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. | 21. J.Gruszczak, M.Heller
Differential Structure of Space-Time and its Prolongations to Singular Boundary
Int J Theor Phys, vol. 32, pp. 625-648 (1993).
| Abstract:
Abstract | 22. J. Gruszczak, M. Heller, Z. Pogoda
Cauchy Boundaries and b-Incompleteness of Space Times
Int J Theor Phys, vol. 30, pp. 555-565 (1991).
| Abstract:
Abstract | 23. J. Gruszczak, M. Heller and P. Muitarzynski
Physics With and Without the Equivalence Principle
Foundations of Physics, vol. 19, pp. 607-618 (1989).
[abstract] | Abstract:
A differential manifold (d-manifold, for short) can be defined as a pair (M, C), where M is any set and C is a family of real functions on M which is (i) closed with respect to localization and (ii) closed with respect to superposition with
smooth Euclidean functions; one also assumes that (iii) M is locally diffeomorphic to R n. These axioms have a traightforward physical interpretation. Axioms (i) and (ii) formalize certain "compatibility conditions" which usually are supposed to be assumed tacitly by' physicists. Axiom (iii) may be though of
as a (nonmetric) version of Einstein's equivalence principle. By dropping axiom (iii), one obtains a more general structure called a differential space (d-space). Every subset of R ~ turns out to be a d-space. Nevertheless it is mathematically a
workable structure. It might be expected that somewhere in the neighborhood of the Big Bang there is a domain in which space-time is not a d-manifold but still continues to be a d-space. In such a domain we would have a physics without the
(usual form of the) equivalence principle. Simple examples of d-spaces which are not d-manifolds elucidate the principal characteristics the resulting physics would
manifest. | 24. J. Gruszczak, M. Heller and P. Multtarzynski
A Generalization of Manifolds as Space-Time Model
Journal of Mathematical Physics, vol. 29, pp. 2576-2580 (1988).
| Abstract:
Abstract | |
