2,440 research outputs found
Cauchy-perturbative matching and outer boundary conditions I: Methods and tests
We present a new method of extracting gravitational radiation from
three-dimensional numerical relativity codes and providing outer boundary
conditions. Our approach matches the solution of a Cauchy evolution of
Einstein's equations to a set of one-dimensional linear wave equations on a
curved background. We illustrate the mathematical properties of our approach
and discuss a numerical module we have constructed for this purpose. This
module implements the perturbative matching approach in connection with a
generic three-dimensional numerical relativity simulation. Tests of its
accuracy and second-order convergence are presented with analytic linear wave
data.Comment: 13 pages, 6 figures, RevTe
Geometrical Hyperbolic Systems for General Relativity and Gauge Theories
The evolution equations of Einstein's theory and of Maxwell's theory---the
latter used as a simple model to illustrate the former--- are written in gauge
covariant first order symmetric hyperbolic form with only physically natural
characteristic directions and speeds for the dynamical variables. Quantities
representing gauge degrees of freedom [the spatial shift vector
and the spatial scalar potential ,
respectively] are not among the dynamical variables: the gauge and the physical
quantities in the evolution equations are effectively decoupled. For example,
the gauge quantities could be obtained as functions of from
subsidiary equations that are not part of the evolution equations. Propagation
of certain (``radiative'') dynamical variables along the physical light cone is
gauge invariant while the remaining dynamical variables are dragged along the
axes orthogonal to the spacelike time slices by the propagating variables. We
obtain these results by taking a further time derivative of the equation
of motion of the canonical momentum, and adding a covariant spatial
derivative of the momentum constraints of general relativity (Lagrange
multiplier ) or of the Gauss's law constraint of electromagnetism
(Lagrange multiplier ). General relativity also requires a harmonic time
slicing condition or a specific generalization of it that brings in the
Hamiltonian constraint when we pass to first order symmetric form. The
dynamically propagating gravity fields straightforwardly determine the
``electric'' or ``tidal'' parts of the Riemann tensor.Comment: 24 pages, latex, no figure
Cauchy-perturbative matching and outer boundary conditions: computational studies
We present results from a new technique which allows extraction of
gravitational radiation information from a generic three-dimensional numerical
relativity code and provides stable outer boundary conditions. In our approach
we match the solution of a Cauchy evolution of the nonlinear Einstein field
equations to a set of one-dimensional linear equations obtained through
perturbation techniques over a curved background. We discuss the validity of
this approach in the case of linear and mildly nonlinear gravitational waves
and show how a numerical module developed for this purpose is able to provide
an accurate and numerically convergent description of the gravitational wave
propagation and a stable numerical evolution.Comment: 20 pages, RevTe
Hamiltonian Time Evolution for General Relativity
Hamiltonian time evolution in terms of an explicit parameter time is derived
for general relativity, even when the constraints are not satisfied, from the
Arnowitt-Deser-Misner-Teitelboim-Ashtekar action in which the slicing density
is freely specified while the lapse is not.
The constraint ``algebra'' becomes a well-posed evolution system for the
constraints; this system is the twice-contracted Bianchi identity when
. The Hamiltonian constraint is an initial value constraint which
determines and hence , given .Comment: 4 pages, revtex, to appear in Phys. Rev. Let
Einstein and Yang-Mills theories in hyperbolic form without gauge-fixing
The evolution of physical and gauge degrees of freedom in the Einstein and
Yang-Mills theories are separated in a gauge-invariant manner. We show that the
equations of motion of these theories can always be written in
flux-conservative first-order symmetric hyperbolic form. This dynamical form is
ideal for global analysis, analytic approximation methods such as
gauge-invariant perturbation theory, and numerical solution.Comment: 12 pages, revtex3.0, no figure
Waveform propagation in black hole spacetimes: evaluating the quality of numerical solutions
We compute the propagation and scattering of linear gravitational waves off a
Schwarzschild black hole using a numerical code which solves a generalization
of the Zerilli equation to a three dimensional cartesian coordinate system.
Since the solution to this problem is well understood it represents a very good
testbed for evaluating our ability to perform three dimensional computations of
gravitational waves in spacetimes in which a black hole event horizon is
present.Comment: 13 pages, RevTeX, to appear in Phys. Rev.
Differential Forms and Wave Equations for General Relativity
Recently, Choquet-Bruhat and York and Abrahams, Anderson, Choquet-Bruhat, and
York (AACY) have cast the 3+1 evolution equations of general relativity in
gauge-covariant and causal ``first-order symmetric hyperbolic form,'' thereby
cleanly separating physical from gauge degrees of freedom in the Cauchy problem
for general relativity. A key ingredient in their construction is a certain
wave equation which governs the light-speed propagation of the extrinsic
curvature tensor. Along a similar line, we construct a related wave equation
which, as the key equation in a system, describes vacuum general relativity.
Whereas the approach of AACY is based on tensor-index methods, the present
formulation is written solely in the language of differential forms. Our
approach starts with Sparling's tetrad-dependent differential forms, and our
wave equation governs the propagation of Sparling's 2-form, which in the
``time-gauge'' is built linearly from the ``extrinsic curvature 1-form.'' The
tensor-index version of our wave equation describes the propagation of (what is
essentially) the Arnowitt-Deser-Misner gravitational momentum.Comment: REVTeX, 26 pages, no figures, 1 macr
Hyperbolic formulations and numerical relativity II: Asymptotically constrained systems of the Einstein equations
We study asymptotically constrained systems for numerical integration of the
Einstein equations, which are intended to be robust against perturbative errors
for the free evolution of the initial data. First, we examine the previously
proposed "-system", which introduces artificial flows to constraint
surfaces based on the symmetric hyperbolic formulation. We show that this
system works as expected for the wave propagation problem in the Maxwell system
and in general relativity using Ashtekar's connection formulation. Second, we
propose a new mechanism to control the stability, which we call the ``adjusted
system". This is simply obtained by adding constraint terms in the dynamical
equations and adjusting its multipliers. We explain why a particular choice of
multiplier reduces the numerical errors from non-positive or pure-imaginary
eigenvalues of the adjusted constraint propagation equations. This ``adjusted
system" is also tested in the Maxwell system and in the Ashtekar's system. This
mechanism affects more than the system's symmetric hyperbolicity.Comment: 16 pages, RevTeX, 9 eps figures, added Appendix B and minor changes,
to appear in Class. Quant. Gra
Collapse of Charge Gap in Random Mott Insulators
Effects of randomness on interacting fermionic systems in one dimension are
investigated by quantum Monte-Carlo techniques. At first, interacting spinless
fermions are studied whose ground state shows charge ordering. Quantum phase
transition due to randomness is observed associated with the collapse of the
charge ordering. We also treat random Hubbard model focusing on the Mott gap.
Although the randomness closes the Mott gap and low-lying states are created,
which is observed in the charge compressibility, no (quasi-) Fermi surface
singularity is formed. It implies localized nature of the low-lying states.Comment: RevTeX with 3 postscript figure
Spin polarization of strongly interacting 2D electrons: the role of disorder
In high-mobility silicon MOSFET's, the inferred indirectly from
magnetoconductance and magnetoresistance measurements with the assumption that
are in surprisingly good agreement with obtained by
direct measurement of Shubnikov-de Haas oscillations. The enhanced
susceptibility exhibits critical behavior of the form
. We examine the significance of the field
scale derived from transport measurements, and show that this field
signals the onset of full spin polarization only in the absence of disorder.
Our results suggest that disorder becomes increasingly important as the
electron density is reduced toward the transition.Comment: 4 pages, 3 figure
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