19. Michael Heller and Wiesław Sasin
Noncommutative structure of singularities in general relativity
J. Math. Phys., vol. 37, p. 5665 (1996).
[abstract] [journal]

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 math of orthonormal frames over space–time and carry out the ‘‘desingularization’’ process. We define the line bundle τ:Ω1/2→math over math 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)).

20. Michael Heller and Wieslaw Sasin
Structured spaces and their application to relativistic physics
J. Math. Phys., vol. 36, p. 3644 (1995).
[abstract] [journal]

Abstract:
A sheaf of functions on a topological space is called a differential structure if it satisfies an axiom of a closure with respect to composition with the Euclidean functions. A differential structure on a nonempty set is called a structured space. It is a generalization of the smooth manifold concept and of an earlier concept of differential space. Differential geometry on structured spaces is developed (tangent space, vector fields, differential forms, exterior algebra, linear connection, curvature, and torsion). Some of its techniques are applied to the classical singularity problem in general relativity. It turns out that Einstein’s equations can be defined on space–times with singularities. This can have important consequences for the search of the quantum theory of gravity.