Metric field solutions to Einstein's Field equations can be put into 4
categories that correspond to the four Pfaff classes over space time [Kiehn 1975].
By considering the properties of Cartan's Topological
Stucture it is apparent that solutions of Pfaff class 1, over a connected
Cartan topology, represent massive objects with gravitational forces that are
"infinitely" extendable over the connected domain, hence have long
range. Such solutions are free of electromagnetism for dA=0, and are parity
preserving as dA^dA=0. The Gravitational force. Solutions of Pfaff class 2, again on a connected Cartan topology hence globally extendable, are representative of long range interctions but now as dA # 0, these massive objects can admit long range electromagnetic interactions. As dA^dA = 0, parity is preserved. The Electromagnetic force. Solutions of Pfaff class 3 are "short" range because the Cartan topology is disconnected and therefore the solutions cannot be infinitely extended. These short range forces, which can support both mass and charge, as dA # 0, are parity preserving, as dA^dA=0. The Nuclear force.
Only solutions of Pfaff class 4 can break parity symmetry, for now dA^dA # 0.
These solutions are representative of interactions which are short range, for
the Cartan topology is disconnected, and can support both mass and charge but do
not preserve parity. The Weak force.
Moreover, there is a conjecture that the symmetry between the {+++} signature
and the {+} signature of propagating electromagnetic discontinuities may be
broken in nature.
