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 4 9 Straight lines, he continued, that are contained within this cone on passing through the origin, are the only ones that can represent the law of motion of a point, since only for these is the velocity of the point with respect to A less than that of light, as is generally known. Straight lines falling outside the cone, on the other hand, represent the geometrical location of events that are simultaneous for an observer, B, possessing a certain velocity with respect to A. Then, by using fundamental invariance, he dealt with the concepts of the elemental in- terval of length and of proper time. The latter would be the time indicated on a fixed clock to an observer traveling along the linear element of the universe. With Lorentz’s equations, he said, we have arrived at an understanding of the fundamental invariant. This, he added, constitutes the most general mathematical formulation of the special theory of relativity. That is, even when it is derived from those equations, it should not be inferred that its con- tent ends there. It is therefore necessary, he said, to determine which mathematical forms satisfy that invariant. This is indispensable if one wishes to extend the reach of this theory. He then explained the significance of linear and homogeneous transformations, which rep- licate the isotropy of space, words that express the fact that the laws of nature are the same, independent of location and direction. Following this, he explained that vectors had been introduced in physics to preserve the invariance of its equations and with respect to changes in the coordinates affecting direc- tion. He went on to consider other systems of magnitude in a certain sense related to these—tensors—which possess that invariant character. Professor Einstein Gave Fourth Lecture on His Theories of Relativity Yesterday [3 April 1925] Operations with Tensors – Invariance of the Maxwell-Lorentz Equations – Equivalence be- tween Mass and Energy Dr. Einstein began by reviewing the concept of a tensor, which, he said, is a mathemat- ical concept of greater complexity than a scalar or a vector, with the help of which the in- variance of some equations can be established when transformed by means of Lorentz’s equations from a system K to another system K’ that moves relative to the first with recti- linear, uniform motion. We will now consider, he went on, the addition and subtraction of tensors and will con- tinue today with other, not so simple, operations. He immediately demonstrated that through multiplication, less simple tensors, that is, of greater range, can be obtained. He also demonstrated the existence of an operation—contraction—which allows us to obtain a simpler tensor from a given one. He defined and differentiated between symmetrical and antisymmetric tensors, catego- ries, he said, that were important in applications, illustrating how each type retained its own character with respect to all systems connected by the relativity transformation equations.