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Abstracts of New Frontiers in Numerical Relativity

Abstract list is subject to change without prior notice. Current state is as of 07/23/2006.

M. Ansorg

Initial Data | Monday, July 17 | 17:45 - 18:30 | Albert-Einstein-Institut

Multi-Domain Spectral Method for Initial Data of Arbitrary Binaries in General Relativity

We present a multi-domain spectral method to compute initial data of binary systems in General Relativity. By utilizing adapted conformal coordinates, the vacuum region exterior to the gravitational sources is divided up into two subdomains within which the spectral expansion of the field quantities is carried out. If a component of the binary is a neutron star, a further subdomain covering the star's interior is added. As such, the method can be used to construct arbitrary initial data corresponding to binary black holes, binary neutron stars or mixed binaries. In particular, it is possible to describe a black hole component by the “puncture” ansatz as well as through an excision technique. First examples are given for binary black hole excision data that fulfill the requirements of the quasi-stationary framework, which combines the thin sandwich formulation of the constraint equations with the isolated horizon conditions for black holes in quasi-equilibrium. These numerical solutions were obtained to extremely high accuracy with moderate computational effort. Moreover, the method proves to be applicable even when tending toward limiting cases such as large mass ratios of the binary components.

L. Baiotti

Fluids I | Wednesday, July 19 | 09:45 - 10:30 | Albert-Einstein-Institut

Challenging the Paradigm of Singularity Excision

A paradigm deeply rooted in modern numerical relativity calculations prescribes the removal of those regions of the computational domain where a physical singularity may develop. We here challenge this paradigm by performing three-dimensional simulations of the collapse of uniformly rotating stars to black holes without excision. We show that this choice, combined with suitable gauge conditions and the use of minute numerical dissipation, improves dramatically the long-term stability of the olutions. In turn, this allows for the calculation of the waveforms well beyond what previously possible, providing information on the black-hole quasi-normal-mode ringing and setting a new mark on the present knowledge of the gravitational-wave emission from the stellar collapse to a rotating black hole.

J. Baker

Evolutions I | Tuesday, July 18 | 09:45 - 10:30 | Albert-Einstein-Institut

Techniques for Binary Black Hole Simulations

Recent advances in techniques for numerical simulation of black hole systems have enabled dramatic progress in astrophysical applications. Our approach to these simulations, which includes new gauge conditions for moving punctures, AMR, and specific tools for analyzing black hole simulations, has been applied to a variety of black hole configurations, typically resulting in simulations lasting several orbits. I will discuss these techniques, what we've learned in applications, and outline some areas for further development.

N. Bishop

Multiblocks | Thursday, July 20 | 17:15 - 17:45 | Albert-Einstein-Institut

Using Analytic Solutions of the Linearized Einstein Equations to Test a Characteristic Code

The characteristic, or Bondi-Sachs, approach to numerical relativity has many positive features, but it has not succeeded in computing the gravitational radiation emitted in a realistic astrophysical scenario. A new class of analytic solutions to the linearized Einstein equations for the Bondi-Sachs metric has been obtained. These solutions radiate indefinitely, and provide a more appropriate testbed than existing solutions that rapidly decay towards a static spacetime. The solutions are used to investigate the accuracy of different versions of the characteristic code, using (a) stereographic coordinates, and (b) six angular patches, with the aim of finding conditions under which the metric and gravitational radiation can be reliably computed.

C. Bona

Formulations | Monday, July 17 | 09:00 - 09:45 | Albert-Einstein-Institut

CSI: Black Holes - Why Do Codes Crash?

There must be a reason for code crashing even when using strongly-hyperbolic first-order formalisms (or their second-order counterparts) and stable (standard) discretization algorithms. The most likely causes of crash are investigated, focusing on singularity avoidance, slice stretching and long-term instabilities linked to current gauge choices. Some suggestions are made in order either to prevent or at least to circumvent these difficulties, that are inherent to Black-Hole simulations.

S. Bonazzola

Fluids I | Wednesday, July 19 | 11:00 - 11:45 | Albert-Einstein-Institut

Magnetohydrodynamics in Strange Stars

An MHD code based on spectral methods is presented in order to study the macroscopic behavior of rotating compact objects in quasi-equilibrium configurations. The following problems can be studied: 1) Equilibrium configurations of the magnetic field (Gravitational wave emission) 2) Stability of magnetic configurations and stabilizing mechanisms 3) Slow evolution of the magnetic field (Magnetars) 4) Fast evolution of the magnetic field (Flares on magnetars). An application to the model of gamma-ray bursts is also presented.

