w₃ s₁ - w₁ s₃,
w₁ s₂ - w₂ s₁,
w₁ s₄ - w₄ s₁,
w₂ s₄ - w₄ s₂,
w₃ s₄ - w₄ s₃,
behaves in case of a Lorentz-transformation as a space-time vector of the II kind. The vector of the second kind with the components (44) are denoted by [w, s]. We see easily that Det½ [w, s] = 0. The dual vector of [w, s] shall be written as [w, s].
If ẇ is a space-time vector of the 1st kind, f of the second kind, w f signifies a 1 × 4 series matrix. In case of a Lorentz-transformation A, w is changed into w′ = wA, f into f′ = A⁻¹ f A,—therefore w′ f′ becomes = (wA A⁻¹ f A) = w f A i.e. w f is transformed as a space-time vector of the 1st kind.[[23]] We can verify, when w is a space-time vector of the 1st kind, f of the 2nd kind, the important identity
(45) [w, wf] + [w, wf*]* = (w] ẇ)f.
The sum of the two space time vectors of the second kind on the left side is to be understood in the sense of the addition of two alternating matrices.
For example, for ω₁ = 0, ω₂ = 0, ω₃ = 0, ω₄ = i,