Functorial Einstein Algebras and the Malicious Singularity
Michael Heller, Tomasz Miller, Leszek Pysiak, Wiesław Sasin
TL;DR
The paper tackles integrating general relativity with supergeometry by introducing Einstein-Grassmann algebras and analyzes how strong, malicious singularities affect the macroscopic body and microscopic soul within this framework. It develops a functorial differential-space approach to lift Einstein algebras to Grassmann stages, yielding Einstein-Grassmann functorial algebras and extending the analysis to Einstein-Grassmann supermanifolds. A key finding is that the soul is annihilated in the smooth, multiplicative setting but can persist when algebraic multiplicativity is relaxed, enabling microstructure to survive at the boundary. The results extend to supercurves and to Einstein-Grassmann supermanifolds, suggesting that singular topology degeneracy coexists with persistent microstructure under relaxed smoothness.
Abstract
Einstein algebra, the concept due to Geroch, is essentially general relativity in an algebraic disguise. We introduce the concept of Einstein-Grassmann algebra as a superalgebra (defining a supermanifold) which is also an Einstein algebra. We employ this concept to confront the supermanifold structure with the structure of strong singularity, the so-called malicious singularity, in general relativity. Einstein-Grassmann algebras consist of two parts: a part called body and a part called soul. For the body part, the singularity theorems apply and the singularities persist as the conclusions of the classical theorems on the existence of singularities require. We prove that, if we relax algebraical requirements, the soul part of the algebra can survive the malicious singularity. In particular, we study the behaviour of supercurves in the presence of malicious singularity.