Note 11
(page 17)
Space-time vectors of the first and the second kind.
As we had already occasion to mention, Sommerfeld has, in two papers on four dimensional geometry (vide, Annalen der Physik, Bd. 32, p. 749; and Bd. 33, p. 649), translated the ideas of Minkowski into the language of four dimensional geometry. Instead of Minkowski’s space-time vector of the first kind, he uses the more expressive term ‘four-vector,’ thereby making it quite clear that it represents a directed quantity like a straight line, a force or a momentum, and has got 4 components, three in the direction of space-axes, and one in the direction of the time-axis.
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 ux, uy, vx, vy of the four vectors in the combination
φx y = uxvy - uyv{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
φy z φx l + φz x φy l + φx y φz 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 = φy z² + φz x² + φx y² + φx l² + φy l² + φz l².