is an invariant in a Lorentz-transformation.
If we divide (ρ₁, ρ₂, ρ₃, ρ₄) by this magnitude, we obtain the four values (ω₁, ω₂, ω₃, ω₄) = (1/√(1 - u²))(ux, uy, uz, i) so that ω₁² + ω₂² + ω₃² + ω₄² = -1.
It is apparent that these four values are determined by the vector u and inversely the vector u of magnitude < 1 follows from the 4 values ω₁, ω₂, ω₃, ω₄; where (ω₁, ω₂, ω₃) are real, -iω₄ real and positive and condition (19) is fulfilled.
The meaning of (ω₁, ω₂, ω₃, ω₄) here is, that they are the ratios of dx₁, dx₂, dx₃, dx₄ to
(20) √(-(dx₁² + dx₂² + dx₃² + dx₄²)) = dt√(1 - u²).
The differentials denoting the displacements of matter occupying the spacetime point (x₁, x₂, x₃, x₄) to the adjacent space-time point.
After the Lorentz-transformation is accomplished the velocity of matter in the new system of reference for the same space-time point (x′ y′ z′ t′) is the vector u′ with the ratios dx′/dt′, dy′/dt′, dz′/dt′, dl′/dt′, as components.
Now it is quite apparent that the system of values
x₁ = ω₁, x₂ = ω₂, x₃ = ω₃, x₄ = ω₄
is transformed into the values