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Mikolaj Korzynski

Publications and source records attributed to Mikolaj Korzynski.

9 recordsLinked to original sources
arXiv2013

Evolution of a family of expanding cubic black-hole lattices in numerical relativity

We present the numerical evolution of a family of conformally-flat, infinite, expanding cubic black-hole lattices. We solve for the initial data using an initial-data prescription presented recently, along with a new multigrid solver developed for this purpose. We then apply the standard tools of numerical relativity to calculate the time development of this initial dataset and derive quantities of cosmological relevance, such as the scaling of proper lengths. Similarly to the case of S3 lattices, we find that the length scaling remains close to the analytical solution for Friedmann-Lemaitre-Robertson-Walker cosmologies throughout our simulations, which span a window of about one order of magnitude in the growth of the scale factor. We highlight, however, a number of important departures from the Friedmann-Lemaitre-Robertson-Walker class.

gr-qc↗
arXiv2012

Evolution of a periodic eight-black-hole lattice in numerical relativity

The idea of black-hole lattices as models for the large-scale structure of the universe has been under scrutiny for several decades, and some of the properties of these systems have been elucidated recently in the context of the problem of cosmological backreaction. The complete, three-dimensional and fully relativistic evolution of these system has, however, never been tackled. We explicitly construct the first of these solutions by numerically integrating Einstein's equation in the case of an eight-black-hole lattice with the topology of S3.

gr-qc↗
arXiv2011

Isometric embeddings of 2-spheres by embedding flow for applications in numerical relativity

We present a numerical method for solving Weyl's embedding problem which consists of finding a global isometric embedding of a positively curved and positive-definite spherical 2-metric into the Euclidean three space. The method is based on a construction introduced by Weingarten and was used in Nirenberg's proof of Weyl's conjecture. The target embedding results as the endpoint of an embedding flow in R^3 beginning at the unit sphere's embedding. We employ spectral methods to handle functions on the surface and to solve various (non)-linear elliptic PDEs. Possible applications in 3+1 numerical relativity range from quasi-local mass and momentum measures to coarse-graining in inhomogeneous cosmological models.

gr-qc↗
arXiv2010

Covariant coarse-graining of inhomogeneous dust flow in General Relativity

A new definition of coarse-grained quantities describing the dust flow in General Relativity is proposed. It assigns the coarse--grained expansion, shear and vorticity to finite-size comoving domains of fluid in a covariant, coordinate-independent manner. The coarse--grained quantities are all quasi-local functionals, depending only on the geometry of the boundary of the considered domain. They can be thought of as relativistic generalizations of simple volume averages of local quantities in a flat space. The procedure is based on the isometric embedding theorem for S^2 surfaces and thus requires the boundary of the domain in question to have spherical topology and positive scalar curvature. We prove that in the limit of infinitesimally small volume the proposed quantities reproduce the local expansion, shear and vorticity. In case of irrotational flow we derive the time evolution for the coarse-grained quantities and show that its structure is very similar to the evolution equation for their local counterparts. Additional terms appearing in it may serve as a measure of the backreacton of small-scale inhomogeneities of the flow on the large-scale motion of the fluid inside the domain and therefore the result may be interesting in the context of the cosmological backreaction problem. We also consider the application of the proposed coarse-graining procedure to a number of known exact solutions of Einstein equations with dust and show that it yields reasonable results.

gr-qc↗
arXiv2010

Coarse-graining of inhomogeneous dust flow in General Relativity via isometric embeddings

We present a new approach to coarse-graining of variables describing dust flow in GR. It is based on assigning quasi-local shear, twist and expansion to 2-dimensional surfaces with the help of isometric embeddings into the 3-dimensional Euclidean space and deriving the time evolution equations for them. In contrast to the popular Buchert's scheme it allows to coarse-grain tensorial quantities in a coordinate-independent way. The framework can be used to estimate backreaction in inhomogeneous cosmological models.

gr-qc↗
arXiv2006

Isolated and dynamical horizons from a common perspective

A framework is developed in which one can write down the constraint equations on a three--dimensional hypersurface of arbitrary signature. It is then applied to isolated and dynamical horizons. The derived equations can be used to extract physicaly relevant quantities describing the horizon irrespective to whether it is isolated (null) or dynamical at a given instant of time. Furthermore, small perturbation of isolated horizons are considered, and finally a family of axially--symmetric exact solution of the constraint equations on a dynamical horizon is presented.

gr-qc↗
arXiv2005

Singular disk of matter in the Cooperstock-Tieu galaxy model

Recently a new model of galactic gravitational field, based on ordinary General Relativity, has been proposed by Cooperstock and Tieu in which no exotic dark matter is needed to fit the observed rotation curve to a reasonable ordinary matter distribution. We argue that in this model the gravitational field is generated not only by the galaxy matter, but by a thin, singular disk as well. The model should therefore be considered unphysical.

astro-ph↗
arXiv2003

The Normal Conformal Cartan Connection and the Bach Tensor

The goal of this paper is to express the Bach tensor of a four dimensional conformal geometry of an arbitrary signature by the Cartan normal conformal (CNC) connection. We show that the Bach tensor can be identified with the Yang-Mills current of the connection. It follows from that result that a conformal geometry whose CNC connection is reducible in an appropriate way has a degenerate Bach tensor. As an example we study the case of a CNC connection which admits a twisting covariantly constant twistor field. This class of conformal geometries of this property is known as given by the Fefferman metric tensors. We use our result to calculate the Bach tensor of an arbitrary Fefferman metric and show it is proportional to the tensorial square of the four-fold eigenvector of the Weyl tensor. Finally, we solve the Yang-Mills equations imposed on the CNC connection for all the homogeneous Fefferman metrics. The only solution is the Nurowski-Plebanski metric.

gr-qc↗