PLUCKER RELATIONS AND THE ELECTROMAGNETIC-FIELD

Every two-dimensional parallelogram in three dimensions casts three parallelogram-shaped shadows onto the three coordinate planes; the signed areas of these shadows are real numbers. Conversely, every set of three real numbers can always be identified with the signed areas of parallelogram-shaped shadows of some one parallelogram. It is never necessary in three dimensions to invoke more than one parallelogram for this purpose. In higher dimensions, however, this converse is not true; to be the signed volumes of the p-dimensional shadows of some one p-dimensional parallelopiped in n dimensions, a set of numbers must satisfy several quadratic equations known collectively as Plucker's relations. When the six electromagnetic field components (three electric, three magnetic) are viewed as signed areas of the shadows of a parallelogram in space time, all inertial observers must agree on the number of parallelogram. Plucker's relations applied to these six numbers must, therefore, produce relativistic invariants.