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Jacek Jezierski

Publications and source records attributed to Jacek Jezierski.

At least 19 recordsLinked to original sources
arXiv2025

Comparison of definitions of angular momentum at null infinity

The supertranslation ambiguity of angular momentum at null infinity creates a technical difficulty in choosing the proper definition for this conserved quantity. As a result, many different approaches are found in the literature. This article aims to review the most relevant of these notions and compare their most significant properties. A new definition of angular momentum at null infinity is also proposed, which is supertranslation invariant under certain assumptions.

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arXiv2024

Quasi-local mass on rigid or round spheres in Kerr spacetime

We introduce and analyze quasi-local mass using Hamiltonian methods. It is based on multipole decomposition for surfaces that are topological spheres. Based on the above model, tests were performed for Kerr spacetime for two arbitrary choices of surfaces: rigid spheres and round spheres.

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arXiv2024

Hopfion-like solutions in de Sitter spacetime

We construct electromagnetic field with non-trivial topological properties on de Sitter background. The field is closely related with Hopf fibration. We analyze energy, angular momentum and topological charges for this solution. The paper is a generalization of CQG \textbf{35} (2018), no. 24, 245010 to de Sitter spacetime.

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arXiv2022

Quasi-local instantaneous charges asymptotics at spatial infinity

The article aims to analyze a construction of charges (conserved quantities) for the gravity field in the (3+1) decomposition. The construction is based on (3+1) splitting of conformal Yano--Killing (CYK) two-form. The splitting leads to charges, defined on Cauchy surface, which are combined from components of Weyl tensor and conformal Killing vector. The relations between the conserved quantities and its classical ADM counterparts are revisited. Asymptotic behavior of the conserved quantities is described. The charges are analyzed for a particular choice of initial data, among others, Bowen -- York spinning black hole.

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arXiv2020

Gauge-invariant quadratic approximation of quasi-local mass and its relation with Hamiltonian for gravitational field

Gauge invariant, Hamiltonian formulation of field dynamics within a compact region $Σ$ with boundary $\partial Σ$ is given for the gravitational field linearized over a Kottler metric. The boundary conditions which make the system autonomous are discussed and the corresponding Hamiltonian functional $\mathcal{H}_\text{Invariant}$ is calculated. It is shown that, under specific boundary conditions, the quasi-local Geroch--Hawking mass $\mathcal{H}_\text{Hawking}$ reduces to $\mathcal{H}_\text{Invariant}$ in the weak field approximation. This observation is a quasi-local version of the classical Brill--Deser result.

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arXiv2020

Localizing Energy in Fierz-Lanczos theory

We calculate energy carried by the massless spin-2 field using Fierz-Lanczos representation of the theory. For this purpose Hamiltonian formulation of the field dynamic is thoroughly analyzed. Final expression for the energy is very much analogous to the Maxwell energy in electrodynamics (spin-1 field) and displays the locality property. Known as a ``super-energy'' in gravity theory, this quantity differs considerably from the well understood gravitational field energy (represented in linear gravity by the quadratic term in Taylor expansion of the A.D.M. mass) which cannot be localized.

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arXiv2019

Gauge invariant description of weak gravitational field on a spherically symmetric background with cosmological constant

We present a formalism for analysis of linear Cauchy data on a Kottler metric. Our method removes redundancy due to gauge transformations and constraints. A set of four gauge-invariant, scalar functions on the Cauchy surface is produced and shown to contain full physical information from the initial data. The symplectic form of the theory and equations of motion are reformulated in terms of these invariants and some simple solutions are explored.

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arXiv2019

Charges of the gravitational field and (3+1) decomposition of CYK tensors part 2

The work describes the method of construction of charges (conserved quantities) for the gravity field in the (3 + 1) decomposition. The presented construction uses tensors of the electrical and magnetic parts of the Weyl tensor and conformal Killing vectors. In the case of conformally flat spatial hypersurfaces, we get twenty local charges, which can be expressed in terms of the initial data (three-dimensional metric and extrinsic curvature tensor). The work shows the relationships between charges which are obtained by this method and the usual ADM approach. In traditional ADM approach Killing vectors are used to construct corresponding charges e.g. time translation corresponds to ADM mass, spatial translations give linear momentum and rotations correspond to angular momentum. Gravito-electric and gravito-magnetic charges need conformal Killing vectors e.g. mass corresponds to dilation, linear momentum is related to rotation and angular momentum needs conformal acceleration. The analyzed example of the Schwarzschild--de Sitter spacetime suggests that in some cases the mass calculated by the presented method has better properties than the traditional ADM mass. Next we discuss asymptotic charges which are no longer rigidly conserved but rather approach finite value at spatial infinity.

