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

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

Charge neutralization in vacuum for non-conducting and isolated objects using directed low-energy electron and ion beams

We propose using ions and electrons of energy 1 eV–10 eV for neutralizing the charges on the non-conducting or isolated surfaces of high-sensitivity experiments. The mirror surfaces of the test masses of the laser interferometer gravitational observatory are used as an example of the implementation of this method. By alternatively directing beams of positive and negative charges towards the mirror surfaces, we ensure the neutralization of the total charge as well as the equalization of the surface charge distribution to within a few eV of the potential of the ground reference of the vacuum system. This method is compatible with operation in high vacuum, does not require measuring the potential of the mirrors and is expected not to damage sensitive optical surfaces

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

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

Regularity of the Einstein Equations at Future Null Infinity

When Einstein's equations for an asymptotically flat, vacuum spacetime are reexpressed in terms of an appropriate conformal metric that is regular at (future) null infinity, they develop apparently singular terms in the associated conformal factor and thus appear to be ill-behaved at this (exterior) boundary. In this article however we show, through an enforcement of the Hamiltonian and momentum constraints to the needed order in a Taylor expansion, that these apparently singular terms are not only regular at the boundary but can in fact be explicitly evaluated there in terms of conformally regular geometric data. Though we employ a rather rigidly constrained and gauge fixed formulation of the field equations, we discuss the extent to which we expect our results to have a more 'universal' significance and, in particular, to be applicable, after minor modifications, to alternative formulations.Comment: 43 pages, no figures, AMS-TeX. Minor revisions, updated to agree with published versio

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

An axisymmetric evolution code for the Einstein equations on hyperboloidal slices

We present the first stable dynamical numerical evolutions of the Einstein equations in terms of a conformally rescaled metric on hyperboloidal hypersurfaces extending to future null infinity. Axisymmetry is imposed in order to reduce the computational cost. The formulation is based on an earlier axisymmetric evolution scheme, adapted to time slices of constant mean curvature. Ideas from a previous study by Moncrief and the author are applied in order to regularize the formally singular evolution equations at future null infinity. Long-term stable and convergent evolutions of Schwarzschild spacetime are obtained, including a gravitational perturbation. The Bondi news function is evaluated at future null infinity.Comment: 21 pages, 4 figures. Minor additions, updated to agree with journal versio

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

Monomeric alpha-synuclein exerts a physiological role in brain ATP synthase

Misfolded α-synuclein is a key factor in the pathogenesis of Parkinson's disease (PD). However, knowledge about a physiological role for the native, unfolded α-synuclein is limited. Using brains of mice lacking α-, β-, and γ-synuclein, we report that extracellular monomeric α-synuclein enters neurons and localizes to mitochondria, interacts with ATP synthase subunit α, and modulates ATP synthase function. Using a combination of biochemical, live-cell imaging and mitochondrial respiration analysis, we found that brain mitochondria of α-, β-, and γ-synuclein knock-out mice are uncoupled, as characterized by increased mitochondrial respiration and reduced mitochondrial membrane potential. Furthermore, synuclein deficiency results in reduced ATP synthase efficiency and lower ATP levels. Exogenous application of low unfolded α-synuclein concentrations is able to increase the ATP synthase activity that rescues the mitochondrial phenotypes observed in synuclein deficiency. Overall, the data suggest that α-synuclein is a previously unrecognized physiological regulator of mitochondrial bioenergetics through its ability to interact with ATP synthase and increase its efficiency. This may be of particular importance in times of stress or PD mutations leading to energy depletion and neuronal cell toxicity