RESEARCH GROUP

MATHEMATICAL STRUCTURES OF THE UNIVERSE

#### Selected publications

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 49. A.Sitarz3 1/2 Lectures on Noncommutative GeometryActa Polytechnica, vol. 48, pp. 34-55 (2008). [abstract] [journal] [download] Abstract:We present a short overview of noncommutative geometry. Starting with C* algebras and noncommutative differential forms we pass to K-theory, K-homology and cyclic (co)homology, and we finish with the notion of spectral triples and the spectral action. 50. D.Essouabri, B.Iochum, C.Levy, A.SitarzSpectral action on noncommutative torusJournal of Noncommutative Geometry, vol. 2, pp. 53-123 (2008). [abstract] [preprint] [journal] [download] Abstract:The spectral action on the noncommutative torus is obtained using a Chamseddine–Connes formula via computations of zeta functions. The importance of a Diophantine condition is outlined. Several results on holomorphic continuation of series of holomorphic functions are obtained in this context. 51. J.GruszczakDiscrete Spectrum of the Deficit Angle and the Differential Structure of a Cosmic StringInt J Theor Phys, vol. 47, pp. 2911-2923 (2008). [abstract] Abstract: Differential properties of Klein-Gordon and electromagnetic fields on the space-time of a straight cosmic string are studied with the help of methods of the differential space theory. It is shown that these fields are smooth in the interior of the cosmic string space-time and that they loose this property at the singular boundary except for the cosmic string space-times with the following deficit angles: \delta = 2π(1 −1/n), n = 1, 2, . . . . A connection between smoothness of fields at the conical singularity and the scalar and electromagnetic conical bremsstrahlung is discussed. It is also argued that the smoothness assumption of fields at the singularity is equivalent to the Aliev and Gal’tsov “quantization” condition leading to the above mentioned discrete spectrum of the deficit angle. 52. Leszek M. Soko³owskiStability of a metric f(R) gravity theory implies the Newtonian limitActa Phys. Polon., vol. B39, pp. 2879-2901 (2008). [abstract] [preprint] [journal] Abstract:We show that the existence of the Newtonian limit cannot work as a selection rule for choosing the correct gravity theory fromm the set of all L=f(R) ones. To this end we prove that stability of the ground state solution in arbitrary purely metric f(R) gravity implies the existence of the Newtonian limit of the theory. And the stability is assumed to be the fundamental viability criterion of any gravity theory. The Newtonian limit is either strict in the mathematical sense if the ground state is flat spacetime or approximate and valid on length scales smaller than the cosmological one if the ground state is de Sitter or AdS space. Hence regarding the Newtonian limit a metric f(R) gravity does not differ from GR with arbitrary Lambda. This is exceptional to Lagrangians solely depending on R and/or Ricci tensor. An independent selection rule is necessary. 53. Michael Heller, Zdzis³aw Odrzygó¼d¼, Leszek Pysiak and Wies³aw SasinGravitational Aharonov-Bohm Effect International Journal of Theoretical Physics, vol. 47, pp. 2566-2575 (2008). [abstract] [journal] [download] Abstract:We study a purely gravitational Aharonov-Bohm effect. The space-time curvature is concentrated in the quasiregular singularity of a cosmic string, outside of which space-time is (locally) flat. The symmetries of this field configuration are described by the groupoid symmetries rather than by the usual group symmetries. The groupoid in question is formed by homotopy classes of piecewise smooth paths in the cosmic string region. A gravitational counterpart of the Aharonov-Bohm effect occurs if the symmetry of the system, with respect to the groupoid action, is broken down. 54. Piotr T. Chru¶ciel, Sebastian J. SzybkaOn the Ernst electro-vacuum equations and ergosurfacesActa Phys. Pol., B , vol. 39, pp. 59-75 (2008). [abstract] [preprint] [journal] Abstract:The question of smoothness at the ergosurface of the space-time metric constructed out of solutions (E,phi) of the Ernst electro-vacuum equations is considered. We prove smoothness of those ergosurfaces at which Re(E) provides the dominant contribution to f=-(Re(E)+|phi|^2) at the zero-level-set of f. Some partial results are obtained in the remaining cases: in particular we give examples of leading-order solutions with singular isolated ergocircles".
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