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arXiv2018

Conformal Yano-Killing tensors for space-times with cosmological constant

We present a new method for constructing conformal Yano-Killing tensors in five-di\-men\-sio\-nal Anti-de Sitter space-time. The found tensors are represented in two different coordinate systems. We also discuss, in terms of CYK tensors, global charges which are well defined for asymptotically (five-dimensional) Anti-de Sitter space-time. Additionally in Appendix we present our own derivation of conformal Killing one-forms in four-dimensional Anti-de Sitter space-time as an application of the Theorem presented in the paper.

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arXiv2016

On the existence of rigid spheres in four-dimensional spacetime manifolds

This paper deals with the generalization of usual round spheres in the flat Minkowski spacetime to the case of a generic four-dimensional spacetime manifold $M$. We consider geometric properties of sphere-like submanifolds in $M$ and introduce conditions on external curvature and torsion, which lead to a definition of a {\em rigid sphere}. The main result is a local existence theorem concernig such spheres. For this purpose we apply the surjective implicit function theorem. The proof is based on a detailed analysis of the linearized problem and leads to an eight-parameter family of solutions in case when the metric tensor $g$ of $M$ is from a certain neighbourhood of the flat Minkowski metric. This contribution continues the study of rigid spheres in (Class. Quantum Grav. \textbf{30} (2013), 175010, doi:10.1088/0264-9381/30/17/175010, 18 pp.).

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arXiv2016

A geometric description of Maxwell field in a Kerr spacetime

We consider the Maxwell field in the exterior of a Kerr black hole. For this system, we propose a geometric construction of generalized Klein-Gordon equation called Fackerell-Ipser equation. Our model is based on conformal Yano-Kiling tensor (CYK tensor). We present non-standard properties of CYK tensors in the Kerr spacetime which are useful in electrodynamics.

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arXiv2015

Proof of Positive Energy Theorem by spacetime foliations

Here we present an alternative approach, requiring only basic tools of differential geometry. This method was conceived by Kijowski and first presented during the IV Marcel Grossman Meeting on General Relativity in Rome (1986). It was later published in a generalized form by J. Jezierski, J. Kijowski in 1987 (Positivity of total energy in general relativity, Physical Review D {\bf 36} (1987), 1041-4). We summarize these results briefly and present a new foliation gauge.

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arXiv2014

The 3+1 decomposition of Conformal Yano-Killing tensors and "momentary" charges for spin-2 field

The "fully charged" spin-2 field solution is presented. This is an analog of the Coulomb solution in electrodynamics and represents the "non-waving" part of the spin-2 field theory. Basic facts and definitions of the spin--2 field and conformal Yano-Killing tensors are introduced. Application of those two objects provides a precise definition of quasi-local gravitational charge. Next, the 3+1 decomposition leads to the construction of the momentary gravitational charges on initial surface which is applicable for Schwarzschild-like spacetimes.

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arXiv2013

Rigid spheres in Riemannian spaces

Choice of an appropriate (3+1)-foliation of spacetime or a (2+1)-foliation of the Cauchy space, leads often to a substantial simplification of various mathematical problems in General Relativity Theory. We propose a new method to construct such foliations. For this purpose we define a special family of topological two-spheres, which we call "rigid spheres". We prove that there is a four-parameter family of rigid spheres in a generic Riemannian three-manifold (in case of the flat Euclidean three-space these four parameters are: 3 coordinates of the center and the radius of the sphere). The rigid spheres can be used as building blocks for various ("spherical", "bispherical" etc.) foliations of the Cauchy space. This way a supertranslation ambiguity may be avoided. Generalization to the full 4D case is discussed. Our results generalize both the Huang foliations (cf. \cite{LHH}) and the foliations used by us (cf. \cite{JKL}) in the analysis of the two-body problem.

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