104 research outputs found
Visualizing Spacetime Curvature via Gradient Flows II: An Example of the Construction of a Newtonian analogue
This is the first in a series of papers in which the gradient flows of
fundamental curvature invariants are used to formulate a visualization of
curvature. We start with the construction of strict Newtonian analogues (not
limits) of solutions to Einstein's equations based on the topology of the
associated gradient flows. We do not start with any easy case. Rather, we start
with the Curzon - Chazy solution, which, as history shows, is one of the most
difficult exact solutions to Einstein's equations to interpret physically. We
show that the entire field of the Curzon - Chazy solution, up to a region very
"close" to the the intrinsic singularity, strictly represents that of a
Newtonian ring, as has long been suspected. In this regard, we consider our
approach very successful. As regrades the local structure of the singularity of
the Curzon - Chazy solution within a fully general relativistic analysis,
however, whereas we make some advances, the full structure of this singularity
remains incompletely resolved.Comment: 12 pages twocolumn revtex 4-1 9 figures. Expanded and correcte
Movable algebraic singularities of second-order ordinary differential equations
Any nonlinear equation of the form y''=\sum_{n=0}^N a_n(z)y^n has a
(generally branched) solution with leading order behaviour proportional to
(z-z_0)^{-2/(N-1)} about a point z_0, where the coefficients a_n are analytic
at z_0 and a_N(z_0)\ne 0. We consider the subclass of equations for which each
possible leading order term of this form corresponds to a one-parameter family
of solutions represented near z_0 by a Laurent series in fractional powers of
z-z_0. For this class of equations we show that the only movable singularities
that can be reached by analytic continuation along finite-length curves are of
the algebraic type just described. This work generalizes previous results of S.
Shimomura. The only other possible kind of movable singularity that might occur
is an accumulation point of algebraic singularities that can be reached by
analytic continuation along infinitely long paths ending at a finite point in
the complex plane. This behaviour cannot occur for constant coefficient
equations in the class considered. However, an example of R. A. Smith shows
that such singularities do occur in solutions of a simple autonomous
second-order differential equation outside the class we consider here
Gap Probabilities for Edge Intervals in Finite Gaussian and Jacobi Unitary Matrix Ensembles
The probabilities for gaps in the eigenvalue spectrum of the finite dimension
random matrix Hermite and Jacobi unitary ensembles on some
single and disconnected double intervals are found. These are cases where a
reflection symmetry exists and the probability factors into two other related
probabilities, defined on single intervals. Our investigation uses the system
of partial differential equations arising from the Fredholm determinant
expression for the gap probability and the differential-recurrence equations
satisfied by Hermite and Jacobi orthogonal polynomials. In our study we find
second and third order nonlinear ordinary differential equations defining the
probabilities in the general case. For N=1 and N=2 the probabilities and
thus the solution of the equations are given explicitly. An asymptotic
expansion for large gap size is obtained from the equation in the Hermite case,
and also studied is the scaling at the edge of the Hermite spectrum as , and the Jacobi to Hermite limit; these last two studies make
correspondence to other cases reported here or known previously. Moreover, the
differential equation arising in the Hermite ensemble is solved in terms of an
explicit rational function of a {Painlev\'e-V} transcendent and its derivative,
and an analogous solution is provided in the two Jacobi cases but this time
involving a {Painlev\'e-VI} transcendent.Comment: 32 pages, Latex2
Constructing Integrable Third Order Systems:The Gambier Approach
We present a systematic construction of integrable third order systems based
on the coupling of an integrable second order equation and a Riccati equation.
