RESEARCH GROUP

MATHEMATICAL STRUCTURES OF THE UNIVERSE

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19. Leszek M. Soko³owski
On the twin paradox in static spacetimes: I. Schwarzschild metric
General Relativity and Gravitation, vol. 44, pp. 1267-1283 (2012).
[abstract] [preprint]

Abstract:
Abstract Motivated by a conjecture put forward by Abramowicz and Bajtlik we reconsider the twin paradox in static spacetimes. According to a well known theorem in Lorentzian geometry the longest timelike worldline between two given points is the unique geodesic line without points conjugate to the initial point on the segment joining the two points. We calculate the proper times for static twins, for twins moving on a circular orbit (if it is a geodesic) around a centre of symmetry and for twins travelling on outgoing and ingoing radial timelike geodesics.We show that the twins on the radial geodesic worldlines are always the oldest ones and we explicitly find the the conjugate points (if they exist) outside the relevant segments. As it is of its own mathematical interest, we find general Jacobi vector fields on the geodesic lines under consideration. In the first part of the work we investigate Schwarzschild geometry. Keywords twin paradox · static spacetimes · Jacobi fields · conjugate points *supported by the grant from The John Templeton Foundation

20. P. Olczykowski, A.Sitarz
K-theory of noncommutative Bieberbach manifolds,
(2012).
[abstract] [preprint]

Abstract:
We compute K-theory of noncommutative Bieberbach manifolds, which quotients of a three-dimensional noncommutative torus by a free action of a cyclic group Z_N, N = 2; 3; 4; 6.

21. Piotr T. Chru¶ciel, Christa R. Ölz, Sebastian J. Szybka
Space-time diagrammatics
Phys. Rev. D: Part. Fields , vol. 86, p. 124041 (2012).
[abstract] [preprint] [journal] [download]

Abstract:
We introduce a new class of two-dimensional diagrams, the \emph{projection diagrams}, as a tool to visualize the global structure of space-times. We construct the diagrams for several metrics of interest, including the Kerr-Newman - (anti) de Sitter family, with or without cosmological constant, and the Emparan-Reall black rings.

22. Andrzej Sitarz
Causality and Noncommutativity
Conference "The Causal Universe", vol. xxx, pp. xxx-xxx (2012).
[abstract] [download]

Abstract:
Noncommutative Geometry o ers a modern mathematical approach to the formulation of physical models, which comprise gravity and gauge theories. We review its basic ideas, applications to models with Lorentzian geometry and challenges it poses to our understanding of causality.

23. A. Sitarz, A. Zaj±c
Spectral action for scalar perturbations of Dirac operators
Lett. Math. Phys., vol. 98, pp. 333-348 (2011).
[abstract] [preprint] [journal] [download]

Abstract:
We investigate the leading terms of the spectral action for odd-dimensional Riemannian spin manifolds with the Dirac operator perturbed by a scalar function. We calculate first two Gilkey–de Witt coefficients and make explicit calculations for the case of n-spheres with a completely symmetric Dirac. In the special case of dimension 3, when such perturbation corresponds to the completely antisymmetric torsion, we carry out the noncommutative calculation following Chamseddine and Connes (J Geom Phys 57:121, 2006) and study the case of SU q (2).

24. B. Iochum, T. Masson, T. Schücker, A. Sitarz
Compact kappa-deformation and spectral triples
Rep. Math. Phys., vol. 68, pp. 37-64 (2011).
[abstract] [journal]

Abstract:
We construct discrete versions of κ-Minkowski space related to a certain compactness of the time coordinate. We show that these models fit into the framework of noncommutative geometry in the sense of spectral triples. The dynamical system of the underlying discrete groups (which include some Baumslag–Solitar groups) is heavily used in order to construct finitely summable spectral triples. This allows to bypass an obstruction to finite-summability appearing when using the common regular representation. The dimension of these spectral triples is unrelated to the number of coordinates defining the κ-deformed Minkowski spaces.

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