Functorial Einstein Algebras and the Malicious Singularity

TL;DR

This paper introduces Einstein-Grassmann algebras, a superalgebraic extension of Einstein algebras, to analyze the persistence of singularities in supermanifolds, revealing that the 'soul' part can survive malicious singularities.

Contribution

It defines Einstein-Grassmann algebras and demonstrates how relaxing algebraic constraints allows the 'soul' component to persist through malicious singularities in supermanifolds.

Findings

The 'body' part of Einstein-Grassmann algebra obeys classical singularity theorems.

Relaxing algebraic requirements enables the 'soul' part to survive singularities.

Supercurves can behave differently near malicious singularities in this framework.

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.