Gauge drivers for the generalized harmonic Einstein equations

The generalized harmonic representation of Einstein's equations is manifestly hyperbolic for a large class of gauge conditions. Unfortunately most of the useful gauges developed over the past several decades by the numerical relativity community are incompatible with the hyperbolicity of the equations in this form. This paper presents a new method of imposing gauge conditions that preserves hyperbolicity for a much wider class of conditions, including as special cases many of the standard ones used in numerical relativity: e.g., K freezing, Gamma freezing, Bona-Massó slicing, conformal Gamma drivers, etc. Analytical and numerical results are presented which test the stability and the effectiveness of this new gauge-driver evolution system

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

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

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

Gowdy waves as a test-bed for constraint-preserving boundary conditions

Gowdy waves, one of the standard 'apples with apples' tests, is proposed as a test-bed for constraint-preserving boundary conditions in the non-linear regime. As an illustration, energy-constraint preservation is separately tested in the Z4 framework. Both algebraic conditions, derived from energy estimates, and derivative conditions, deduced from the constraint-propagation system, are considered. The numerical errors at the boundary are of the same order than those at the interior points.Comment: 5 pages, 1 figure. Contribution to the Spanish Relativity Meeting 200

Solving Einstein's Equations With Dual Coordinate Frames

A method is introduced for solving Einstein's equations using two distinct coordinate systems. The coordinate basis vectors associated with one system are used to project out components of the metric and other fields, in analogy with the way fields are projected onto an orthonormal tetrad basis. These field components are then determined as functions of a second independent coordinate system. The transformation to the second coordinate system can be thought of as a mapping from the original ``inertial'' coordinate system to the computational domain. This dual-coordinate method is used to perform stable numerical evolutions of a black-hole spacetime using the generalized harmonic form of Einstein's equations in coordinates that rotate with respect to the inertial frame at infinity; such evolutions are found to be generically unstable using a single rotating coordinate frame. The dual-coordinate method is also used here to evolve binary black-hole spacetimes for several orbits. The great flexibility of this method allows comoving coordinates to be adjusted with a feedback control system that keeps the excision boundaries of the holes within their respective apparent horizons.Comment: Updated to agree with published versio

A point process framework for modeling electrical stimulation of the auditory nerve

Model-based studies of auditory nerve responses to electrical stimulation can provide insight into the functioning of cochlear implants. Ideally, these studies can identify limitations in sound processing strategies and lead to improved methods for providing sound information to cochlear implant users. To accomplish this, models must accurately describe auditory nerve spiking while avoiding excessive complexity that would preclude large-scale simulations of populations of auditory nerve fibers and obscure insight into the mechanisms that influence neural encoding of sound information. In this spirit, we develop a point process model of the auditory nerve that provides a compact and accurate description of neural responses to electric stimulation. Inspired by the framework of generalized linear models, the proposed model consists of a cascade of linear and nonlinear stages. We show how each of these stages can be associated with biophysical mechanisms and related to models of neuronal dynamics. Moreover, we derive a semi-analytical procedure that uniquely determines each parameter in the model on the basis of fundamental statistics from recordings of single fiber responses to electric stimulation, including threshold, relative spread, jitter, and chronaxie. The model also accounts for refractory and summation effects that influence the responses of auditory nerve fibers to high pulse rate stimulation. Throughout, we compare model predictions to published physiological data and explain differences in auditory nerve responses to high and low pulse rate stimulation. We close by performing an ideal observer analysis of simulated spike trains in response to sinusoidally amplitude modulated stimuli and find that carrier pulse rate does not affect modulation detection thresholds.Comment: 1 title page, 27 manuscript pages, 14 figures, 1 table, 1 appendi

Reliability and validity of the Dutch dimensional assessment of personality pathology-short form(DAPP-SF), a shortened version of the DAPP-Basic questionnaire

