arXiv subjects

Richard A. Matzner

Publications and source records attributed to Richard A. Matzner.

35 records · Page 2Linked to original sources

arXiv2004

Hyperbolicity and Constrained Evolution in Linearized Gravity

Solving the 4-d Einstein equations as evolution in time requires solving equations of two types: the four elliptic initial data (constraint) equations, followed by the six second order evolution equations. Analytically the constraint equations remain solved under the action of the evolution, and one approach is to simply monitor them ({\it unconstrained} evolution). Since computational solution of differential equations introduces almost inevitable errors, it is clearly "more correct" to introduce a scheme which actively maintains the constraints by solution ({\it constrained} evolution). This has shown promise in computational settings, but the analysis of the resulting mixed elliptic hyperbolic method has not been completely carried out. We present such an analysis for one method of constrained evolution, applied to a simple vacuum system, linearized gravitational waves. We begin with a study of the hyperbolicity of the unconstrained Einstein equations. (Because the study of hyperbolicity deals only with the highest derivative order in the equations, linearization loses no essential details.) We then give explicit analytical construction of the effect of initial data setting and constrained evolution for linearized gravitational waves. While this is clearly a toy model with regard to constrained evolution, certain interesting features are found which have relevance to the full nonlinear Einstein equations.

gr-qc↗

arXiv2003

Tips for implementing multigrid methods on domains containing holes

As part of our development of a computer code to perform 3D `constrained evolution' of Einstein's equations in 3+1 form, we discuss issues regarding the efficient solution of elliptic equations on domains containing holes (i.e., excised regions), via the multigrid method. We consider as a test case the Poisson equation with a nonlinear term added, as a means of illustrating the principles involved, and move to a "real world" 3-dimensional problem which is the solution of the conformally flat Hamiltonian constraint with Dirichlet and Robin boundary conditions. Using our vertex-centered multigrid code, we demonstrate globally second-order-accurate solutions of elliptic equations over domains containing holes, in two and three spatial dimensions. Keys to the success of this method are the choice of the restriction operator near the holes and definition of the location of the inner boundary. In some cases (e.g. two holes in two dimensions), more and more smoothing may be required as the mesh spacing decreases to zero; however for the resolutions currently of interest to many numerical relativists, it is feasible to maintain second order convergence by concentrating smoothing (spatially) where it is needed most. This paper, and our publicly available source code, are intended to serve as semi-pedagogical guides for those who may wish to implement similar schemes.

gr-qc↗

arXiv2003

Adaptive Event Horizon Tracking and Critical Phenomena in Binary Black Hole Coalescence

This work establishes critical phenomena in the topological transition of black hole coalescence. We describe and validate a computational front tracking event horizon solver, developed for generic studies of the black hole coalescence problem. We then apply this to the Kastor - Traschen axisymmetric analytic solution of the extremal Maxwell - Einstein black hole merger with cosmological constant. The surprising result of this computational analysis is a power law scaling of the minimal throat proportional to time. The minimal throat connecting the two holes obeys this power law during a short time immediately at the beginning of merger. We also confirm the behavior analytically. Thus, at least in one axisymmetric situation a critical phenomenon exists. We give arguments for a broader universality class than the restricted requirements of the Kastor - Traschen solution.

gr-qc↗

arXiv2003

Tracking Black Holes in Numerical Relativity

This work addresses and solves the problem of generically tracking black hole event horizons in computational simulation of black hole interactions. Solutions of the hyperbolic eikonal equation, solved on a curved spacetime manifold containing black hole sources, are employed in development of a robust tracking method capable of continuously monitoring arbitrary changes of topology in the event horizon, as well as arbitrary numbers of gravitational sources. The method makes use of continuous families of level set viscosity solutions of the eikonal equation with identification of the black hole event horizon obtained by the signature feature of discontinuity formation in the eikonal's solution. The method is employed in the analysis of the event horizon for the asymmetric merger in a binary black hole system. In this first such three dimensional analysis, we establish both qualitative and quantitative physics for the asymmetric collision; including: 1. Bounds on the topology of the throat connecting the holes following merger, 2. Time of merger, and 3. Continuous accounting for the surface of section areas of the black hole sources.

