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

(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.^[1] 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.

1. That material velocities exceeding that of light are not possible, follows from the appearance of the radical ${\sqrt {1-v^{2}}}$ in the special Lorentz transformation (29).