The Principle of Relativity

by Albert Einstein; H. Minkowski

Through inner, outer and mixed multiplications of a tensor with the fundamental tensor, tensors of other kinds and of other ranks can be formed.

Example:—

A^μ = g^μσ A_σ

A = g_μν A^μν

We would point out specially the following combinations:

A^μν = g^μα g^νβ A_αβ

A_μν = g_μα g_νβ A^αβ

(complement to the co-variant or contravariant tensors)

and B_μν = g_μν g^αβ A_αβ

We can call B_μν the reduced tensor related to A_μν.