9 6 0 A P P E N D I X F This can be done for all regions, taking each one separately, that is to say, choosing a system of reference for each location, but not simultaneously for all locations using a single system of reference which means, in physical terms, that it is not possible to cancel out the gravitational field simultaneously in all areas by means of a single accelerated system. (With an elevator in free fall, we cancel out the effects of gravity at the location of the fall, but not in other, distant locations where the magnitude and direction of acceleration are dif- ferent. If the gravitational field were homogeneous, it would be canceled out, of course, with a single accelerated system.) Speaking now of the space-time continuum from a four-dimensional geometrical point of view, we will say that it is Euclidean within small regions, but not within finite regions, which means that the distance between two of its points will be given, in a general way, by the following expression: (5) where, as before, the g values depend on the system of reference and vary continuously with the coordinates. Although those magnitudes depend on the coordinates, Dr. Einstein said, they are not arbitrary. They are—he went on to demonstrate—the components of a tensor that is invariant for the geometrical type we are dealing with, although it varies from one location to another. As in the case of two-dimensional geometric continua, the space-time continuum of gen- eral relativity is perfectly defined, which means that it is possible to locate events in space and time if the fundamental tensor whose components are g values is known. It is clear that in the case we have been studying, the g values have a distinctly physical meaning. They are called gravitational potentials. These potentials provide, as is recognized, the metrical and gravitational properties of space with which we are concerned. Now, he continued, we must establish the differential equations that the g values obey. We have the means for establishing them, he said. In order to do so, we must state the prin- ciple of general relativity in a more exact way, more consistent with the localization method we have used, by saying: “All of the Gaussian coordinate systems are equivalent for the for- mulation of natural laws.” That is, they retain their form with respect to all of them. He then dealt with the extension of the concept of tensor and its operations. He referred specifically to the fundamental tensor (whose components are the g values) and those formed by its determinants. Professor Einstein Gave His Next-to-Last Lecture Yesterday [17 April 1925] Dr. Einstein developed some of Levi-Civita’s geometrical concepts and established var- ious mathematical entities that would be essential for establishing gravitational equations. We will reserve the outline of these questions for the next lecture. ds2 g11dx1 2 2g12dx1dx2 … g44dx4 2 + + + =