The Principle of Relativity

by Albert Einstein; H. Minkowski

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is a co-variant tensor of the second rank. We have thus got the result that out of the co-variant tensor of the first rank A_μ = ∂φ/∂x_μ we can get by differentiation a co-variant tensor of 2nd rank

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We call the tensor A_μν the “extension” of the tensor A_μ. Then we can easily show that this combination also leads to a tensor, when the vector A_μ is not representable as a gradient. In order to see this we first remark that ψ (dφ/∂x_μ) is a co-variant four-vector when ψ and φ are scalars. This is also the case for a sum of four such terms :—

when ψ⁽¹⁾, φ⁽¹⁾ ... ψ⁽⁴⁾, φ⁽⁴⁾ are scalars. Now it is however clear that every co-variant four-vector is representable in the form of S_μ.

If for example, A_μ is a four-vector whose components are any given functions of x_ν, we have, (with reference to the chosen co-ordinate system) only to put

ψ⁽¹⁾ = A₁ φ⁽¹⁾ = x₁