Page:The principle of relativity (1920).djvu/223

 equations (57a) together with the given equation between p and ρ, and the equation

g_{αβ} dx_α/ds dx_β/ds = 1,

are sufficient, with the given values of g_{αβ}, for finding out the six unknowns

p, ρ, dx_1/ds, dx_2/ds, dx_3/ds dx_4/ds.

If g_{μν}'s are unknown we have also to take the equtions (53). There are now 11 equations for finding out 10 functions g, so that the number is more than sufficient. Now it is be noticed that the equation (57a) is already contained in (53), so that the latter only represents (7) independent equations. This indefiniteness is due to the wide freedom in the choice of co-ordinates, so that mathematically the problem is indefinite in the sense that three of the space-functions can be arbitrarily chosen.

§20. Maxwell's Electro-Magnetic field-equations.

Let φ_ν be the components of a covariant four-vector, the electro-magnetic potential; from it let us form according to (36) the components F_{ρσ} of the covariant six-vector of the electro-magnetic field according to the system of equations

(59) F_{ρσ} = [part]φ_ρ/[part]x_σ - [part]φ_σ/[part]x_ρ.