gr-qc↗

arXiv2000

Solving the Initial Value Problem of two Black Holes

We solve the elliptic equations associated with the Hamiltonian and momentum constraints, corresponding to a system composed of two black holes with arbitrary linear and angular momentum. These new solutions are based on a Kerr-Schild spacetime slicing which provides more physically realistic solutions than the initial data based on conformally flat metric/maximal slicing methods. The singularity/inner boundary problems are circumvented by a new technique that allows the use of an elliptic solver on a Cartesian grid where no points are excised, simplifying enormously the numerical problem.

gr-qc↗

arXiv2000

Grazing Collisions of Black Holes via the Excision of Singularities

We present the first simulations of non-headon (grazing) collisions of binary black holes in which the black hole singularities have been excised from the computational domain. Initially two equal mass black holes $m$ are separated a distance $\approx10m$ and with impact parameter $\approx2m$. Initial data are based on superposed, boosted (velocity $\approx0.5c$) solutions of single black holes in Kerr-Schild coordinates. Both rotating and non-rotating black holes are considered. The excised regions containing the singularities are specified by following the dynamics of apparent horizons. Evolutions of up to $t \approx 35m$ are obtained in which two initially separate apparent horizons are present for $t\approx3.8m$. At that time a single enveloping apparent horizon forms, indicating that the holes have merged. Apparent horizon area estimates suggest gravitational radiation of about 2.6% of the total mass. The evolutions end after a moderate amount of time because of instabilities.

gr-qc↗

arXiv2000

Generic Tracking of Multiple Apparent Horizons with Level Flow

We report the development of the first apparent horizon locator capable of finding multiple apparent horizons in a ``generic'' numerical black hole spacetime. We use a level-flow method which, starting from a single arbitrary initial trial surface, can undergo topology changes as it flows towards disjoint apparent horizons if they are present. The level flow method has two advantages: 1) The solution is independent of changes in the initial guess and 2) The solution can have multiple components. We illustrate our method of locating apparent horizons by tracking horizon components in a short Kerr-Schild binary black hole grazing collision.

gr-qc↗

arXiv2000

Locating Boosted Kerr and Schwarzschild Apparent Horizons

We describe a finite-difference method for locating apparent horizons and illustrate its capabilities on boosted Kerr and Schwarzschild black holes. Our model spacetime is given by the Kerr-Schild metric. We apply a Lorentz boost to this spacetime metric and then carry out a 3+1 decomposition. The result is a slicing of Kerr/Schwarzschild in which the black hole is propagated and Lorentz contracted. We show that our method can locate distorted apparent horizons efficiently and accurately.

gr-qc↗

arXiv1998

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.

gr-qc↗

arXiv1998

Light Propagation in inhomogeneous Universes

astro-ph↗

arXiv1998

Light Propagation in inhomogeneous Universes

Using a multi-plane lensing method that we have developed, we follow the evolution of light beams as they propagate through inhomogeneous universes. We use a P3M code to simulate the formation and evolution of large-scale structure. The resolution of the simulations is increased to sub-Megaparsec scales by using a Monte Carlo method to locate galaxies inside the computational volume according to the underlying particle distribution. The galaxies are approximated by isothermal spheres, with each morphological type having its own distribution of masses and core radii. The morphological types are chosen in order to reproduce the observed morphology-density relation. This algorithm has an effective resolution of 9 orders of magnitudes in length, from the size of superclusters down to the core radii of the smallest galaxies. We consider cold dark matter models normalized to COBE, and perform a large parameter survey by varying the cosmological parameters Omega_0, lambda_0, H_0, and n (the tilt of the primordial power spectrum). The values of n are chosen by imposing particular values or sigma_8, the rms mass fluctuation at a scale of 8/h Mpc. We use the power spectrum given by Bunn & White. This is the largest parameter survey ever done is this field.

astro-ph↗

arXiv1998

Initial Data and Coordinates for Multiple Black Hole Systems

We present here an alternative approach to data setting for spacetimes with multiple moving black holes generalizing the Kerr-Schild form for rotating or non-rotating single black holes to multiple moving holes. Because this scheme preserves the Kerr-Schild form near the holes, it selects out the behaviour of null rays near the holes, may simplify horizon tracking, and may prove useful in computational applications. For computational evolution, a discussion of coordinates (lapse function and shift vector) is given which preserves some of the properties of the single-hole Kerr-Schild form.

