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

Abstract:
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.

8. 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]

Abstract:
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.

9. 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.

10. 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.

11. 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.

12. 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.