The Principle of Relativity

by Albert Einstein; H. Minkowski

A_β^δ = A_αβ^αδ = (∑_α A_αβ^αδ)

and from it again by “reduction” the tensor of the zero rank

A = A_β^β = A_αβ^αβ.

The proof that the result of reduction retains a truly tensorial character, follows either from the representation of tensor according to the generalisation of (12) in combination with (6) or out of the generalisation of (13).

Inner and mixed multiplication of Tensors.

This consists in the combination of outer multiplication with reduction. Examples:—From the co-variant tensor of the second rank A_μν and the contravariant tensor of the first rank B^σ we get by outer multiplication the mixed tensor

D^σ_μν = A_μν B^σ .

Through reduction according to indices ν and σ (i.e., putting ν = σ), the co-variant four vector

D_μ = D^ν_μν = A_μν B^ν is generated.

These we denote as the inner product of the tensor A_μν and B^σ. Similarly we get from the tensors A_μν and B^στ through outer multiplication and two-fold reduction the inner product A_μν B^μν. Through outer multiplication and one-fold reduction we get out of A_μν and B^στ, the mixed tensor of the second rank D^τ_μ = A_μν B^τν. We can fitly call this operation a mixed one; for it is outer with reference to the indices μ and τ and inner with respect to the indices ν and σ.