The Principle of Relativity

by Albert Einstein; H. Minkowski

B_x = (B_{x x}, B_{x y}, B_{x z}) }

B_y = (B_{y x}, B_{y y}, B_{y z}) } (B_{i k} = - B_{k i}, B_{i i} = 0).

B_z = (B_{z x}, B_{z y}, B_{z z}) }

Since B_{j j} is zero, B_j is perpendicular to the j-axis. The j-component of the vector-product of A and B is equivalent to the scalar product of A and B_j, i.e.,

(A B_j,) = A_x B_{j x} + A_y B_{j y} + A_z B_{j z}.

We see easily that this coincides with the usual rule for the vector-product; e. g., for j = x.

(AB_x) = A_y B_{x y} - A_z B_{z x}.

Correspondingly let us define in the four-dimensional case the product (Pf) of any four-vector P and the six-vector f. The j-component (j = x, y, z, or l) is given by

(Pf_j) = P_xf_{j x} + P_yf_{j y} + P_wf_{j z} + P_zf_{j l}

Each one of these components is obtained as the scalar product of P, and the vector f_j which is perpendicular to j-axis, and is obtained from f by the rule f_j = [(f_{j x}, f_{j y}, f_{j z}, f_{j l}) f_{j j} = 0.]