The Principle of Relativity

by Albert Einstein; H. Minkowski

Let us now on the other hand take the case of the unit plane φ^* normal to φ; we can call this plane the Complement of φ. Then we have the following relations between the components of the two plane:—

φ_{y z}^* = φ_{x l}, φ_{z x}^* = φ_{y l}, φ_{x y}^* = φ_{z l} φ_{z l}^* = φ_{y x} ...

The proof of these assertions is as follows. Let u^*, v^* be the four vectors defining φ^*. Then we have the following relations:—

u_x^* u_x + u_y^* u_y + u_z^* u_z + u_l^* u_l = 0

u_x^* v_x + u_y^* v_y + u_z^* v_z + u_l^* v_l = 0

v_x^* u_x + v_y^* u_y + v_z^* u_z + v_l^* u_l = 0

v_x^* v_x + v_y^* v_y + v_z^* v_z + v_l^* v_l = 0

If we multiply these equations by v_l, u_l, v_s, and subtract the second from the first, the fourth from the third we obtain

u_x^* φ_{x l} + u_y^* φ_{y l} + u_z^* φ_{z l} = 0

v_x^* φ_{z l} + v_y^* φ_{y l} + v_z^* φ_{z l} = 0