arXiv subjects

Adam Chudecki

Publications and source records attributed to Adam Chudecki.

18 recordsLinked to original sources

arXiv2025

An example of algebraically general para-K\"ahler Einstein space

Algebraically general para-K\"ahler Einstein spaces equipped with 3D algebras of infinitesimal symmetries are considered. It is shown that if the algebra contains 2D trivial subalgebra then vacuum Einstein field equations with cosmological constant can be reduced to a single, first-order differential equation. One of the cases is solved explicitly. Hence, a first example of an algebraically general para-K\"ahler Einstein space is given.

math-ph↗

arXiv2025

On a certain class of para-Hermite Einstein spaces

A special class of (complex) para-Hermite Einstein spaces is analyzed. It is well-known that the self-dual Weyl tensor in para-Hermite Einstein spaces is of the Petrov-Penrose type [D]. In what follows we assume that the anti-self-dual Weyl tensor is algebraically degenerate. It is equivalent to the existence of an anti-self-dual congruence of null strings which is assumed not to be parallely propagated. Hence, spaces analyzed here are not Walker spaces. A classification of such spaces is given and the explicit metrics are found.

math-ph↗

arXiv2025

On Walker and para-Hermite Einstein spaces

A special class of (complex) para-Hermite Einstein spaces is analyzed. For this class of spaces the self-dual Weyl tensor is type-[D] in the Petrov-Penrose classification. The anti-self-dual Weyl tensor is algebraically degenerate, equivalently, there exists an anti-self-dual congruence of null strings. It is assumed that this congruence is parallely propagated. Thus, the spaces are not only para-Hermite but also Walker. A classification of the spaces according to three criteria is given. Finally, explicit metrics of all admitted Petrov-Penrose types are found.

math-ph↗

arXiv2025

Curvature and conformal curvature dynamics formalisms and their applications in linearized gravity

Tensorial, spinorial and helicity formalisms of the curvature and conformal curvature dynamics are developed. Equations of linearized gravity within that formalisms are given. Gravitational radiation in linearized gravity in terms of curvature dynamics is investigated. Equivalence of the Bia\l{}ynicki-Birula formula for the gravitational energy in linearized gravity and the Landau-Lifschitz formula is proved. Analogous result is found for the momentum in linearized gravity.

gr-qc↗

arXiv2022

Hyperheavenly spaces and their application in Walker and para-K\"ahler geometries: part II

4-dimensional spaces equipped with congruences of null strings are considered. It is assumed that a space admits a congruence of expanding self-dual null strings and its self-dual part of the Weyl tensor is algebraically degenerate. Different Petrov-Penrose types of such spaces are analyzed. A special attention is paid to para-K\"ahler Einstein spaces. All para-K\"ahler Einstein metrics of spaces with algebraically degenerate self-dual Weyl spinor are found in all the generality.

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arXiv2021

Hyperheavenly spaces and their application in Walker and para-K\"ahler geometries: Part I

Spaces equipped with congruences of null strings are considered. A special attention is paid to the spaces which belong to the two-sided Walker class and para-K\"ahler class. Properties of an intersection of self-dual and anti-self-dual congruences of null strings are used as an additional criterion for a classification of such spaces. Finally, a few examples of para-K\"ahler and para-K\"ahler-Einstein spaces are presented.

math-ph↗

arXiv2020

Two-sided conformally recurrent self-dual spaces

Two-sided conformally recurrent 4-dimensional self-dual spaces are considered. It is shown that such spaces are equipped with nonexpanding congruences of null strings. The general structure of weak nonexpanding hyperheavenly spaces is given. Finally, the general metrics of Petrov-Penrose type $[D] x [-]$ spaces are presented.

math-ph↗

arXiv2018

Classification of complex and real, vacuum spaces of the type $[\textrm{N}] \otimes [\textrm{N}]$

Complex and real, vacuum spaces with both self-dual and anti-self-dual parts of the Weyl tensor being of the type [N] are considered. Such spaces are classified according to two criteria. The first one takes into account the properties of the congruences of totally null, geodesic 2-dimensional surfaces (the null strings). The second criterion is based on investigations of the properties of the intersection of these congruences. It is proved that there exist six distinct types of the $[\textrm{N}] \otimes [\textrm{N}]$ spaces. New examples of the Lorentzian slices of the complex metrics are presented. Also, some type $[\textrm{N}] \otimes [\textrm{N}]$ spaces which do not posses Lorentzian slices are considered.

gr-qc↗

arXiv2018

On twisting type $[\textrm{N}] \otimes [\textrm{N}]$ Ricci flat complex spacetimes with two homothetic symmetries

$\mathcal{HH}$ spaces of type $[\textrm{N}] \otimes [\textrm{N}]$ with twisting congruence of null geodesics defined by the 4-fold undotted and dotted Penrose spinors are investigated. It is assumed that these spaces admit two homothetic symmetries. The general form of the homothetic vector fields are found. New coordinates are introduced which enable us to reduce the $\mathcal{HH}$ system of PDEs to one ODE on one holomorphic function. In a special case this is a second-order ODE and its general solution is explicitly given. In the generic case one gets rather involved fifth-order ODE.

gr-qc↗

arXiv2016

Classification of the traceless Ricci tensor in 4-dimensional pseudo-Riemannian spaces of neutral signature

