RESEARCH GROUP

MATHEMATICAL STRUCTURES OF THE UNIVERSE

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7. Nicolas Franco
Temporal Lorentzian Spectral Triples
Rev. Math. Phys., vol. 26, 8, p. 14300076 (2014).
[abstract] [preprint] [journal]

Abstract:
We present the notion of temporal Lorentzian spectral triple which is an extension of the notion of pseudo-Riemannian spectral triple with a way to ensure that the signature of the metric is Lorentzian. A temporal Lorentzian spectral triple corresponds to a specific 3+1 decomposition of a possibly noncommutative Lorentzian space. This structure introduces a notion of global time in noncommutative geometry. As an example, we construct a temporal Lorentzian spectral triple over a Moyal--Minkowski spacetime. We show that, when time is commutative, the algebra can be extended to unbounded elements. Using such an extension, it is possible to define a Lorentzian distance formula between pure states with a well-defined noncommutative formulation.

8. Nicolas Franco, Michał Eckstein
Exploring the Causal Structures of Almost Commutative Geometries
SIGMA, vol. 10, 010 (2014).
[abstract] [preprint] [journal] [download]

Abstract:
We investigate the causal relations in the space of states of almost commutative Lorentzian geometries. We fully describe the causal structure of a simple model based on the algebra $\mathcal{S}(\mathbb{R}^{1,1}) \otimes M_2(\mathbb{C})$, which has a non-trivial space of internal degrees of freedom. It turns out that the causality condition imposes restrictions on the motion in the internal space. Moreover, we show that the requirement of causality favours a unitary evolution in the internal space.

9. Sebastian J. Szybka
On gravitational interactions between two bodies
In "Mathematical Structures of the Universe", eds. M. Eckstein, M. Heller, S. J. Szybka, CCPress, pp. 137-151 (2014).
[abstract] [preprint] [journal]

Abstract:
Many physicists, following Einstein, believe that the ultimate aim of theoretical physics is to find a unified theory of all interactions which would not depend on any free dimensionless constant, i.e., a dimensionless constant that is only empirically determinable. We do not know if such a theory exists. Moreover, if it exists, there seems to be no reason for it to be comprehensible for the human mind. On the other hand, as pointed out in Wigner's famous paper, human mathematics is unbelievably successful in natural science. This seeming paradox may be mitigated by assuming that the mathematical structure of physical reality has many `layers'. As time goes by, physicists discover new theories that correspond to the physical reality on the deeper and deeper level. In this essay, I will take a narrow approach and discuss the mathematical structure behind a single physical phenomenon - gravitational interaction between two bodies. The main aim of this essay is to put some recent developments of this topic in a broader context. For the author it is an exercise - to investigate history of his scientific topic in depth.

10. Sebastian J. Szybka, Krzysztof Głód, Michał J. Wyrębowski, Alicja Konieczny
Inhomogeneity effect in Wainwright-Marshman space-times
Phys. Rev. D: Part. Fields , vol. 89, p. 044033 (2014).
[abstract] [preprint] [journal] [download]

Abstract:
Green and Wald have presented a mathematically rigorous framework to study, within general relativity, the effect of small scale inhomogeneities on the global structure of space-time. The framework relies on the existence of a one-parameter family of metrics that approaches the effective background metric in a certain way. Although it is not necessary to know this family in an exact form to predict properties of the backreaction effect, it would be instructive to find explicit examples. In this paper, we provide the first example of such a family of exact non-vacuum solutions to the Einstein's equations. It belongs to the Wainwright-Marshman class and satisfies all of the assumptions of the Green-Wald framework.

11. 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.

12. 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'.

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