Noncommutative structure of singularities in general relativity

Abstract

Initial and final singularities in the closed Friedman world model are typical examples of malicious singularities. They form the single point of Schmidt’s b-boundary of this model and are not Hausdorff separated from the rest of space–time. The method of noncommutative geometry, developed by A. Connes and his co-workers, is applied to this case. We rephrase Schmidt’s construction in terms of the groupoid Ḡ of orthonormal frames over space–time and carry out the ‘‘desingularization’’ process. We define the line bundle τ:Ω1/2→Ḡ over Ḡ and change the space of its cross sections into an involutive algebra. This algebra is represented in the space of operators on a Hilbert space and, with the norm inherited from these operators, it becomes a C*-algebra. The initial and final singularities of the closed Friedman model are given by two distinct representations of this C*-algebra in the space of operators acting on the Hilbert space L2(O(3,1)).