Electromagnetism and Gravitation: A Conformal Jigsaw Puzzle

Abstract

It is now known that homological algebra and double duality have brought a revolution in pure mathematics after 1950, particularly in algebraic geometry with the use of extension modules. The aim of this paper is to prove that differential homological algebra and differential double duality bring a similar revolution in physics, particularly in general relativity (GR) and gauge theory (GT) with the use of differential extension modules. Combining differential sequences and their adjoint sequences, we prove that GR is based on two confusions made by Einstein and followers. Indeed, one has been done between the Cauchy = ad(Killing) operator and the “div” operator induced from the Bianchi operator but one has also been done between the deformation of the metric and the stress functions allowing to parametrize the Cauchy operator by means of ad(Ricci) in arbitrary dimension n. We also prove that the two sets of Maxwell equations only depend on the non-linear elations of the conformal group of space-time when n = 4 by using the Spencer δ -cohomology that has only been introduced fifty years later. Like in a puzzle of difficulty increasing with n , these unifying results cannot be even imagined until the ultimate step has been achieved in all cases, a reason for which they have been ignored during one century, even in the cases of low dimension.