Page:Grundgleichungen (Minkowski).djvu/16

I shall add a few more remarks here in order to elucidate the conception of space-time vector of the 2nd kind. Clearly, the following are invariants for such a vector when subjected to a group of Lorentz transformation.

(25)	${\mathfrak {m}}^{2}-{\mathfrak {e}}^{2}=f_{23}^{2}+f_{31}^{2}+f_{12}^{2}+f_{14}^{2}+f_{24}^{2}+f_{34}^{2}$,

(26)	${\mathfrak {me}}=i(f_{23}f_{14}+f_{31}f_{24}+f_{12}f_{34})$,

A space-time vector of the second kind ${\mathfrak {m}},\ -i{\mathfrak {e}}$, where ${\mathfrak {m}}$ and ${\mathfrak {e}}$ are real magnitudes, may be called singular, when the scalar square $({\mathfrak {m}}-i{\mathfrak {e}})^{2}=0$, i.e. ${\mathfrak {m}}^{2}-{\mathfrak {e}}^{2}=0$, and at the same time $({\mathfrak {me}})=0$, i.e. the vector ${\mathfrak {m}}$ and ${\mathfrak {e}}$ are equal and perpendicular to each other; when such is the case, these two properties remain conserved for the space-time vector of the 2nd kind in every Lorentz-transformation.

If the space-time vector of the 2nd kind is not singular, we rotate the spacial co-ordinate system in such a manner that the vector-product $[{\mathfrak {me}}]$ coincides with the z-axis, i.e. ${\mathfrak {m}}_{z}=0,{\mathfrak {e}}_{z}=0$. Then

${\mathfrak {m}}_{x}-i{\mathfrak {e}}_{x})^{2}+({\mathfrak {m}}_{y}-i{\mathfrak {e}}_{y})^{2}\neq 0$

Therefore ${\frac {{\mathfrak {e}}_{y}+i{\mathfrak {m}}_{y}}{{\mathfrak {e}}_{x}+i{\mathfrak {m}}_{x}}}$ is different from $\pm i$, and we can therefore define a complex argument $\varphi +i\psi$ in such a manner that

$tg(\varphi +i\psi )={\frac {{\mathfrak {e}}_{y}+i{\mathfrak {m}}_{y}}{{\mathfrak {e}}_{x}+i{\mathfrak {m}}_{x}}}$

If then, by referring back to equations (9), we carry out the transformation (1) through the angle $\psi$ and a subsequent rotation round the z-axis through the angle $\varphi$, we perform a Lorentz-transformation at the end of which ${\mathfrak {m}}_{y}=0,\ {\mathfrak {e}}_{y}=0$, and therefore ${\mathfrak {m}}$ and ${\mathfrak {e}}$ shall both coincide with the new x-axis. Then by means of the invariants ${\mathfrak {m}}^{2}+{\mathfrak {e}}^{2}$ and $({\mathfrak {me}})$ the final values of these vectors, whether they are of the same or of opposite directions, or whether one of them is equal to zero, would be at once settled.

§ 6. Concept of Time.

By the Lorentz transformation, we are allowed to effect certain changes of the time parameter. In consequence of this fact, it is no longer permissible to speak of the absolute simultaneity of two events. The ordinary idea of simultaneity rather presupposes that six independent parameters, which are evidently required for defining a system of space and time axes, are somehow reduced to three.