The Principle of Relativity

by Albert Einstein; H. Minkowski

ψ⁽²⁾ = A₂ φ⁽²⁾ = x₂

ψ⁽³⁾ = A₃ φ⁽³⁾ = x₃

ψ⁽⁴⁾ = A₄ φ⁽⁴⁾ = x₄.

in order to arrive at the result that S_μ is equal to A_μ.

In order to prove then that A_μν is a tensor when on the right side of (26) we substitute any co-variant four-vector for A_μ we have only to show that this is true for the four-vector S_μ. For this latter case, however, a glance on the right hand side of (26) will show that we have only to bring forth the proof for the case when

A_μ = ψ ∂φ/∂x_μ.

Now the right hand side of (25) multiplied by ψ is

which has a tensor character. Similarly, (∂φ/∂x_μ) (∂φ/∂x_ν) is also a tensor (outer product of two four-vectors).

Through addition follows the tensor character of