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13. Nicolas Franco, Michał Eckstein
Causality in noncommutative two-sheeted space-times
, vol. xxx, pp. xxx-xxx (2015).
[abstract] [preprint]

We investigate the causal structure of two-sheeted space-times using the tools of Lorentzian spectral triples. We show that the noncommutative geometry of these spaces allows for causal relations between the two sheets. The computation is given in details when the sheet is a 2- or 4-dimensional globally hyperbolic spin manifold. The conclusions are then generalised to a point-dependent distance between the two sheets resulting from the fluctuations of the Dirac operator.

14. Michał Eckstein, Nicolas Franco
Causal structure for noncommutative geometry
Frontiers of Fundamental Physics, vol. 14, pp. 138-xxx (2015).
[abstract] [journal]

Noncommutative Geometry à la Connes offers a promising framework for models of fundamental interactions. To guarantee the correct signature, the theory of Lorentzian spectral triples has been developed. We will briefly summarise its main elements and show that it can accommodate a sensible notion of causality understood as a partial order relation on the space of states on an algebra. For almost-commutative algebras of the form $C^\infty \otimes \A_F$, with $\A_F$ being finite-dimensional, the space of (pure) states is a simple product of space-time $\M$ and an internal space. The exploration of causal structures in this context leads to a surprising conclusion: The motion in both space-time and internal space is restricted by a "finite speed of light" constraint. We will present this phenomena on 2 simple toy-models.

15. Edited by James Ladyman, Stuart Presnell, Gordon McCabe, Michał Eckstein, Sebastian J. Szybka
Road to Reality with Roger Penrose
CCPress [abstract] [preprint] [journal]

Where does the road to reality lie? This fundamental question is addressed in this collection of essays by physicists and philosophers, inspired by the original ideas of Sir Roger Penrose. The topics range from black holes and quantum information to the very nature of mathematical cognition itself.

16. A. Woszczyna, W. Czaja, K. Głód, Z. A. Golda, R. A. Kycia, A. Odrzywołek, P. Plaszczyk, L. M. Sokołowski, S. J. Szybka
ccgrg: The symbolic tensor analysis package with tools for general relativity
Wolfram Library Archive, vol. 8848 (2014).
[abstract] [journal]

Riemann and Weyl curvature, covariant derivative, Lie derivative, the first and the second fundamental form on hyper-surfaces, as well as basic notions of relativistic hydrodynamics (expansion, vorticity, shear) are predefined functions of the package. New tensors are easy to define. Instructions, basic examples, and some more advanced examples are attached to the package. Characteristic feature of the ccgrg package is the specific coupling between the functional programming and the Parker-Christensen index convention. This causes that no particular tools to rising/lowering tensor indices neither to the tensor contractions are needed. Tensor formulas are written in the form close to that of classical textbooks in GRG, with the only difference that the summation symbol appears explicitly. Tensors are functions, not matrixes, and their components are evaluated lazily. This means that only these components which are indispensable to realize the final task are computed. The memoization technique prevents repetitive evaluation of the same quantities. This saves both, time and memory.

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

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.

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

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.

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