Improved outer boundary conditions for Einstein's field equations

In a recent article, we constructed a hierarchy B_L of outer boundary conditions for Einstein's field equations with the property that, for a spherical outer boundary, it is perfectly absorbing for linearized gravitational radiation up to a given angular momentum number L. In this article, we generalize B_2 so that it can be applied to fairly general foliations of spacetime by space-like hypersurfaces and general outer boundary shapes and further, we improve B_2 in two steps: (i) we give a local boundary condition C_2 which is perfectly absorbing including first order contributions in 2M/R of curvature corrections for quadrupolar waves (where M is the mass of the spacetime and R is a typical radius of the outer boundary) and which significantly reduces spurious reflections due to backscatter, and (ii) we give a non-local boundary condition D_2 which is exact when first order corrections in 2M/R for both curvature and backscatter are considered, for quadrupolar radiation.Comment: accepted Class. Quant. Grav. numerical relativity special issue; 17 pages and 1 figur

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

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

Schwarzschild Tests of the Wahlquist-Estabrook-Buchman-Bardeen Tetrad Formulation for Numerical Relativity

A first order symmetric hyperbolic tetrad formulation of the Einstein equations developed by Estabrook and Wahlquist and put into a form suitable for numerical relativity by Buchman and Bardeen (the WEBB formulation) is adapted to explicit spherical symmetry and tested for accuracy and stability in the evolution of spherically symmetric black holes (the Schwarzschild geometry). The lapse and shift which specify the evolution of the coordinates relative to the tetrad congruence are reset at frequent time intervals to keep the constant-time hypersurfaces nearly orthogonal to the tetrad congruence and the spatial coordinate satisfying a kind of minimal rate of strain condition. By arranging through initial conditions that the constant-time hypersurfaces are asymptotically hyperbolic, we simplify the boundary value problem and improve stability of the evolution. Results are obtained for both tetrad gauges (``Nester'' and ``Lorentz'') of the WEBB formalism using finite difference numerical methods. We are able to obtain stable unconstrained evolution with the Nester gauge for certain initial conditions, but not with the Lorentz gauge.Comment: (accepted by Phys. Rev. D) minor changes; typos correcte

Electrostatic Patch Effect in Cylindrical Geometry. I. Potential and Energy between Slightly Non-Coaxial Cylinders

We study the effect of any uneven voltage distribution on two close cylindrical conductors with parallel axes that are slightly shifted in the radial and by any length in the axial direction. The investigation is especially motivated by certain precision measurements, such as the Satellite Test of the Equivalence Principle (STEP). By energy conservation, the force can be found as the energy gradient in the vector of the shift, which requires determining potential distribution and energy in the gap. The boundary value problem for the potential is solved, and energy is thus found to the second order in the small transverse shift, and to lowest order in the gap to cylinder radius ratio. The energy consists of three parts: the usual capacitor part due to the uniform potential difference, the one coming from the interaction between the voltage patches and the uniform voltage difference, and the energy of patch interaction, entirely independent of the uniform voltage. Patch effect forces and torques in the cylindrical configuration are derived and analyzed in the next two parts of this work.Comment: 26 pages, 1 Figure. Submitted to Classical and Quantum Gravit

Testing outer boundary treatments for the Einstein equations

Various methods of treating outer boundaries in numerical relativity are compared using a simple test problem: a Schwarzschild black hole with an outgoing gravitational wave perturbation. Numerical solutions computed using different boundary treatments are compared to a `reference' numerical solution obtained by placing the outer boundary at a very large radius. For each boundary treatment, the full solutions including constraint violations and extracted gravitational waves are compared to those of the reference solution, thereby assessing the reflections caused by the artificial boundary. These tests use a first-order generalized harmonic formulation of the Einstein equations. Constraint-preserving boundary conditions for this system are reviewed, and an improved boundary condition on the gauge degrees of freedom is presented. Alternate boundary conditions evaluated here include freezing the incoming characteristic fields, Sommerfeld boundary conditions, and the constraint-preserving boundary conditions of Kreiss and Winicour. Rather different approaches to boundary treatments, such as sponge layers and spatial compactification, are also tested. Overall the best treatment found here combines boundary conditions that preserve the constraints, freeze the Newman-Penrose scalar Psi_0, and control gauge reflections.Comment: Modified to agree with version accepted for publication in Class. Quantum Gra

Combinations of motor measures more strongly predict adverse health outcomes in old age: the rush memory and aging project, a community-based cohort study

