Page:The Meaning of Relativity - Albert Einstein (1922).djvu/54

42 (event) represented in the four-dimensional space of the $$x_1,x_2,x_3,l$$, then all the "points" which can be connected to $$P$$ by means of a light signal lie upon the cone $$s^2 = 0$$ (compare Fig. 1, in which the dimension $$x_3$$ is suppressed). The "upper" half of the cone may contain the "points" to which light signals can be sent from $$P$$; then the "lower" half of the cone will contain the "points" from which light signals can be sent to $$P$$. The points $$P'$$ enclosed by the conical surface furnish, with $$P$$, a negative $$s^2$$; as well as $$P'P$$ is then, according to Minkowski, of the nature of a time. Such intervals represent elements of possible paths of motion, the velocity being less than that of light. In this case the $$l$$-axis may be drawn in the direction of $$PP'$$ by suitably choosing the state of motion of the inertial system. If $$P'$$ lies outside of the "light-cone" then $$PP'$$ is of the nature of a space; in this case, by properly choosing the inertial system, $$\Delta l$$ can be made to vanish.

By the introduction of the imaginary time variable, $$x_4 = il$$, Minkowski has made the theory of invariants for the four-dimensional continuum of physical phenomena fully analogous to the theory of invariants for the three-dimensional continuum of Euclidean space. The theory of four-dimensional tensors of special relativity differs from the theory of tensors in three-dimensional space, therefore, only in the number of dimensions and the relations of reality.