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 $\beta^{i}(t,x^{j})$ and the spatial scalar potential $\phi(t,x^{j})$, 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 $(t,x^{j})$ 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 $(1)$ taking a further time derivative of the equation of motion of the canonical momentum, and $(2)$ adding a covariant spatial derivative of the momentum constraints of general relativity (Lagrange multiplier $\beta^{i}$) or of the Gauss's law constraint of electromagnetism (Lagrange multiplier $\phi$). 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 $\alpha(x,t)$ is freely specified while the lapse $N=\alpha g^{1/2}$ is not. The constraint ``algebra'' becomes a well-posed evolution system for the constraints; this system is the twice-contracted Bianchi identity when $R_{ij}=0$. The Hamiltonian constraint is an initial value constraint which determines $g^{1/2}$ and hence $N$, given $\alpha$.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 "$\lambda$-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 $g^*m^*$ inferred indirectly from magnetoconductance and magnetoresistance measurements with the assumption that $g^*\mu_BH_s=2E_F$ are in surprisingly good agreement with $g^*m^*$ obtained by direct measurement of Shubnikov-de Haas oscillations. The enhanced susceptibility $\chi^* \propto (g^*m^*)$ exhibits critical behavior of the form $\chi^* \propto (n - n_0)^{-\alpha}$. We examine the significance of the field scale $H_s$ 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