On the commutator of two 2-forms in Minkowski space-time

Abstract

In the space-time of general relativity, 2-forms play several important roles. But their commutators are generally seen as elements related to their algebraic aspects, rather than to their geometric ones. Here a clear geometric meaning of the commutator of two 2-forms is given. To obtain it, some simple notions on the Minkowski tangent space are needed. The characteristic tetrad and the characteristic planes of two generic planes are introduced. For non generic planes (those that cut each other or coincide), the matrix of their intersections and of the intersections of their orthogonals is considered, and some of its properties analyzed. The orthogonal cut of a plane is defined and the planes that orthogonally cut two given planes are studied. These results allow one to relate very easily the invariant planes of the commutator of two 2-forms to the invariant planes of the 2-forms: the first ones are the single orthogonal cut of the last ones. Some applications are indicated.