Page:The principle of relativity (1920).djvu/110

 the 1st kind. We can verify, when w below] is a space-time vector of the 1st kind, [function] of the 2nd kind, the important identity

(45) [w, w[function]] + [w, w[function]*]* = (w] [=w])[function].

The sum of the two space time vectors of the second kind on the left side is to be understood in the sense of the addition of two alternating matrices.

For example, for ω_{1} = 0, ω_{2} = 0, ω_{3} = 0, ω_{4} = i,

ω[function] = | i[function]_{4 1}, i[function]_{4 2}, i[function]_{4 3}, 0 |; ω[function]* = | i[function]_{3 2}, i[function]_{1 3}, i[function]_{2 1}, 0 |

[ω · ω[function]] = 0, 0, 0, [function]_{4 1}, [function]_{4 2}, [function]_{4 3}; [ω · ω[function]*]* = 0, 0, 0, [function]_{3 2}, [function]_{1 3}, [function]_{2 1}.

The fact that in this special case, the relation is satisfied, suffices to establish the theorem (45) generally, for this relation has a covariant character in case of a Lorentz transformation, and is homogeneous in (ω_{1}, ω_{2}, ω_{3}, ω_{4}).

After these preparatory works let us engage ourselves with the equations (C,) (D,) (E) by means which the constants μ, σ will be introduced.

Instead of the space vector u, the velocity of matter, we shall introduce the space-time vector of the first kind ω with the components.

ω_{1} = u_{x}/[sqrt](1 - u_{2}), ω_{2} = u_{y}/[sqrt](1 - u^2), ω_{3} = u_{z}/[sqrt](1 - u^2), ω_{4} = i/[sqrt](1 - u^2).

(40) where ω_{1}^2 + ω_{2}^2 + ω_{3}^2 + ω_{4}^2 = -1 and -iω_{4} > 0.

By F and [function] shall be understood the space time vectors of the second kind M - iE, m - ie.

In Φ = ωF, we have a space time vector of the first kind with components

Φ_{1} = ω_{2}F_{1 2} + ω_{3}F_{1 3} + ω_{4}F_{1 4} }

Φ_{1}_{2}'] = ω_{1}F_{2 1} + ω_{3}F_{2 3} + ω_{4}F_{2 4} }

Φ_{3} = ω_{1}F_{3 1} + ω_{2}F_{3 2} + ω_{4}F_{3 4} }

Φ_{4} = ω_{1}F_{4 1} + ω_{2}F_{4 2} + ω_{3}F_{4 3} }