The Dimensional Assessment of Personality Pathology-Basic Questionnaire (DAPP-BQ) appears to be a good choice for the assessment of personality pathology. However, due to its length, administration of the instrument is rather time-consuming, hindering standard inclusion of the DABB-BQ in a battery of assessment instruments at intake. We developed the 136-item DAPP-SF (Short Form), and investigated its psychometric characteristics in various samples, i.e., a community-based sample (n = 487), patients with mood-, anxiety-, and somatoform disorders (n = 1,329), and patients with personality disorders (n = 1,393). Results revealed high internal consistency for almost all dimensions. The factor structure appeared almost identical as compared to the factor structure of the original DAPP-BQ, and was shown to be invariant across the various patient and community samples. Indices for convergent, discriminant and criterion related validity were satisfactory. It is concluded that the good psychometric characteristics of the original DAPP-BQ were preserved in the shortened version of the instrument

Wurzelverlauf von Buchen (Fagus sylvatica) unter Rückegassen

Das Befahren von Waldböden im Zuge der mechanisierten Holzernte führt häufig zu deutlich erkennbaren Fahrspuren. Damit einher gehen Verdichtung und Belüftungsstörung des Bodens mit einer Reduktion der Feinwurzeldichte. Es stellt sich die Frage, ob eine Fahrspur im Wald als ein für Pflanzen toter Raum angesehen werden muss und ob es Bäumen möglich ist, den Raum zwischen den beiden Fahrspuren zu nutzen Um dies herauszufinden, wurde der Wurzelverlauf mehrerer am Rand einer Rückegasse stehender Bäume (Fagus sylvatica) in einem Waldgebiet auf Löß geprägter Braunerde untersucht. Die Wurzelanläufe der Bäume dienten bei der Wurzelfreilegung als Anfangspunkt. Mit diversen Kleinwerkzeugen wurde der Wurzelraum unter der Rückegasse vom Stammfuß bis zur Wurzelspitze einer jeden Wurzel freigelegt. Mit Hilfe eines Schnurgerüstes wurden die Tiefen der Wurzeln, deren Lage sowie deren Zustand und Anzahl, Dicke und Verzweigung vermessen und dokumentiert. In den Rückegassen gab es lebende Grobwurzeln, aber auch deutliche Verdichtungsanzeichen (Rostflecken, Bleichzonen). Der Einfluss der Fahrspuren ist deutlich zu erkennen. Viele Wurzeln verkümmerten in den Spuren, andere verweigerten den Wuchs in sie hinein und wählten einen Weg parallel zur Fahrspur. Wieder andere wuchsen unter der Spur hindurch, kamen im Mittelstreifen empor und drangen bei der zweiten Spur wieder tiefer in den Boden. Auffällig war ebenfalls die hohe Anzahl an Wurzeln, die nicht in den verdichteten Boden gingen, sondern in der neu gebildeten Humus-/ Laubschicht wuchsen. Somit lässt sich festhalten, dass eine Rückegasse, bzw. eine Fahrspur, kein toter Raum für Baumwurzeln ist, aber eine durchaus starke Behinderung darstellt. Die Bäume nutzen Möglichkeiten, der Verdichtung auszuweichen (Humusschicht, Parallelwuchs) oder sind kräftig genug sie zu überwinden und den Mittelstreifen als Lebensraum zu erobern

Black Hole Mergers and Unstable Circular Orbits

We describe recent numerical simulations of the merger of a class of equal mass, non-spinning, eccentric binary black hole systems in general relativity. We show that with appropriate fine-tuning of the initial conditions to a region of parameter space we denote the threshold of immediate merger, the binary enters a phase of close interaction in a near-circular orbit, stays there for an amount of time proportional to logarithmic distance from the threshold in parameter space, then either separates or merges to form a single Kerr black hole. To gain a better understanding of this phenomena we study an analogous problem in the evolution of equatorial geodesics about a central Kerr black hole. A similar threshold of capture exists for appropriate classes of initial conditions, and tuning to threshold the geodesics approach one of the unstable circular geodesics of the Kerr spacetime. Remarkably, with a natural mapping of the parameters of the geodesic to that of the equal mass system, the scaling exponent describing the whirl phase of each system turns out to be quite similar. Armed with this lone piece of evidence that an approximate correspondence might exist between near-threshold evolution of geodesics and generic binary mergers, we illustrate how this information can be used to estimate the cross section and energy emitted in the ultra relativistic black hole scattering problem. This could eventually be of use in providing estimates for the related problem of parton collisions at the Large Hadron Collider in extra dimension scenarios where black holes are produced.Comment: 16 pages, 12 figures; updated to coincide with journal versio