Page:The Meaning of Relativity - Albert Einstein (1922).djvu/52

that the clock goes slower than if it were at rest relatively to $K'$. These two consequences, which hold, mutatis mutandis, for every system of reference, form the physical content, free from convention, of the Lorentz transformation.

Addition Theorem for Velocities. If we combine two special Lorentz transformations with the relative velocities $v_{1}$ and $v_{2}$, then the velocity of the single Lorentz transformation which takes the place of the two separate ones is, according to (27), given by

$v_{12}=i\tan(\psi _{1}+\psi _{2})=i{\frac {\tan \psi _{1}+\tan \psi _{2}}{1-\tan \psi _{1}\tan \psi _{2}}}={\frac {v_{1}+v_{2}}{1+v_{1}v_{2}}}.$

(30)

General Statements about the Lorentz Transformation and its Theory of Invariants. The whole theory of invariants of the special theory of relativity depends upon the invariant $s^{2}$ (23). Formally, it has the same rôle in the four-dimensional space-time continuum as the invariant $\Delta x_{1}^{2}+\Delta x_{2}^{2}+\Delta x_{3}^{2}$ in the Euclidean geometry and in the pre-relativity physics. The latter quantity is not an invariant with respect to all the Lorentz transformations; the quantity $s^{2}$ of equation (23) assumes the rôle of this invariant. With respect to an arbitrary inertial system, $s^{2}$ may be determined by measurements; with a given unit of measure it is a completely determinate quantity, associated with an arbitrary pair of events.

The invariant $s^{2}$ differs, disregarding the number of dimensions, from the corresponding invariant of the Euclidean geometry in the following points. In the Euclidean geometry $s^{2}$ is necessarily positive; it vanishes