lor ṡ = ∂s₁/∂x₁ + ∂s₂/∂x₂ + ∂s₃/∂x₃ + ∂s₄/∂x₄.
In case of a Lorentz transformation A, we have
lor ′ṡ′ = lor A. Ās = lor s.
i.e., lor s is an invariant in a Lorentz-transformation.
In all these operations the operator lor plays the part of a space-time vector of the first kind.
If f represents a space-time vector of the second kind,—lor f denotes a space-time vector of the first kind with the components
∂f₁₂/∂x₂ + ∂f₁₃/∂x₃ + ∂f₁₄/∂x₄,
∂f₂₁/∂x₁ + ∂f₂₃/∂x₃ + ∂f₂₄/∂x₄,
∂f₃₁/∂x₁ + ∂f₃₂/∂x₂ + ∂f₃₄/∂x₄,
∂f₄₁/∂x₁ + ∂f₄₂/∂x₂ + ∂f₄₃/∂x₃