Page:The principle of relativity (1920).djvu/225

 The first Maxwellian system is obtained by a generalisation of the form given by Minkowski.

We introduce the contra-variant six-vector F_{αβ} by the equation

(62) F^{μν} = g^{μα} g^{νβ} F_{αβ},

and also a contra-variant four-vector J^μ, which is the electrical current-density in vacuum. Then remembering (40) we can establish the system of equations, which remains invariant for any substitution with determinant 1 (according to our choice of co-ordinates).

(63) [part]F^{μν}/[part]x_{ν} = J^μ

If we put

{ F^{2 3} = H´_{x} F^{1 4} = -E´_{x} { (64) { F^{3 1} = H´_{y} F^{2 4} = -E´_{y} {    { F^{1 2} = H´_{z} F^{3 4} = -E´_{z}

which quantities become equal to H_{x} E_{x} in the case of the special relativity theory, and besides

J^1 = i_{x} J^4 = ρ

we get instead of (63)

{ rot H´ - [part]E´/[part]t = i (63a) { { div E´ = ρ