The traceless Ricci tensor $C_{ab}$ in 4-dimensional pseudo-Riemannian spaces equipped with the metric of the neutral signature is analyzed. Its algebraic classification is given. This classification uses the properties of $C_{ab}$ treated as a matrix. The Petrov-Penrose types of Pleba\'nski spinors associated with the traceless Ricci tensor are given. Finally, the classification is compared with a similar classification in the complex case.

math-ph↗

arXiv2016

On geometry of congruences of null strings in 4-dimensional complex and real pseudo-Riemannian spaces

4-dimensional spaces equipped with 2-dimensional (complex holomorphic or real smooth) completely integrable distributions are considered. The integral manifolds of such distributions are totally null and totally geodesics 2-dimensional surfaces which are called the null strings. Properties of congruences (foliations) of such 2-surfaces are studied. Some relations between properties of congruences of null strings, Petrov-Penrose type of SD Weyl spinor and algebraic types of the traceless Ricci tensor are analyzed.

gr-qc↗

arXiv2016

On some examples of para-Hermite and para-K\"{a}hler Einstein spaces with $\Lambda \ne 0$

Spaces equipped with two complementary (distinct) congruences of self-dual null strings and at least one congruence of anti-self-dual null strings are considered. It is shown that if such spaces are Einsteinian then the vacuum Einstein field equations can be reduced to a single nonlinear partial differential equation of the second order. Different forms of these equations are analyzed. Finally, several new explicit metrics of the para-Hermite and para-K\"{a}hler Einstein spaces with $\Lambda \ne 0$ are presented. Some relation of that metrics to a modern approach to mechanical issues is discussed.

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arXiv2014

Proper conformal symmetries in SD Einstein spaces

Proper conformal symmetries in self-dual (SD) Einstein spaces are considered. It is shown, that such symmetries are admitted only by the Einstein spaces of the type [N]x[N]. Spaces of the type [N]x[-] are considered in details. Existence of the proper conformal Killing vector implies existence of the isometric, covariantly constant and null Killing vector. It is shown, that there are two classes of [N]x[-]-metrics admitting proper conformal symmetry. They can be distinguished by analysis of the associated anti-self-dual (ASD) null strings. Both classes are analyzed in details. The problem is reduced to single linear PDE. Some general and special solutions of this PDE are presented.

gr-qc↗

arXiv2013

All ASD complex and real 4-dimensional Einstein spaces with $\Lambda \ne 0$ admitting a nonnull Killing vector

Anti-self-dual (ASD) 4-dimensional complex Einstein spaces with nonzero cosmological constant $\Lambda$ equipped with a nonnul Killing vector are considered. It is shown, that any conformally nonflat metric of such spaces can be always brought to a special form and the Einstein field equations can be reduced to the Boyer-Finley-Pleba\'nski equation (Toda field equation). Some alternative form of the metric are discussed. All possible real slices (neutral, Euclidean and Lorentzian) of ASD complex spaces admitting a nonnull Killing vector are found.

math-ph↗

arXiv2013

Null Killing vectors and geometry of null strings in Einstein spaces

Einstein complex spacetimes admitting null Killing or null homothetic Killing vectors are studied. These vectors define totally null and geodesic 2-surfaces called the null strings or twistor surfaces. Geometric properties of these null strings are discussed. It is shown, that spaces considered are hyperheavenly spaces (HH-spaces) or, if one of the parts of the Weyl tensor vanishes, heavenly spaces (H-spaces). The explicit complex metrics admitting null Killing vectors are found. Some Lorentzian and ultrahyperbolic slices of these metrics are discussed.

gr-qc↗

arXiv2013

Killing Symmetries in $\mathcal{H}$-Spaces with $\Lambda$

All Killing symmetries in complex $\mathcal{H}$-spaces with $\Lambda$ in terms of the Pleba\'nski - Robinson - Finley coordinate system are found. All $\mathcal{H}$-metrics with $\Lambda$ admitting a null Killing vector are explicitly given. It is shown that the problem of non-null Killing vector reduces to looking for solution of the Boyer - Finley - Pleba\'nski (Toda field) equation

gr-qc↗

arXiv2011

Classification of the Killing Vectors in Nonexpanding HH-Spaces with Lambda

Conformal Killing equations and their integrability conditions for nonexpanding hyperheavenly spaces with Lambda are studied. Reduction of ten Killing equations to one master equation is presented. Classification of homothetic and isometric Killing vectors in nonexpanding hyperheavenly spaces with Lambda and homothetic Killing vectors in heavenly spaces is given. Some nonexpanding complex metrics of types [III,N]x[N] are found. A simple example of Lorentzian real slice of the type [N]x[N] is explicitly given.

gr-qc↗

arXiv2011

Homothetic Killing Vectors in Expanding HH-Spaces with Lambda

Conformal Killing equations and their integrability conditions for expanding hyperheavenly spaces with Lambda in spinorial formalism are studied. It is shown that any conformal Killing vector reduces to homothetic or isometric Killing vector. Reduction of respective Killing equation to one master equation is presented. Classification of homothetic and isometric Killing vectors is given. Type [D]x[any] is analysed in details and some expanding HH complex metrics of types [III, N]x[III, N] with Lambda admitting isometric Killing vectors are found.

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