gr-qc↗

arXiv1997

Non-intercommuting Cosmic Strings

We perform the numerical field evolution for the collision of two Abelian type I cosmic strings. We present evidence that, for collisions at small but characteristic relative velocities and angles, these cosmic strings do not exchange ends and separate. Rather, local higher winding number bound states are formed close to the collision point, which promote multiple local scatterings at right angles and prevent intercommutation from happening. This constitutes the simplest example of the breakdown of the intercommutation rule, usually assumed in the construction of effective models for cosmic string network evolution.

hep-ph↗

arXiv1996

Analysis of ``Gauge Modes'' in Linearized Relativity

By writing the complete set of $3 + 1$ (ADM) equations for linearized waves, we are able to demonstrate the properties of the initial data and of the evolution of a wave problem set by Alcubierre and Schutz. We show that the gauge modes and constraint error modes arise in a straightforward way in the analysis, and are of a form which will be controlled in any well specified convergent computational discretization of the differential equations.

gr-qc↗

arXiv1994

Spin Dynamics of the LAGEOS Satellite in Support of a Measurement of the Earth's Gravitomagnetism

LAGEOS is an accurately-tracked, dense spherical satellite covered with 426 retroreflectors. The tracking accuracy is such as to yield a medium term (years to decades) inertial reference frame determined via relatively inexpensive observations. This frame is used as an adjunct to the more difficult and data intensive VLBI absolute frame measurements. There is a substantial secular precession of the satellite's line of nodes consistent with the classical, Newtonian precession due to the non-sphericity of the earth. Ciufolini has suggested the launch of an identical satellite (LAGEOS-3) into an orbit supplementary to that of LAGEOS-1: LAGEOS-3 would then experience an equal and opposite classical precession to that of LAGEOS-1. Besides providing a more accurate real-time measurement of the earth's length of day and polar wobble, this paired-satellite experiment would provide the first direct measurement of the general relativistic frame-dragging effect. Of the five dominant error sources in this experiment, the largest one involves surface forces on the satellite, and their consequent impact on the orbital nodal precession. The surface forces are a function of the spin dynamics of the satellite. Consequently, we undertake here a theoretical effort to model the spin ndynamics of LAGEOS. In this paper we present our preliminary results.

gr-qc↗

arXiv1994

Anisotropies of the Cosmic Background Radiation in a Reionized Universe

We trace the evolution of cosmic microwave background photons propagating through a reionized model universe. The reionization of the intergalactic medium is achieved by UV photons emitted from the decaying `hot' dark matter neutrinos. The model universe is represented by a comoving cube through which we propagate photons and subject them to the Sachs-Wolfe effect and Thomson scattering off the ionized fraction of the IGM, beginning at z=1900 until the present z=0. The simulation follows the evolving matter inhomogeneities and microwave background photons into the present nonlinear regime and yields temperature maps of the microwave sky as a comoving observer would detect. Our temperature maps display anisotropies on scales ranging from 2 to 8 degrees with rms amplitudes ranging from DT/T_{rms} =2.8 to 3.4 times 10^{-5} without reionization. If we allow the neutrinos to decay and subsequently reionize the IGM, the level of the temperature anisotropies is reduced to DT/T=0.8 to 1.1 times 10^{-5}. This includes the Doppler shifts suffered by the CMB photons as they scatter off the moving ionized fraction of the IGM. If, however, reionization is caused by a late and sudden energy input into the IGM, we find that some reduction in $ΔT/T$ occurs, but not enough to make this scenario consistent with observations. We find that the decaying neutrino hypothesis is more efficient at reducing primordial anisotropies and generates smaller Doppler anisotropies than a model reionized via a late and sudden energy input into the IGM.

astro-ph↗

arXiv1993

Inhomogeneous Inflation: Numerical Evolution

We describe our 3-dimensional numerical relativity code for the evolution of inhomogeneous cosmologies. During the evolution the constraint equations are monitored but not solved. The code has been tested against perturbation theory with good results. We present some runs of inhomogeneous inflation with strongly inhomogeneous initial data.

astro-ph↗