CYK Tensors, Maxwell Field and Conserved Quantities for Spin-2 Field

Starting from an important application of Conformal Yano--Killing tensors for the existence of global charges in gravity, some new observations at \scri^+ are given. They allow to define asymptotic charges (at future null infinity) in terms of the Weyl tensor together with their fluxes through \scri^+. It occurs that some of them play a 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 obtained formulae are similar to the functionals obtained from the Bel--Robinson tensor.Comment: 20 pages, LaTe

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.Comment: 23 page

Energy and angular momentum of the weak gravitational waves on the Schwarzschild background -- quasilocal gauge-invariant formulation

It is shown that the axial and polar perturbations of the spherically symmetric black hole can be described in a gauge-invariant way. The reduced phase space describing gravitational waves outside of the horizon is described by the gauge-invariant quantities. Both degrees of freedom fulfill generalized scalar wave equation. For the axial degree of freedom the radial part of the equation corresponds to the Regge-Wheeler result (Phys. Rev. 108, 1063-1069 (1957)) and for the polar one we get Zerilli result (Phys. Rev. D2, 2141-2160 (1970)), see also Chandrasekhar (The Mathematical Theory of Black Holes,(Clarendon Press Oxford, 1983)), Moncrief (Annals of Physics 88, 323-342 (1974)) for both. An important ingredient of the analysis is the concept of quasilocality which does duty for the separation of the angular variables in the usual approach. Moreover, there is no need to represent perturbations by normal modes (with time dependence $\exp(-ikt)$), we have fields in spacetime and the Cauchy problem for them is well defined outside of the horizon. The reduced symplectic structure explains the origin of the axial and polar invariants. It allows to introduce an energy and angular momentum for the gravitational waves which is invariant with respect to the gauge transformations. Both generators represent quadratic approximation of the ADM nonlinear formulae in terms of the perturbations of the Schwarzschild metric. We also discuss the boundary-initial value problem for the linearized Einstein equations on a Schwarzschild background outside of the horizon.Comment: 23 page

Asymptotic Conformal Yano--Killing Tensors for Schwarzschild Metric

The asymptotic conformal Yano--Killing tensor proposed in J. Jezierski, On the relation between metric and spin-2 formulation of linearized Einstein theory [GRG, in print (1994)] 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/r^2) 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.Comment: 8 pages, latex, no figure

Dynamics of a self gravitating light-like matter shell: a gauge-invariant Lagrangian and Hamiltonian description

A complete Lagrangian and Hamiltonian description of the theory of self-gravitating light-like 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. Energy-momentum tensor-density of a light-like matter shell is unambiguously defined in terms of an invariant matter Lagrangian density. Noether identity and Belinfante-Rosenfeld theorem for such a tensor-density are proved. Finally, the Hamiltonian dynamics of the interacting system: ``gravity + matter'' is derived from the total Lagrangian, the latter being an invariant scalar density.Comment: 20 pages, RevTeX4, no figure

Semimetalic antiferromagnetism in the half-Heusler compound CuMnSb

The half-Heusler compound CuMnSb, the first antiferromagnet (AFM) in the Mn-based class of Heuslers and half-Heuslers that contains several conventional and half metallic ferromagnets, shows a peculiar stability of its magnetic order in high magnetic fields. Density functional based studies reveal an unusual nature of its unstable (and therefore unseen) paramagnetic state, which for one electron less (CuMnSn, for example) would be a zero gap semiconductor (accidentally so) between two sets of very narrow, topologically separate bands of Mn 3d character. The extremely flat Mn 3d bands result from the environment: Mn has four tetrahedrally coordinated Cu atoms whose 3d states lie well below the Fermi level, and the other four tetrahedrally coordinated sites are empty, leaving chemically isolated Mn 3d states. The AFM phase can be pictured heuristically as a self-doped Cu$^{1+}$Mn$^{2+}$Sb$^{3-}$ compensated semimetal with heavy mass electrons and light mass holes, with magnetic coupling proceeding through Kondo and/or antiKondo coupling separately through the two carrier types. The ratio of the linear specific heat coefficient and the calculated Fermi level density of states indicates a large mass enhancement $m^*/m \sim 5$, or larger if a correlated band structure is taken as the reference

Unexpected coexisting solid solutions in the quasi-binary Ag(II)F2/Cu(II)F2 phase diagram

High-temperature solid-state reaction between orthorhombic AgF2 and monoclinic CuF2 (y = 0.15, 0.3, 0.4, 0.5) in a fluorine atmosphere resulted in coexisting solid solutions of Cu-poor orthorhombic and Cu-rich monoclinic phases with stoichiometry Ag1-xCuxF2. Based on X-ray powder diffraction analyses, the mutual solubility in the orthorhombic phase (AgF2 doped with Cu) appears to be at an upper limit of Cu concentration of 30 mol % (Ag0.7Cu0.3F2), while the monoclinic phase (CuF2 doped with Ag) can form a nearly stoichiometric Cu : Ag = 1 : 1 solid solution (Cu0.56Ag0.44F2), preserving the CuF2 crystal structure. Experimental data and DFT calculations showed that AgF2 doped with Cu and CuF2 doped with Ag solid solutions deviate from the classical Vegards law. Magnetic measurements of Ag1-xCuxF2 showed that the Neel temperature (TN) decreases with increasing Cu content in both phases. Likewise, theoretical DFT+U calculations for Ag1-xCuxF2 showed that the progressive substitution of Ag by Cu decreases the magnetic interaction strength (J2D) in both structures. Electrical conductivity measurements of Ag0.85Cu0.15F2 showed a ca. 2-fold increase in specific ionic conductivity (3.71 x 10-13 plus/minus 2.6 x 10-15 S/cm) as compared to pure AgF2 (1.85 x 10-13 plus/minus 1.2 x 10-15 S/cm), indicating the formation of a vacancy- or F adatom-free metal difluoride sample.Comment: 9 pages, 4 figures, 1 Table, and electronic supplement of 14 page

Trapped surfaces and the Penrose inequality in spherically symmetric geometries

We demonstrate that the Penrose inequality is valid for spherically symmetric geometries even when the horizon is immersed in matter. The matter field need not be at rest. The only restriction is that the source satisfies the weak energy condition outside the horizon. No restrictions are placed on the matter inside the horizon. The proof of the Penrose inequality gives a new necessary condition for the formation of trapped surfaces. This formulation can also be adapted to give a sufficient condition. We show that a modification of the Penrose inequality proposed by Gibbons for charged black holes can be broken in early stages of gravitational collapse. This investigation is based exclusively on the initial data formulation of General Relativity.Comment: plain te

Conserved superenergy currents

We exploit once again the analogy between the energy-momentum tensor and the so-called ``superenergy'' tensors in order to build conserved currents in the presence of Killing vectors. First of all, we derive the divergence-free property of the gravitational superenergy currents under very general circumstances, even if the superenergy tensor is not divergence-free itself. The associated conserved quantities are explicitly computed for the Reissner-Nordstrom and Schwarzschild solutions. The remaining cases, when the above currents are not conserved, lead to the possibility of an interchange of some superenergy quantities between the gravitational and other physical fields in such a manner that the total, mixed, current may be conserved. Actually, this possibility has been recently proved to hold for the Einstein-Klein-Gordon system of field equations. By using an adequate family of known exact solutions, we present explicit and completely non-obvious examples of such mixed conserved currents.Comment: LaTeX, 19 pages; improved version adding new content to the second section and some minor correction