RESEARCH GROUP

MATHEMATICAL STRUCTURES OF THE UNIVERSE

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37. Michael Heller, Leszek Pysiak, and Wiesław Sasin
Geometry of non-Hausdorff spaces and its significance for physics
J. Math. Phys., vol. 52, p. 043506 (2011).
[abstract] [preprint] [journal]

Abstract:
Hausdorff relation, topologically identifying points in a given space, belongs to elementary tools of modern mathematics. We show that if subtle enough mathematical methods are used to analyze this relation, the conclusions may be far-reaching and illuminating. Examples of situations in which the Hausdorff relation is of the total type, i.e., when it identifies all points of the considered space, are the space of Penrose tilings and space-times of some cosmological models with strong curvature singularities. With every Hausdorff relation a groupoid can be associated, and a convolutive algebra defined on it allows one to analyze the space that otherwise would remain intractable. The regular representation of this algebra in a bundle of Hilbert spaces leads to a von Neumann algebra of random operators. In this way, a probabilistic description (in a generalized sense) naturally takes over when the concept of point looses its meaning. In this situation counterparts of the position and momentum operators can be defined, and they satisfy a commutation relation which, in the suitable limiting case, reproduces the Heisenberg indeterminacy relation. It should be emphasized that this is neither an additional assumption nor an effect of a quantization process, but simply the consequence of a purely geometric analysis.

38. P. Olczykowski, A. Sitarz
On spectral action over Bieberbach manifolds
Acta Phys. Pol., B , vol. 42, p. 1189 (2011).
[abstract] [preprint] [journal] [download]

Abstract:
We compute the leading terms of the spectral action for orientable three dimensional Bieberbach manifolds using two different methods: the Poisson summation formula and the perturbative expansion. Assuming that the cut-off function is not necessarily symmetric we find that the scale invariant part of the perturbative expansion might only differ from the spectral action of the flat three-torus by the value of the eta invariant.

39. Piotr T. Chru¶ciel, Michał Eckstein, Luc Nguyen and Sebastian J. Szybka
Existence of singularities in two-Kerr black holes
Class. Quantum Grav., vol. 28, p. 245017 (2011).
[abstract] [preprint] [journal]

Abstract:
We show that the angular momentum—area inequality 8π|J| ≤ A for weakly stable minimal surfaces would apply to I+-regular many-Kerr solutions, if any existed. Hence, we remove the undesirable hypothesis in the Hennig–Neugebauer proof of non-existence of well-behaved two-component solutions.
*supported by the grant from The John Templeton Foundation

40. Piotr T. Chru¶ciel, Sebastian J. Szybka
Stable causality of the Pomeransky-Senkov black holes
Adv. Theor. Math. Phys., vol. 15, pp. 175-178 (2011).
[abstract] [preprint] [journal] [download]

Abstract:
We show stable causality of the Pomeransky-Senkov black rings.

41. Sebastian J. Szybka
On light propagation in Swiss-Cheese cosmologies
Phys. Rev. D: Part. Fields , vol. 84, p. 044011 (2011).
[abstract] [preprint] [journal] [download]

Abstract:
We study the effect of inhomogeneities on light propagation. The Sachs equations are solved numerically in the Swiss-Cheese models with inhomogeneities modelled by the Lemaitre-Tolman solutions. Our results imply that, within the models we study, inhomogeneities may partially mimic the accelerated expansion of the Universe, but the effect is small.

42. Sebastian J. Szybka
Stable causality of Black Saturns
Journal of High Energy Physics, vol. 2011, pp. 1-8 (2011).
[abstract] [preprint] [journal]

Abstract:
We prove that the Black Saturns are stably causal on the closure of the domain of outer communications.

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