13. Editors: Michał Eckstein, Michael Heller, Sebastian J. Szybka
Mathematical Structures of the Universe
Copernicus Center Press (2014) [abstract] [journal]

Abstract:
The book contains a collection of essays on mathematical structures that serve us to model the Universe. The authors discuss such topics as: the interplay between mathematics and physics, geometrical structures in physical models, observational and conceptual aspects of cosmology. The reader can also contemplate the scientific method on the verge of its limits.

14. Nicolas Franco, Michał Eckstein
Noncommutative geometry, Lorentzian structures and causality
in “Mathematical Structures of the Universe”, eds. M. Eckstein, M. Heller, S.J. Szybka, Copernicus Center Press (2014), pp. 315-340 [abstract] [preprint] [journal]

Abstract:
The theory of noncommutative geometry provides an interesting mathematical background for developing new physical models. In particular, it allows one to describe the classical Standard Model coupled to Euclidean gravity. However, noncommutative geometry has mainly been developed using the Euclidean signature, and the typical Lorentzian aspects of space-time, the causal structure in particular, are not taken into account. We present an extension of noncommutative geometry \`a la Connes suitable the for accommodation of Lorentzian structures. In this context, we show that it is possible to recover the notion of causality from purely algebraic data. We explore the causal structure of a simple toy model based on an almost commutative geometry and we show that the coupling between the space-time and an internal noncommutative space establishes a new `speed of light constraint'.

15. Mikko Lavinto, Syksy Rasanen, Sebastian J. Szybka
Average expansion rate and light propagation in a cosmological Tardis spacetime
JCAP, vol. 12, p. 051 (2013).
[abstract] [preprint] [journal] [download]

Abstract:
We construct the first exact statistically homogeneous and isotropic cosmological solution in which inhomogeneity has a significant effect on the expansion rate. The universe is modelled as a Swiss Cheese, with Einstein-de Sitter background and inhomogeneous holes. We show that if the holes are described by the quasispherical Szekeres solution, their average expansion rate is close to the background under certain rather general conditions. We specialise to spherically symmetric holes and violate one of these conditions. As a result, the average expansion rate at late times grows relative to the background, i.e. backreaction is significant. The holes fit smoothly into the background, but are larger on the inside than a corresponding background domain: we call them Tardis regions. We study light propagation, find the effective equations of state and consider the relation of the spatially averaged expansion rate to the redshift and the angular diameter distance.

16. Nicolas Franco, Michał Eckstein
An algebraic formulation of causality for noncommutative geometry
Class. Quantum Grav., vol. 30, p. 135007 (2013).
[abstract] [preprint] [journal]

Abstract:
We propose an algebraic formulation of the notion of causality for spectral triples corresponding to globally hyperbolic manifolds with a well defined noncommutative generalization. The causality is given by a specific cone of Hermitian elements respecting an algebraic condition based on the Dirac operator and a fundamental symmetry. We prove that in the commutative case the usual notion of causality is recovered. We show that, when the dimension of the manifold is even, the result can be extended in order to have an algebraic constraint suitable for a Lorentzian distance formula.

17. Christa R. Ölz, Sebastian J. Szybka
Conformal and projection diagrams in LaTeX
(2013).
[abstract] [preprint]

Abstract:
In general relativity, the causal structure of space-time may sometimes be depicted by conformal Carter-Penrose diagrams or a recent extension of these - the projection diagrams. The introduction of conformal diagrams in the sixties was one of the progenitors of the golden age of relativity. They are the key ingredient of many scientific papers. Unfortunately, drawing them in the form suitable for LaTeX documents is time-consuming and not easy. We present below a library that allows one to draw an arbitrary conformal diagram in a few simple steps.

18. A. Gierzkiewicz, K. Wójcik
Lefschetz sequences and detecting of periodic points
Discrete and Continuous Dynamical Systems - Series A, vol. 32, pp. 81-100 (2012).

Abstract:
Abstract