallelism in any kind of manifold. In Euclidean geometry, if we have two points and a vector, drawn from one of them to the other, one can draw another vector par- allel to the first that is, a parallel displacement of the vector can be made. This cannot be done in non-Euclidean continua, for example, on a sphere. But if we take a small part of the continuum, it can be supposed to be Euclidean, and in it to perform the parallel displace- ment of the vector, repeating the operation as many times as is necessary to carry on the (in a certain sense parallel) displacement of the vector all along the curve. If one takes another curve with the same ends, one obtains a different vector, and one arrives at a general for- mula for the parallel displacement: . The first term represents the change of the components of the vector as a consequence of the change of the coordinates of its origin. In these formulas enter forty Γ ’s which define the law of parallel displacement, that is, the affine structure of space. If we have a space of given metric, we find that the Γ ’s are determined by the g’s. It is sufficient to stip- ulate the condition that the modulus of the vector not vary, a natural thing since it involves a repeated Euclidean process that is, we must write that is invariant, which gives the conditions that the Γ ’s must satisfy, and one finds that they are precisely the three-index Christoffel symbols. It is easy to see that Riemann’s fundamental tensor can be obtained with the parallel displacement. In a closed cycle, a vector returns [to its starting point] with a different value, and the difference of value allows to calculate the Christoffel symbols. We can say that Levi-Civita and Weyl have found the meaning of an intermediate quantity that is of utmost importance, and Weyl has generalized geometry with respect to Riemann’s geometry, for [now] we can imagine geometries without metrics, but with a definite law of parallel displacement, that is, with an affine law more general than [that of] the metric. In order to modify the theory as little as possible, he [Einstein] thinks in the following way: the can be measured by solids and clocks that are not immediate things. He is convinced that one cannot do without basic facts. There is a basic fact: the propagation of light given by in this equation the ratios of the g’s appear it turns out thus that the g’s do not, in themselves, have any physical meaning, only their ratios, and therefore we are left with a factor devoid of physical meaning. One finds a new geometry, for if two vectors ds2 dAm Γik m Aldxk –= dxk A2 gikAiAk = ds2 ds 0=