Page:The principle of relativity (1920).djvu/249

 The representation of the plane (defined by two straight lines) is much more difficult. In three dimensions, the plane can be represented by the vector perpendicular to itself. But that artifice is not available in four dimensions. For the perpendicular to a plane, we now have not a single line, but an infinite number of lines constituting a plane. This difficulty has been overcome by Minkowski in a very elegant manner which will become clear later on. Meanwhile we offer the following extract from the above mentioned work of Sommerfeld.

(Pp. 755, Bd. 32, Ann. d. Physik.)

"In order to have a better knowledge about the nature of the six-vector (which is the same thing as Minkowski's space-time vector of the 2nd kind) let us take the special case of a piece of plane, having unit area (contents), and the form of a parallelogram, bounded by the four-vectors u, v, passing through the origin. Then the projection of this piece of plane on the xy plane is given by the projections u_{x}, u_{y}, v_{x}, v_{y} of the four vectors in the combination

φ_{x y} = u_{x}v_{y} - u_{y}v{x}.

Let us form in a similar manner all the six components of this plane φ. Then six components are not all independent but are connected by the following relation

φ_{y z} φ_{x l} + φ_{z x} φ_{y l} + φ_{x y} φ_{z l} = 0

Further the contents | φ | of the piece of a plane is to be defined as the square root of the sum of the squares of these six quantities. In fact,


 * φ |^2 = φ_{y z}^2 + φ_{z x}^2 + φ_{x y}^2 + φ_{x l}^2 + φ_{y l}^2 + φ_{z l}^2.

Let us now on the other hand take the case of the unit plane φ^* normal to φ; we can call this plane the