These facts can be more concisely expressed in these words: the system of equations (I, and II) as well as the system of equations (III) (IV) are covariant in all cases of Lorentz-transformation, where (ρu, iρ) is to be transformed as a space time vector of the 1st kind, (m - ie) is to be treated as a vector of the 2nd kind, or more significantly,—
(ρu, iρ) is a space time vector of the 1st kind, (m - ie)[1] is a space-time vector of the 2nd kind.
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.
(i) m^2 - e^2 = [function]_{2 3}^2 + [function]_{3 1}^2 + [function]_{1 2}^2 + [function]_{1 4}^2 + [function]_{2 4}^2 ] + [function]_{2 4}^2
me = i([function]_{2 3}[function]_{1 4} + [function]_{3 1}[function]_{2 4} + [function]_{1 2}[function]_{3 4}).
A space-time vector of the second kind (m - ie), where (m, and e) are real magnitudes, may be called singular, when the scalar square (m - ie)^2 = 0, ie m^2 - e^2 = 0, and at the same time (m e) = 0, ie the vector m and 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 [me] coincides with the Z-axis, i.e. m_{x}, = 0, e_{x} = 0. Then
(m_{x}, -ie_{x})^2 + (m_{y}, -ie_{y})^2 [/=] 0,
Therefore (e_{y} + im_{y},)/(e_{x} + ie_{x}) is different from +i, and we can therefore define a complex argument φ + i) in such a manner that
tan (φ + iψ) = (e_{y} + im_{y})/(e_{x} + im_{x}),
- ↑ Vide Note.
