Explicit solution of the linearized Einstein equations in TT gauge for all multipoles

We write out the explicit form of the metric for a linearized gravitational wave in the transverse-traceless gauge for any multipole, thus generalizing the well-known quadrupole solution of Teukolsky. The solution is derived using the generalized Regge-Wheeler-Zerilli formalism developed by Sarbach and Tiglio.Comment: 9 pages. Minor corrections, updated references. Final version to appear in Class. Quantum Gra

Atomic multipole relaxation rates near surfaces

The spontaneous relaxation rates for an atom in free space and close to an absorbing surface are calculated to various orders of the electromagnetic multipole expansion. The spontaneous decay rates for dipole, quadrupole and octupole transitions are calculated in terms of their respective primitive electric multipole moments and the magnetic relaxation rate is calculated for the dipole and quadrupole transitions in terms of their respective primitive magnetic multipole moments. The theory of electromagnetic field quantization in magnetoelectric materials is used to derive general expressions for the decay rates in terms of the dyadic Green function. We focus on the decay rates in free space and near an infinite half space. For the decay of atoms near to an absorbing dielectric surface we find a hierarchy of scaling laws depending on the atom-surface distance z.Comment: Updated to journal version. 16 page

Black Hole--Scalar Field Interactions in Spherical Symmetry

We examine the interactions of a black hole with a massless scalar field using a coordinate system which extends ingoing Eddington-Finkelstein coordinates to dynamic spherically symmetric-spacetimes. We avoid problems with the singularity by excising the region of the black hole interior to the apparent horizon. We use a second-order finite difference scheme to solve the equations. The resulting program is stable and convergent and will run forever without problems. We are able to observe quasi-normal ringing and power-law tails as well an interesting nonlinear feature.Comment: 16 pages, 26 figures, RevTex, to appear in Phys. Rev.

Constructing hyperbolic systems in the Ashtekar formulation of general relativity

Hyperbolic formulations of the equations of motion are essential technique for proving the well-posedness of the Cauchy problem of a system, and are also helpful for implementing stable long time evolution in numerical applications. We, here, present three kinds of hyperbolic systems in the Ashtekar formulation of general relativity for Lorentzian vacuum spacetime. We exhibit several (I) weakly hyperbolic, (II) diagonalizable hyperbolic, and (III) symmetric hyperbolic systems, with each their eigenvalues. We demonstrate that Ashtekar's original equations form a weakly hyperbolic system. We discuss how gauge conditions and reality conditions are constrained during each step toward constructing a symmetric hyperbolic system.Comment: 15 pages, RevTeX, minor changes in Introduction. published as Int. J. Mod. Phys. D 9 (2000) 1

Symmetric hyperbolic system in the Ashtekar formulation

We present a first-order symmetric hyperbolic system in the Ashtekar formulation of general relativity for vacuum spacetime. We add terms from constraint equations to the evolution equations with appropriate combinations, which is the same technique used by Iriondo, Leguizam\'on and Reula [Phys. Rev. Lett. 79, 4732 (1997)]. However our system is different from theirs in the points that we primarily use Hermiticity of a characteristic matrix of the system to characterize our system "symmetric", discuss the consistency of this system with reality condition, and show the characteristic speeds of the system.Comment: 4 pages, RevTeX, to appear in Phys. Rev. Lett., Comments added, refs update

Approaching the Heisenberg limit with two mode squeezed states

Two mode squeezed states can be used to achieve Heisenberg limit scaling in interferometry: a phase shift of $\delta \phi \approx 2.76 /$ can be resolved. The proposed scheme relies on balanced homodyne detection and can be implemented with current technology. The most important experimental imperfections are studied and their impact quantified.Comment: 4 pages, 7 figure

Energy Norms and the Stability of the Einstein Evolution Equations

The Einstein evolution equations may be written in a variety of equivalent analytical forms, but numerical solutions of these different formulations display a wide range of growth rates for constraint violations. For symmetric hyperbolic formulations of the equations, an exact expression for the growth rate is derived using an energy norm. This expression agrees with the growth rate determined by numerical solution of the equations. An approximate method for estimating the growth rate is also derived. This estimate can be evaluated algebraically from the initial data, and is shown to exhibit qualitatively the same dependence as the numerically-determined rate on the parameters that specify the formulation of the equations. This simple rate estimate therefore provides a useful tool for finding the most well-behaved forms of the evolution equations.Comment: Corrected typos; to appear in Physical Review

Implementation of higher-order absorbing boundary conditions for the Einstein equations

We present an implementation of absorbing boundary conditions for the Einstein equations based on the recent work of Buchman and Sarbach. In this paper, we assume that spacetime may be linearized about Minkowski space close to the outer boundary, which is taken to be a coordinate sphere. We reformulate the boundary conditions as conditions on the gauge-invariant Regge-Wheeler-Zerilli scalars. Higher-order radial derivatives are eliminated by rewriting the boundary conditions as a system of ODEs for a set of auxiliary variables intrinsic to the boundary. From these we construct boundary data for a set of well-posed constraint-preserving boundary conditions for the Einstein equations in a first-order generalized harmonic formulation. This construction has direct applications to outer boundary conditions in simulations of isolated systems (e.g., binary black holes) as well as to the problem of Cauchy-perturbative matching. As a test problem for our numerical implementation, we consider linearized multipolar gravitational waves in TT gauge, with angular momentum numbers l=2 (Teukolsky waves), 3 and 4. We demonstrate that the perfectly absorbing boundary condition B_L of order L=l yields no spurious reflections to linear order in perturbation theory. This is in contrast to the lower-order absorbing boundary conditions B_L with L<l, which include the widely used freezing-Psi_0 boundary condition that imposes the vanishing of the Newman-Penrose scalar Psi_0.Comment: 25 pages, 9 figures. Minor clarifications. Final version to appear in Class. Quantum Grav

Stable radiation-controlling boundary conditions for the generalized harmonic Einstein equations

This paper is concerned with the initial-boundary value problem for the Einstein equations in a first-order generalized harmonic formulation. We impose boundary conditions that preserve the constraints and control the incoming gravitational radiation by prescribing data for the incoming fields of the Weyl tensor. High-frequency perturbations about any given spacetime (including a shift vector with subluminal normal component) are analyzed using the Fourier-Laplace technique. We show that the system is boundary-stable. In addition, we develop a criterion that can be used to detect weak instabilities with polynomial time dependence, and we show that our system does not suffer from such instabilities. A numerical robust stability test supports our claim that the initial-boundary value problem is most likely to be well-posed even if nonzero initial and source data are included.Comment: 27 pages, 4 figures; more numerical results and references added, several minor amendments; version accepted for publication in Class. Quantum Gra