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13. 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. 137151 (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.  14. Sebastian J. Szybka, Krzysztof Głód, Michał J. Wyrębowski, Alicja Konieczny Inhomogeneity effect in WainwrightMarshman spacetimes 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 spacetime. The framework relies on the existence of a oneparameter 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 nonvacuum solutions to the Einstein's equations. It belongs to the WainwrightMarshman class and satisfies all of the assumptions of the GreenWald framework.  15. 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.  16. 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 Einsteinde 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.  17. Christa R. Ölz, Sebastian J. Szybka Conformal and projection diagrams in LaTeX (2013). [abstract] [preprint]  Abstract: In general relativity, the causal structure of spacetime may sometimes be depicted by conformal CarterPenrose 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 timeconsuming and not easy. We present below a library that allows one to draw an arbitrary conformal diagram in a few simple steps.  18. Leszek M. Sokołowski On the twin paradox in static spacetimes: I. Schwarzschild metric General Relativity and Gravitation, vol. 44, pp. 12671283 (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  
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