This approach is the extension of the Gambier method that led to the equation
that bears his name. Our study is carried through for both continuous and
discrete systems. In both cases the investigation is based on the study of the
singularities of the system (the Painlev\'e method for ODE's and the
singularity confinement method for mappings).Comment: 14 pages, TEX FIL
Morgan-Morgan-NUT disk space via the Ehlers transformation
Using the Ehlers transformation along with the gravitoelectromagnetic
approach to stationary spacetimes we start from the Morgan-Morgan disk
spacetime (without radial pressure) as the seed metric and find its
corresponding stationary spacetime. As expected from the Ehlers transformation
the stationary spacetime obtained suffers from a NUT-type singularity and the
new parameter introduced in the stationary case could be interpreted as the
gravitomagnetic monopole charge (or the NUT factor). As a consequence of this
singularity there are closed timelike curves (CTCs) in the singular region of
the spacetime. Some of the properties of this spacetime including its particle
velocity distribution, gravitational redshift, stability and energy conditions
are discussed.Comment: 18 pages, 5 figures, RevTex 4, replaced with the published versio
Non-Schlesinger Deformations of Ordinary Differential Equations with Rational Coefficients
We consider deformations of and matrix linear ODEs with
rational coefficients with respect to singular points of Fuchsian type which
don't satisfy the well-known system of Schlesinger equations (or its natural
generalization). Some general statements concerning reducibility of such
deformations for ODEs are proved. An explicit example of the general
non-Schlesinger deformation of -matrix ODE of the Fuchsian type with
4 singular points is constructed and application of such deformations to the
construction of special solutions of the corresponding Schlesinger systems is
discussed. Some examples of isomonodromy and non-isomonodromy deformations of
matrix ODEs are considered. The latter arise as the compatibility
conditions with linear ODEs with non-singlevalued coefficients.Comment: 15 pages, to appear in J. Phys.
Exact relativistic models of thin disks around static black holes in a magnetic field
The exact superposition of a central static black hole with surrounding thin
disk in presence of a magnetic field is investigated. We consider two models of
disk, one of infinite extension based on a Kuzmin-Chazy-Curzon metric and other
finite based on the first Morgan-Morgan disk. We also analyze a simple model of
active galactic nuclei consisting of black hole, a Kuzmin-Chazy-Curzon disk and
two rods representing jets, in presence of magnetic field. To explain the
stability of the disks we consider the matter of the disk made of two
pressureless streams of counterrotating charged particles (counterrotating
model) moving along electrogeodesic. Using the Rayleigh criterion we derivate
for circular orbits the stability conditions of the particles of the streams.
The influence of the magnetic field on the matter properties of the disk and on
its stability are also analyzed.Comment: 17 pages, 14 figures. arXiv admin note: text overlap with
arXiv:gr-qc/0409109 by other author
Integrable systems without the Painlev\'e property
We examine whether the Painlev\'e property is a necessary condition for the
integrability of nonlinear ordinary differential equations. We show that for a
large class of linearisable systems this is not the case. In the discrete
domain, we investigate whether the singularity confinement property is
satisfied for the discrete analogues of the non-Painlev\'e continuous
linearisable systems. We find that while these discrete systems are themselves
linearisable, they possess nonconfined singularities
Square lattice Ising model susceptibility: Series expansion method and differential equation for
In a previous paper (J. Phys. A {\bf 37} (2004) 9651-9668) we have given the
Fuchsian linear differential equation satisfied by , the
``three-particle'' contribution to the susceptibility of the isotropic square
lattice Ising model. This paper gives the details of the calculations (with
some useful tricks and tools) allowing one to obtain long series in polynomial
time. The method is based on series expansion in the variables that appear in
the -dimensional integrals representing the -particle contribution to
the isotropic square lattice Ising model susceptibility . The
integration rules are straightforward due to remarkable formulas we derived for
these variables. We obtain without any numerical approximation as
a fully integrated series in the variable , where , with the conventional Ising model coupling constant. We also
give some perspectives and comments on these results.Comment: 28 pages, no figur
Disks in Expanding FRW Universes
We construct exact solutions to Einstein equations which represent
relativistic disks immersed into an expanding FRW Universe. It is shown that
the expansion influences dynamical characteristics of the disks such as
rotational curves, surface mass density, etc. The effects of the expansion is
exemplified with non-static generalizations of Kuzmin-Curzon and generalized
Schwarzschild disks.Comment: Revised version to appear in ApJ, Latex, 17 pages, 10 figures, uses
aaspp4 and epsf style file
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