(16) _r__{_v_} = (_r′__{_v_} + _qt′_)/√(1 - _q²_),
_r__{_ṽ_} = _r′__{_ṽ_},
_t_ = (_qr′__{_v_} + _t′_)/√(1 - _q²_),
Now we shall make a very important observation about the vectors _u_ and
_u′_. We can again introduce the indices 1, 2, 3, 4, so that we write
(_x₁_′, _x₂_′, _x₃_′, _x₄_′) instead of (_x′_, _y′_, _z′_, _it′_) and
ρ₁′, ρ₂′, ρ₃′, ρ₄′ instead of (ρ′_u′_{_x′_}, ρ′_u′_{_y′_}, ρ′_u′_{_z′_},
_i_ρ′).
Like the rotation round the Z-axis, the transformation (4), and more
generally the transformations (10), (11), (12), are also linear
transformations with the determinant + 1, so that
(17) _x₁²_ + _x₂²_ + _x₃²_ + _x₄²_ _i. e._ _x²_ + _y²_ + _z²_ -
_t²_,
is transformed into
_x₁′²_ + _x₂′²_ + _x₃′²_ + _x₄′²_ _i. e._ _x′²_ + _y′²_ + _z′²_ -
_t′²_.
On the basis of the equations (13), (14), we shall have (ρ₁² + ρ₂² + ρ₃²
+ ρ₄²) = ρ²(1 - _u__{_x²_}, -_u__{_y²_}, -_u__{_z²_}) = ρ²(1 - _u²_)
transformed into ρ²(1 - _u²_) or in other words,
(18) ρ√(1 - _u²_)
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_))(_u__{_x_}, _u__{_y_},
_u__{_z_}, _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, -_i_ω₄ 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.