L E C T U R E S A T U N I V E R S I T Y O F B U E N O S A I R E S 9 5 9 (Riemann was the first to demonstrate that our ordinary geometry of space could be con- sidered a special case of a three-dimensional, non-Euclidean geometry. He also showed that what works for three-dimensional geometry also works for all geometry of any number of dimensions. The reader will see what this means in the following section.) The Time-Space Continuum in Special and General Relativity In the special theory of relativity—he said—we saw that Minkowski deduced this essen- tial content from the Lorentz transformation equations: That the square of the distance between locations where two events take place mi- nus the square of the speed of light times the square of the time elapsed between them is invariant for all systems moving with rectilinear, uniform motion. If we call that difference , we get: (3) if dx, dy, dz are the differences in the coordinates of the points. Minkowski writes with which, if instead of x, y, z the letters x1, x2, x3 are used, (3) becomes: (4) By means of (4) the location and time of the events can be expressed geometrically. It is the static schematization of the events. Just as a point in ordinary space can be given by three coordinates and the distance between the differences of the similar coordinates, here the spatial-temporal localization is given by the four coordinates , and the spatial-temporal distance between two events by (4). It is, of course, a formal re- presentation. In Minkowski’s universe, made up of that four-dimensional geometric type, Euclidean geometry therefore functions (this is an extension by analogy), since the distance between two of its points is expressed throughout by (4), which is in the same form as that which corresponds to two- and three-dimensional Euclidean continua. I have already demonstrated that four-dimensional gravitational “space” is not Euclide- an in one particular case, he said. It can also be demonstrated in a general way. We saw— he went on—that a gravitational field is equivalent to an acceleration. (This means not only that “the perspective nature offers an observer at rest in a gravitational field is the same one it offers in a field free from that condition to an observer moving at a certain acceleration,” but also that the modifications produced by gravity in a gravitational field will not be no- ticed by an observer moving with a certain acceleration. This observer would see the phe- nomena as they occur in a system without gravitation or acceleration, that is, an inertial system.) Thus, for example, the effect of weight disappears with respect to a system in free fall. (A spring-loaded scale on which a person inside an elevator is standing will produce no reading if the elevator is in free fall. It would be easy to offer other examples.) That is—he continued—with respect to a system falling with the acceleration of gravity, the principle of special relativity applies, so that the fundamental invariant retains form (4). ds2 ds2 dx2 dy2 dz2 c2dt2 – + + = –1ct ds2 dx2 dy2 dz2 c2dt2 – + + = x1, x2, x3, x4