Page:Grundgleichungen (Minkowski).djvu/13

If in particular the vector ${\mathfrak {v}}$ of the special Lorentz-transformation be equal to the velocity vector ${\mathfrak {w}}$ of matter at the space-time point $x_{1},\ x_{2},\ x_{3},\ x_{4}$, then it follows out of (10), (11), (12) that

$w'_{1}=0,\ w'_{2}=0,\ w'_{3}=0,\ w'_{4}=i$,

Under these circumstances therefore, the corresponding space-time point has the velocity ${\mathfrak {w}}'=0$ after the transformation, it is as if we transform to rest. We may call the invariant $\varrho {\sqrt {1-{\mathfrak {w}}^{2}}}$ as the rest-density of Electricity.

§ 5. Space-time Vectors. Of the 1st and 2nd kind.

If we take the principal result of the Lorentz transformation together with the fact that the system (A) as well as the system (B) is covariant with respect to a rotation of the coordinate-system round the null point, we obtain the general relativity theorem. In order to make the facts easily comprehensible, it may be more convenient to define a series of expressions, for the purpose of expressing the ideas in a concise form, while on the other hand I shall adhere to the practice of using complex magnitudes, in order to render certain symmetries quite evident.

Let us take a linear homogeneous transformation,

(21)

${\begin{array}{c}x_{1}=\alpha _{11}x'_{1}+\alpha _{12}x'_{2}+\alpha _{13}x'_{3}+\alpha _{14}x'_{4},\\\\x_{2}=\alpha _{21}x'_{1}+\alpha _{22}x'_{2}+\alpha _{23}x'_{3}+\alpha _{24}x'_{4},\\\\x_{3}=\alpha _{31}x'_{1}+\alpha _{32}x'_{2}+\alpha _{33}x'_{3}+\alpha _{34}x'_{4},\\\\x_{4}=\alpha _{41}x'_{1}+\alpha _{42}x'_{2}+\alpha _{43}x'_{3}+\alpha _{44}x'_{4}\end{array}}$

the Determinant of the matrix is +1, all co-efficients without the index 4 occurring once are real, while $\alpha _{14},\ \alpha _{24},\ \alpha _{34}$, are purely imaginary, but $\alpha _{44}$ is real and $>0$, and $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}$ transforms into $x_{1}^{'2}+x_{2}^{'2}+x_{3}^{'2}+x_{4}^{'2}$. The operation shall be called a general Lorentz transformation.

If we put $x'_{1}=x',\ x'_{2}=y',\ x'_{3}=z',\ x'_{4}=it'$ then immediately there occurs a homogeneous linear transformation