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Jacek Tafel

Publications and source records attributed to Jacek Tafel.

12 recordsLinked to original sources
arXiv2022

The Einstein equations and multipole moments at null infinity

We consider vacuum metrics admitting conformal compactification which is smooth up to the scri $\mathscr{I^+}$. We write metric in the Bondi-Sachs form and expand it into power series in the inverse affine distance $1/r$. Like in the case of the luminosity distance, given the news tensor and initial data for a part of metric the Einstein equations define coefficients of the series in a recursive way. This is also true in the stationary case however now the news tensor vanishes and the role of initial data is taken by multipole moments which are equivalent to moments of Thorne. We find an approximate form of metric and show that in the case of vanishing mass the mass dipole may be different from zero. Then the known result about the Kerr like behaviour of a stationary metric is violated. Finally we find an approximate (up to the quadrupole moment) Bondi-Sachs form of the Kerr metric.

gr-qc↗
arXiv2021

The Einstein metrics with smooth scri

We consider solutions of the Einstein equations with cosmological constant $Λ\neq 0$ admitting conformal compactification with smooth scri $\mathscr{I^+}$. Metrics are written in the Bondi-Sachs coordinates and expanded into inverse powers of the affine distance $r$. Unlike in the case $Λ=0$ all free data are located on the scri. There are linear differential constraints on the Bondi mass and angular momentum aspects. All other components of metrics are defined in a recursive way.

gr-qc↗
arXiv2019

The Penrose inequality for perturbations of the Schwarzschild initial data

We show that in the conformally flat case the Penrose inequality is satisfied for the Schwarzschild initial data with a small addition of the axially symmetric traceless exterior curvature. In this class the inequality is saturated only for data related to special sections of the Schwarzschild spacetime.

gr-qc↗
arXiv2016

Decoupling the momentum constraints in general relativity

We present a 2+1 decomposition of the vacuum initial conditions in general relativity. For a constant mean curvature one of the momentum constraints decouples in quasi isotropic coordinates and it can be solved by quadrature. The remaining momentum constraints are written in the form of the tangential Cauchy-Riemann equation. Under additional assumptions its solutions can be written in terms of integrals of known functions. We show how to obtain initial data with a marginally outer trapped surface. A generalization of the Kerr data is presented.

gr-qc↗
arXiv2016

Covariant description of isothermic surfaces

We present a covariant formulation of the Gauss-Weingarten equations and the Gauss-Mainardi-Codazzi equations for surfaces in 3-dimensional curved spaces. We derive a coordinate invariant condition on the first and second fundamental form which is locally necessary and sufficient for the surface to be isothermic. We show how to construct isothermic coordinates.

gr-qc↗
arXiv2016

A lower bound of the Trautman-Bondi energy

We obtain an energy inequality on null surfaces $u=const$ in the Bondi-Sachs formalism. We show that for a sufficiently regular event horizon $H$ there is an affine radial coordinate which is constant on $H$. Then the energy inequality can be prolongated to the horizon giving an estimation which is closely related to the Penrose inequality. We test it for the Kerr solution written in the Fletcher-Lun coordinates.

gr-qc↗
arXiv2015

On the energy of a null cone

We derive a formula for the Bondi mass aspect in terms of asymptotic data of the Bondi-Sachs metric in the affine gauge. We prove the positivity of the total energy of a regular null cone in agreement with a recent result of Chruściel and Paetz.

gr-qc↗
arXiv2014

Exact solutions and their interpretation - session A1

We report on the oral contributions and give a list of posters presented in the session A1 "Exact solutions and their interpretation" at the 20th International Conference on General Relativity and Gravitation (GR 20) in Warsaw, July 7-13, 2013.

gr-qc↗
arXiv2006

Symmetries of the Robinson-Trautman equation

We study point symmetries of the Robinson--Trautman equation. The cases of one- and two-dimensional algebras of infinitesimal symmetries are discussed in detail. The corresponding symmetry reductions of the equation are given. Higher dimensional symmetries are shortly discussed. It turns out that all known exact solutions of the Robinson--Trautman equation are symmetric.

gr-qc↗
arXiv2003

From 2-Dimensional Surfaces to Cosmological Solutions

We construct perfect fluid metrics corresponding to spacelike surfaces invariant under a 1-dimensional group of isometries in 3-dimensional Minkowski space. Under additional assumptions we obtain new cosmological solutions of Bianchi type II, VI_0 and VII_0. The solutions depend on an arbitrary function of time, which can be specified in order to satisfy an equation of state.

gr-qc↗
arXiv2003

Perfect Fluid Spacetimes With Two Symmetries

A method of solving perfect fluid Einstein equations with two commuting spacelike Killing vectors is presented. Given a spacelike 2-dimensional surface in the 3-dimensional nonphysical Minkowski space the field equations reduce to a single nonlinear differential equation. An example is discussed.

gr-qc↗