189 research outputs found
Nielsen equalizer theory
We extend the Nielsen theory of coincidence sets to equalizer sets, the
points where a given set of (more than 2) mappings agree. On manifolds, this
theory is interesting only for maps between spaces of different dimension, and
our results hold for sets of k maps on compact manifolds from dimension (k-1)n
to dimension n. We define the Nielsen equalizer number, which is a lower bound
for the minimal number of equalizer points when the maps are changed by
homotopies, and is in fact equal to this minimal number when the domain
manifold is not a surface.
As an application we give some results in Nielsen coincidence theory with
positive codimension. This includes a complete computation of the geometric
Nielsen number for maps between tori.Comment: + addendum, sync with published versio
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
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
Towards uniqueness of degenerate axially symmetric Killing horizon
We examine the linearized equations around extremal Kerr horizon and give
some arguments towards stability of the horizon with respect to generic
(non-symmetric) linear perturbation of near horizon geometry.Comment: 17 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 ), 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
Pair of null gravitating shells II. Canonical theory and embedding variables
The study of the two shell system started in our first paper ``Pair of null
gravitating shells I'' (gr-qc/0112060) is continued. An action functional for a
single shell due to Louko, Whiting and Friedman is generalized to give
appropriate equations of motion for two and, in fact, any number of spherically
symmetric null shells, including the cases when the shells intersect. In order
to find the symplectic structure for the space of solutions described in paper
I, the pull back to the constraint surface of the Liouville form determined by
the action is transformed into new variables. They consist of Dirac
observables, embeddings and embedding momenta (the so-called Kucha\v{r}
decomposition). The calculation includes the integration of a set of coupled
partial differential equations. A general method of solving the equations is
worked out.Comment: 20 pages, Latex file using amstex, some references correcte
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
The Trautman-Bondi mass of hyperboloidal initial data sets
We give a definition of mass for conformally compactifiable initial data sets. The asymptotic conditions are compatible with existence of gravitational radiation, and the compactifications are allowed to be polyhomogeneous. We show that the resulting mass is a geometric invariant, and we prove positivity thereof in the case of a spherical conformal infinity. When R(g) - or, equivalently, the trace of the extrinsic curvature tensor - tends to a negative constant to order one at infinity, the definition is expressed purely in terms of three-dimensional or two-dimensional objects
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
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
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