The Principle of Relativity

by Albert Einstein; H. Minkowski

g_μσ g_ντ g^στ dx_μ dx_ν

Now according to the rules of multiplication, of the fore-going paragraph, the magnitudes

dξ_σ = g_μσ dx_μ

forms a co-variant four-vector, and in fact (on account of the arbitrary choice of dx_μ) any arbitrary four-vector.

If we introduce it in our expression, we get

ds² = g^στ dξ_σ dξ_τ.

For any choice of the vectors dξ_σ dξ_τ this is scalar, and g^στ, according to its definition is a symmetrical thing in σ and τ, so it follows from the above results, that g^στ is a contravariant tensor. Out of (16) it also follows that δ^ν_μ is a tensor which we may call the mixed fundamental tensor.

Determinant of the fundamental tensor.

According to the law of multiplication of determinants, we have

| g_μα g^αν | = | g_μα | | g^αν |