Jacek Jezierski

Conformal Yano-Killing tensor for the Kerr metric and conserved quantities

Properties of (skew-symmetric) conformal Yano–Killing tensors are reviewed. Explicit forms of three symmetric conformal Killing tensors in Kerr spacetime are obtained from the Yano–Killing tensor. The relation between spin-2 fields and solutions to the Maxwell equations is used in the construction of a new conserved quantity which is quadratic in terms of the Weyl tensor. The formula obtained is similar to the functional obtained from the Bel–Robinson tensor and is examined in Kerr spacetime. A new interpretation of the conserved quantity obtained is proposed.

Dynamics of a self-gravitating lightlike matter shell: A gauge-invariant Lagrangian and Hamiltonian description

A complete Lagrangian and Hamiltonian description of the theory of self-gravitating lightlike matter shells is given in terms of gauge-independent geometric quantities. For this purpose the notion of an extrinsic curvature for a null-like hypersurface is discussed and the corresponding Gauss-Codazzi equations are proved. These equations imply Bianchi identities for spacetimes with null-like, singular curvature. The energymomentum tensor density of a lightlike matter shell is unambiguously defined in terms of an invariant matter Lagrangian density. The Noether identity and Belinfante-Rosenfeld theorem for such a tensor density are proved. Finally, the Hamiltonian dynamics of the interacting ‘‘gravity1matter’’ system is derived from the total Lagrangian, the latter being an invariant scalar density.

Energy and Angular Momentum ofthe Weak Gravitational Waves on the Schwarzschild Background — Quasilocal Gauge-invariant Formulation

A four-dimensional spherically covariantgauge-invariant quasilocal framework for theperturbation of the Schwarzschild metric is given. Animportant ingredient of the analysis is the concept ofquasilocality, which does duty for the separation ofangular variables in the usual approach. A precise andfull analysis for the “mono-dipole” part ofthe theory is presented. Direct construction (from theconstraints) of the reduced canonical structure for theinitial data and explicit formulae for thegaugeinvariants are proposed. The reduced symplecticstructure explains the origin of the axial and polarinvariants. This enables one to introduce an energy andangular momentum for the gravitational waves, which isinvariant with respect to the gauge transformations. Anexplicit expression for the energy and new proposition for angular momentum is introduced, inparticular, compatibility of theChristodoulou-Klainerman S.A.F. condition withwell-possedness of our functionals is checked. Bothgenerators (energy and angular momentum) represent quadratic approximation ofthe adm nonlinear formulae in terms of the perturbationsof the Schwarzschild metric. The previously knownresults are presented in a new geometric andself-consistent way. Both degrees of freedom fulfill thegeneralized scalar wave equation. For the axial degreeof freedom the radial part of the equation correspondsto the Regge-Wheeler result and for the polar one we get the Zerilli result.

Uniqueness of the Trautman-Bondi mass

It is shown that the only functionals, within a natural class, which are monotonic in time for all solutions of the vacuum Einstein equations admitting a smooth “piece” of conformal null infinity J ( , are those depending on the metric only through a specific combination of the Bondi ‘mass aspect’ and other next–to–leading order terms in the metric. Under the extra condition of passive BMS invariance, the unique such functional (up to a multiplicative factor) is the Trautman–Bondi energy. It is also shown that this energy remains well-defined for a wide class of ‘polyhomogeneous’ metrics.

Spacetimes foliated by Killing horizons

It seems to be expected that a horizon of a quasi-local type, such as a Killing or an isolated horizon, by analogy with a globally defined event horizon, should be unique in some open neighbourhood in the spacetime, provided the vacuum Einstein or the Einstein–Maxwell equations are satisfied. The aim of our paper is to verify whether that intuition is correct. If one can extend a so-called Kundt metric, in such a way that its null, shear-free surfaces have spherical spacetime sections, the resulting spacetime is foliated by socalled non-expanding horizons. The obstacle is Kundt’s constraint induced at the surfaces by the Einstein or the Einstein–Maxwell equations, and the requirement that a solution be globally defined on the sphere. We derived a transformation (reflection) that creates a solution to Kundt’s constraint out of data defining an extremal isolated horizon. Using that transformation, we derived a class of exact solutions to the Einstein or Einstein–Maxwell equations of very special properties. Each spacetime we construct is foliated by a family of the Killing horizons. Moreover, it admits another, transversal Killing horizon. The intrinsic and extrinsic geometries of the transversal Killing horizon coincide with the one defined on the event horizon of the extremal Kerr–Newman solution. However, the Killing horizon in our example admits yet another Killing vector tangent to and null at it. The geometries of the leaves are given by the reflection.

The localization of energy in gauge field theories and in linear gravitation

It is shown how the energy-positivity criterion enables us to localize the energy in various field theories. For this purpose the role of surface integrals in a canonical formalism is investigated. The same techniques are applied to linearized gravity, where the mixed Cauchy-boundary value problem in a finite volume is analyzed. Unconstrained degrees of freedom and boundary data which have to be controlled are found. This paper is part of a program to analyze the possibility of localization of gravitational energy in complete General Relativity.

CYK tensors, Maxwell field and conserved quantities for the spin-2 field

Starting from an important application of conformal Yano–Killing tensors for the existence of global charges in gravity (which has been performed in [17] and [18]), some new observations at + are given. They allow us to define asymptotic charges (at future null infinity) in terms of the Weyl tensor together with their fluxes through +. It happens that some of them play the role of obstructions for the existence of angular momentum. Moreover, new relations between solutions of the Maxwell equations and the spin-2 field are given. They are used in the construction of new conserved quantities which are quadratic in terms of the Weyl tensor. The formulae obtained are similar to the functionals obtained from the Bel–Robinson tensor.

Geometry of null-like surfaces in general relativity and its application to dynamics of gravitating matter

Abstract Geometric tools describing the structure of a null-like surface S (wave front) are constructed. They are applied to the analysis of interaction between a light-like matter shell and the surrounding gravitational field. It is proved that the Einstein tensor Gab describing such a situation may be written in terms of external curvature of S. Conservation laws (Bianchi identities) for G are proved. Also geometry of isolated horizons (surfaces surrounding black holes) is analyzed in terms of the constructed tools. The possibility of application of these results to the problem of motion of isolated objects in general relativity is discussed.

Conformal Yano - Killing tensors and asymptotic CYK tensors for the Schwarzschild metric

The asymptotic conformal Yano–Killing tensor proposed in [1] is analyzed for Schwarzschild metric and tensor equations defining this object are given. The result shows that the Schwarzschild metric (and other metrics which are asymptotically “Schwarzschildean” up to O(1/r2) at spatial infinity) is among the metrics fullfilling stronger asymptotic conditions and supertranslations ambiguities disappear. It is also clear from the result that 14 asymptotic gravitational charges are well defined on the “Schwarzschildean” background.

The relation between metric and spin-2 formulations of linearized Einstein theory

A twenty-dimensional space of charged solutions of spin-2 equations is proposed. The relation with extended (via dilatation) Poincaré group is analyzed. Locally, each solution of the theory may be described in terms of a potential, which can be interpreted as a metric tensor satisfying linearized Einstein equations. Globally, the nonsingular metric tensor exists if and only if 10 among the above 20 charges do vanish. The situation is analogous to that in classical electrodynamics, where vanishing of magnetic monopole implies the global existence of the electromagnetic potentials. The notion ofasymptotic conformal Yano-Killing tensor is defined and used as a basic concept to introduce an inertial frame in General Relativity via asymptotic conditions at spatial infinity. The introduced class of asymptotically flat solutions is free of supertranslation ambiguities.