The Principle of Relativity

by Albert Einstein; H. Minkowski

We can also find out here the geometrical significance of vectors of the third type, when f = φ, i.e., f represents only one plane.

We replace φ by the parallelogram defined by the two four-vectors U, V, and let us pass over to the conjugate plane φ^*, which is formed by the perpendicular four-vectors U^*, V^*. The components of (Pφ) are then equal to the 4 three-rowed under-determinants D_x D_y D_z D_l of the matrix

| P_x P_y P_z P_l |

| |

| U_x^* U_y^* U_z^* U_l^* |

| |

| V_x^* V_y^* V_z^* V_l^* |

Leaving aside the first column we obtain

D_x = P_y(U_z^* V_l^* - U_l^* V_z^*) + P_z(U_l^* V_y^* - U_y^* V_l^*)

+ P_l(U_y^* V_z^* - U_z^* V_y^*)