The Principle of Relativity

by Albert Einstein; H. Minkowski

for the system (x₁ x₂ x₃ x₄) on the boundary of the sichel, (δx₁ δx₂ δx₃ δx₄) shall vanish for every value of λ and therefore ξ₁, ξ₂, ξ₃, ξ₄ are nil. Then by partial integration, the integral is transformed into the form

∫∫∫∫ ∑ ξ_h(∂νω_hω₁/∂x₁ + ∂νω_hω₂/∂x₂ + ∂νω_hω₃/∂x₃ + ∂νω_hω₄/∂x₄)

dx dy dz dt

the expression within the bracket may be written as

= ω_h ∑ ∂νω_k/∂x_k + ν∑ω_k∂ω_h/∂x_k.

The first sum vanishes in consequence of the continuity equation (b). The second may be written as

(∂ω_h/∂x₁)(dx₁/dτ) + (∂ω_h/∂x₂)(dx₂/dτ) + (∂ω_h/∂x₃)(dx₃/dτ) + (∂ω_h/∂x₄)(dx₄/dτ)

= dω_h/dτ = (d/dτ)(dx_h/dτ)

whereby (d/dτ) is meant the differential quotient in the direction of the space-time line at any position. For the differential quotient (12), we obtain the final expression

(14) ∫∫∫∫ ν((∂ω₁/∂τ)ξ₁ + (∂ω₂/∂τ)ξ₂ + (∂ω₃/∂τ)ξ₃ + (∂ω₄/∂τ)ξ₄)