Shear-free Null Quasi-Spherical Spacetimes

The residual gauge freedom within the null quasi-spherical coordinate condition is studied, for spacetimes admitting an expanding, shear-free null foliation. The freedom consists of a boost and rotation at each coordinate sphere, corresponding to a specification of inertial frame at each sphere. Explicit formulae involving arbitrary functions of two variables are obtained for the accelerated Minkowski, Schwarzschild, and Robinson-Trautman spacetimes. These examples will be useful as test metrics in numerical relativity.Comment: 20 pages, revte

On Existence of Static Metric Extensions in General Relativity

Motivated by problems related to quasi-local mass in general relativity, we study the static metric extension conjecture proposed by R. Bartnik \cite{Bartnik_energy}. We show that, for any metric on $\bar{B}_1$ that is close enough to the Euclidean metric and has reflection invariant boundary data, there always exists an asymptotically flat and scalar flat {\em static} metric extension in $M = \R^3 \setminus B_1$ such that it satisfies Bartnik's geometric boundary condition \cite{Bartnik_energy} on $\partial B_1$.Comment: 20 page

The Fountains of Łódź: Their Relevance to the Lives of its Inhabitants

This paper presents the characteristics of the fountains of Łódź, their location in the public spaces of the city and changes in various time periods. Special attention is drawn to the function of fountains in contemporary cities and their social perception. Moreover, in the last part, the presumed reasons for their present distribution and typological variety are given

Spherically symmetric dynamical horizons

We determine sufficient and necessary conditions for a spherically symmetric initial data set to satisfy the dynamical horizon conditions in the spacetime development. The constraint equations reduce to a single second order linear master equation, which leads to a systematic construction of all spherically symmetric dynamical horizons (SSDH) satisfying certain boundedness conditions. We also find necessary and sufficient conditions for a given spherically symmetric spacetime to contain a SSDH.Comment: latex, 19 pages, no figure

Maximum discharges and maximum runoffs in Poland

Published in: Natural environment of Poland and its protection in Łódź University Geographical Research, edited by E. Kobojek and T.Marsza