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

Energy-minimizing two black holes initial data

An attempt to construct the ``ground state'' vacuum initial data for the gravitational field surrounding two black holes is presented. The ground state is defined as the gravitational initial data minimizing the ADM mass within the class of data for which the masses of the holes and their distance are fixed. To parameterize different geometric arrangements of the two holes (and, therefore, their distance) we use an appropriately chosen scale factor. A method for analyzing the variations of the ADM mass and the masses (areas) of the horizons in terms of gravitational degrees of freedom is proposed. The Misner initial data are analyzed in this context: it is shown that they do not minimize the ADM mass.Comment: Minor corrections, 2 references adde

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.Comment: 29 page

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

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

Electronic Structure and Charge Dynamics of Huesler Alloy Fe2TiSn Probed by Infrared and Optical Spectroscopy

We report on the electrodynamics of a Heusler alloy Fe2TiSn probed over four decades in energy: from the far infrared to the ultraviolet. Our results do not support the suggestion of Kondo-lattice behavior inferred from specific heat measurements. Instead, we find a conventional Drude-like response of free carriers, with two additional absorption bands centered at around 0.1 and 0.87 eV. The latter feature can be interpreted as excitations across a pseudogap, in accord with band structure calculations.Comment: 3 pages, 4 figure

Construction and enlargement of traversable wormholes from Schwarzschild black holes

Analytic solutions are presented which describe the construction of a traversable wormhole from a Schwarzschild black hole, and the enlargement of such a wormhole, in Einstein gravity. The matter model is pure radiation which may have negative energy density (phantom or ghost radiation) and the idealization of impulsive radiation (infinitesimally thin null shells) is employed.Comment: 22 pages, 7 figure

Unconstrained Hamiltonian formulation of General Relativity with thermo-elastic sources

A new formulation of the Hamiltonian dynamics of the gravitational field interacting with(non-dissipative) thermo-elastic matter is discussed. It is based on a gauge condition which allows us to encode the six degrees of freedom of the ``gravity + matter''-system (two gravitational and four thermo-mechanical ones), together with their conjugate momenta, in the Riemannian metric q_{ij} and its conjugate ADM momentum P^{ij}. These variables are not subject to constraints. We prove that the Hamiltonian of this system is equal to the total matter entropy. It generates uniquely the dynamics once expressed as a function of the canonical variables. Any function U obtained in this way must fulfil a system of three, first order, partial differential equations of the Hamilton-Jacobi type in the variables (q_{ij},P^{ij}). These equations are universal and do not depend upon the properties of the material: its equation of state enters only as a boundary condition. The well posedness of this problem is proved. Finally, we prove that for vanishing matter density, the value of U goes to infinity almost everywhere and remains bounded only on the vacuum constraints. Therefore the constrained, vacuum Hamiltonian (zero on constraints and infinity elsewhere) can be obtained as the limit of a ``deep potential well'' corresponding to non-vanishing matter. This unconstrained description of Hamiltonian General Relativity can be useful in numerical calculations as well as in the canonical approach to Quantum Gravity.Comment: 29 pages, TeX forma