<p>Abstract</p> <p>Objective</p> <p>Motor impairment in old age is a growing public-health concern, and several different constructs have been used to identify motor impairments in older people. We tested the hypothesis that combinations of motor constructs more strongly predict adverse health outcomes in older people.</p> <p>Methods</p> <p>In total, 949 people without dementia, history of stroke or Parkinson's disease, who were participating in the Rush Memory and Aging Project (a longitudinal community-based cohort study), underwent assessment at study entry. From this, three constructs were derived: 1) physical frailty based on grip strength, timed walk, body mass index and fatigue; 2) Parkinsonian Signs Score based on the modified motor section of the Unified Parkinson's Disease Rating Scale; and 3) a motor construct, based on nine strength measures and nine motor performances. Disability and cognitive status were assessed annually. A series of Cox proportional-hazards models, controlling for age, sex and education, were used to examine the association of each of these three constructs alone and in various combinations with death, disability and Alzheimer's disease (AD).</p> <p>Results</p> <p>All three constructs were related (mean <it>r </it>= 0.50, all <it>P </it>< 0.001), and when considered individually in separate proportional-hazards models, were associated with risk of death, incident disability and AD. However, when considered together, combinations of these constructs more strongly predicted adverse health outcomes.</p> <p>Conclusions</p> <p>Physical frailty, parkinsonian signs score and global motor score are related constructs that capture different aspects of motor function. Assessments using several motor constructs may more accurately identify people at the highest risk of adverse health consequences in old age.</p

Deletion of alpha-synuclein decreases impulsivity in mice

The presynaptic protein alpha-synuclein, associated with Parkinson's Disease (PD), plays a role in dopaminergic neurotransmission and is implicated in impulse control disorders (ICDs) such as drug addiction. In this study we investigated a potential causal relationship between alpha-synuclein and impulsivity, by evaluating differences in motor impulsivity in the 5-choice serial reaction time task (5-CSRTT) in strains of mice that differ in the expression of the alpha-synuclein gene. C57BL/6JOlaHsd mice differ from their C57BL/6J ancestors in possessing a chromosomal deletion resulting in the loss of two genes, snca, encoding alpha-synuclein, and mmrn1, encoding multimerin-1. C57BL/6J mice displayed higher impulsivity (more premature responding) than C57BL/6JOlaHsd mice when the pre-stimulus waiting interval was increased in the 5-CSRTT. In order to ensure that the reduced impulsivity was indeed related to snca, and not adjacent gene deletion, wild type (WT) and mice with targeted deletion of alpha-synuclein (KO) were tested in the 5-CSRTT. Similarly, WT mice were more impulsive than mice with targeted deletion of alpha-synuclein. Interrogation of our ongoing analysis of impulsivity in BXD recombinant inbred mouse lines revealed an association of impulsive responding with levels of alpha-synuclein expression in hippocampus. Expression of beta- and gamma-synuclein, members of the synuclein family that may substitute for alpha-synuclein following its deletion, revealed no differential compensations among the mouse strains. These findings suggest that alpha-synuclein may contribute to impulsivity and potentially, to ICDs which arise in some PD patients treated with dopaminergic medication

Numerical Relativity Using a Generalized Harmonic Decomposition

A new numerical scheme to solve the Einstein field equations based upon the generalized harmonic decomposition of the Ricci tensor is introduced. The source functions driving the wave equations that define generalized harmonic coordinates are treated as independent functions, and encode the coordinate freedom of solutions. Techniques are discussed to impose particular gauge conditions through a specification of the source functions. A 3D, free evolution, finite difference code implementing this system of equations with a scalar field matter source is described. The second-order-in-space-and-time partial differential equations are discretized directly without the use first order auxiliary terms, limiting the number of independent functions to fifteen--ten metric quantities, four source functions and the scalar field. This also limits the number of constraint equations, which can only be enforced to within truncation error in a numerical free evolution, to four. The coordinate system is compactified to spatial infinity in order to impose physically motivated, constraint-preserving outer boundary conditions. A variant of the Cartoon method for efficiently simulating axisymmetric spacetimes with a Cartesian code is described that does not use interpolation, and is easier to incorporate into existing adaptive mesh refinement packages. Preliminary test simulations of vacuum black hole evolution and black hole formation via scalar field collapse are described, suggesting that this method may be useful for studying many spacetimes of interest.Comment: 18 pages, 6 figures; updated to coincide with journal version, which includes some expanded discussions and a new appendix with a stability analysis of a simplified problem using the same discretization scheme described in the pape

Constraint-preserving boundary treatment for a harmonic formulation of the Einstein equations

We present a set of well-posed constraint-preserving boundary conditions for a first-order in time, second-order in space, harmonic formulation of the Einstein equations. The boundary conditions are tested using robust stability, linear and nonlinear waves, and are found to be both less reflective and constraint preserving than standard Sommerfeld-type boundary conditions.Comment: 18 pages, 7 figures, accepted in CQ