52 research outputs found
On the variational principle for dust shells in General Relativity
The variational principle for a thin dust shell in General Relativity is
constructed. The principle is compatible with the boundary-value problem of the
corresponding Euler-Lagrange equations, and leads to ``natural boundary
conditions'' on the shell. These conditions and the gravitational field
equations which follow from an initial variational principle, are used for
elimination of the gravitational degrees of freedom. The transformation of the
variational formula for spherically-symmetric systems leads to two natural
variants of the effective action. One of these variants describes the shell
from a stationary interior observer's point of view, another from the exterior
one. The conditions of isometry of the exterior and interior faces of the shell
lead to the momentum and Hamiltonian constraints. The canonical equivalence of
the mentioned systems is shown in the extended phase space. Some particular
cases are considered.Comment: 25 pages, RevTeX, no figures, revised version, typos corrected,
accepted for publication in Journal of Mathematical Physic
Some peculiarities of motion of neutral and charged test particles in the field of a spherically symmetric charged object in General Relativity
We propose the method of investigation of radial motions for charged and
neutral test particles in the Reissner-Nordstr\"{o}m field by means of mass
potential. In this context we analyze special features of interaction of
charges and their motions in General Relativity and construct the radial motion
classification. For test particles and a central source with charges and
, respectively, the conditions of attraction (when ) and repulsion
(when ) are obtained. The conditions of motionless test particle states
with respect to the central source are investigated and, in addition, stability
conditions for such static equilibrium states are found. It is shown that
stable states are possible only for the bound states of weakly charged
particles in the field of a naked singularity. Frequencies of small
oscillations of test particles near their equilibrium positions are also found.Comment: 15 pages, 9 figure
Split structures in general relativity and the Kaluza-Klein theories
We construct a general approach to decomposition of the tangent bundle of
pseudo-Riemannian manifolds into direct sums of subbundles, and the associated
decomposition of geometric objects. An invariant structure {\cal H}^r defined
as a set of r projection operators is used to induce decomposition of the
geometric objects into those of the corresponding subbundles. We define the
main geometric objects characterizing decomposition. Invariant non-holonomic
generalizations of the Gauss-Codazzi-Ricci's relations have been obtained. All
the known types of decomposition (used in the theory of frames of reference, in
the Hamiltonian formulation for gravity, in the Cauchy problem, in the theory
of stationary spaces, and so on) follow from the present work as special cases
when fixing a basis and dimensions of subbundles, and parameterization of a
basis of decomposition. Various methods of decomposition have been applied here
for the Unified Multidimensional Kaluza-Klein Theory and for relativistic
configurations of a perfect fluid. Discussing an invariant form of the
equations of motion we have found the invariant equilibrium conditions and
their 3+1 decomposed form. The formulation of the conservation law for the curl
has been obtained in the invariant form.Comment: 30 pages, RevTeX, aps.sty, some additions and corrections, new
references adde
Problems with Tunneling of Thin Shells from Black Holes
It is shown that is not invariant under canonical
transformations in general. Specifically for shells tunneling out of black
holes, this quantity is not invariant under canonical transformations. It can
be interpreted as the transmission coefficient only in the cases in which it is
invariant under canonical transformations. Although such cases include alpha
decay, they do not include the tunneling of shells from black holes. The
simplest extension to this formula which is invariant under canonical
transformations is proposed. However it is shown that this gives half the
correct temperature for black holes.Comment: 25 pages, 3 figures; v4: Made changes for publicatio
Wave patterns generated by a supersonic moving body in a binary Bose-Einstein condensate
Generation of wave structures by a two-dimensional object (laser beam) moving
in a two-dimensional two-component Bose-Einstein condensate with a velocity
greater than both sound velocities of the mixture is studied by means of
analytical methods and systematic simulations of the coupled Gross-Pitaevskii
equations. The wave pattern features three regions separated by two Mach cones.
Two branches of linear patterns similar to the so-called "ship waves" are
located outside the corresponding Mach cones, and oblique dark solitons are
found inside the wider cone. An analytical theory is developed for the linear
patterns. A particular dark-soliton solution is also obtained, its stability is
investigated, and two unstable modes of transverse perturbations are
identified. It is shown that, for a sufficiently large flow velocity, this
instability has a convective character in the reference frame attached to the
moving body, which makes the dark soliton effectively stable. The analytical
findings are corroborated by numerical simulations.Comment: 13 pages, 6 figure
Emission spectra and intrinsic optical bistability in a two-level medium
Scattering of resonant radiation in a dense two-level medium is studied
theoretically with account for local field effects and renormalization of the
resonance frequency. Intrinsic optical bistability is viewed as switching
between different spectral patterns of fluorescent light controlled by the
incident field strength. Response spectra are calculated analytically for the
entire hysteresis loop of atomic excitation. The equations to describe the
non-linear interaction of an atomic ensemble with light are derived from the
Bogolubov-Born-Green-Kirkwood-Yvon hierarchy for reduced single particle
density matrices of atoms and quantized field modes and their correlation
operators. The spectral power of scattered light with separated coherent and
incoherent constituents is obtained straightforwardly within the hierarchy. The
formula obtained for emission spectra can be used to distinguish between
possible mechanisms suggested to produce intrinsic bistability.Comment: 18 pages, 5 figure
Nonlinear diffraction of light beams propagating in photorefractive media with embedded reflecting wire
The theory of nonlinear diffraction of intensive light beams propagating
through photorefractive media is developed. Diffraction occurs on a reflecting
wire embedded in the nonlinear medium at relatively small angle with respect to
the direction of the beam propagation. It is shown that this process is
analogous to the generation of waves by a flow of a superfluid past an
obstacle. The ``equation of state'' of such a superfluid is determined by the
nonlinear properties of the medium. On the basis of this hydrodynamic analogy,
the notion of the ``Mach number'' is introduced where the transverse component
of the wave vector plays the role of the fluid velocity. It is found that the
Mach cone separates two regions of the diffraction pattern: inside the Mach
cone oblique dark solitons are generated and outside the Mach cone the region
of ``ship waves'' is situated. Analytical theory of ``ship waves'' is developed
and two-dimensional dark soliton solutions of the equation describing the beam
propagation are found. Stability of dark solitons with respect to their decay
into vortices is studied and it is shown that they are stable for large enough
values of the Mach number.Comment: 18 page
Dark solitons in atomic Bose-Einstein condensates: from theory to experiments
This review paper presents an overview of the theoretical and experimental
progress on the study of matter-wave dark solitons in atomic Bose-Einstein
condensates. Upon introducing the general framework, we discuss the statics and
dynamics of single and multiple matter-wave dark solitons in the quasi
one-dimensional setting, in higher-dimensional settings, as well as in the
dimensionality crossover regime. Special attention is paid to the connection
between theoretical results, obtained by various analytical approaches, and
relevant experimental observations.Comment: 82 pages, 13 figures. To appear in J. Phys. A: Math. Theor
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