B. Brügmann

Evolutions II | Tuesday, July 18 | 14:30 - 15:15 | Albert-Einstein-Institut

Orbiting Black Holes and Moving Punctures

Numerical simulations of black hole binaries have recently become possible using the moving puncture method. We critically evaluate the validity and the potential of that approach and discuss results on wave extraction for two orbiting punctures shortly before the merger.

L. Buchman

Boundaries | Friday, July 21 | 11:00 - 11:45 | Albert-Einstein-Institut

Absorbing Boundary Conditions in Linearized Gravity

We construct exact solutions to the Bianchi equations on a flat spacetime background. These solutions, which are closely related to the Teukolsky waves, represent in- and outgoing linearized gravitational radiation. We then consider the Bianchi equations on a subset of flat spacetime of the form [0,T] x B_R, where B_R is a ball of radius R, and analyze different kinds of boundary conditions. Our main results are: (1) We give an explicit analytic example that shows that boundary conditions obtained from freezing the incoming characteristic fields to their initial values is not compatible with the constraints. (2) With the help of the exact solutions constructed, we determine the amount of artificial reflection introduced by imposing constraint-preserving boundary conditions that freeze the Weyl scalar Psi_0. (3) For each L greater or equal to 2, we construct local boundary conditions that are perfectly absorbing for linearized waves with angular momentum number l=L. Implications of these results for numerical relativity on finite domains are also discussed.

P. Csizmadia

Multiblocks | Thursday, July 20 | 17:45 - 18:15 | Albert-Einstein-Institut

Testing a High Precision Mesh Refinement Code in the Evolution of Massive Spherically Symmetric Fields

Numerical evolution of spherically symmetric fields is presented using a new adaptive mesh refinement (AMR) code with fourth order discretization in space and time, along with compactification in space. The non-interacting massive Klein-Gordon field along with other, non-linear systems are investigated, including various SU(2) Yang-Mills-Higgs systems. A common property of these dynamical systems is that initial disturbances are radiated away in the form of expanding shells built up of higher frequency oscillations. The frequency of these oscillations grows in time, hence the proper numerical description requires AMR.

P. Diener

Multiblocks | Thursday, July 20 | 14:30 - 15:15 | Albert-Einstein-Institut

Time Evolution with Multiple Grid Blocks

I will describe the ingredients of a multi-block infrastructure for time evolution in numerical relativity. These ingredients include high-order summation by parts (SBP) finite differencing operators and compatible dissipation operators, the use of penalty methods for the inter-block boundaries and adaptive time stepping. The interplay between these techniques will be illustrated with a simple example and the generalization to more complicated scenarios will be discussed.

H. Dimmelmeier

Fluids II | Tuesday, July 20 | 11:30 - 12:00 | Albert-Einstein-Institut

Rotating Collapse of Realistic Stellar Iron Cores in General Relativity - I. The Emergence of a Generic Gravitational Wave Signal from Core Bounce

We present the first-ever 3D simulations of the rotating collapse of realistic stellar cores in general relativity employing a finite-temperature equation of state and an approximate treatment of deleptonization. Part I: We show results for the gravitational-wave emission from the rotating collapse, core bounce, and early postbounce phases. They indicate that the waveform signature of these phases is much more generic than previously predicted by studies in Newtonian gravity and/or using a simple equation of state. In particular, we do not observe the collapse type characterized by multiple centrifugal bounces. This leads to a very narrow spread of the gravitational wave signal in frequency space.

J.A. Font

Fluids II | Thursday, July 20 | 09:45 - 10:30 | Albert-Einstein-Institut

Towards Relativistic Magnetized Core Collapse Simulations

In this talk, recent axisymmetric simulations of relativistic magneto-rotational core collapse are discussed and compared with existing work in the framework of Newtonian physics. Our simulations, performed adopting the magnetic field test approximation (no back-reaction on the dynamics), allow us to measure the effects of magnetic fields on the collapse dynamics and on the gravitational radiation waveforms. Amplification mechanisms for the magnetic field (radial compression, Ω-dynamo, and magneto-rotational instability) are discussed, as well as the long-term evolution of the magnetic field in proto-neutron stars.

D. Garfinkle

Evolutions II | Tuesday, July 18 | 16:45 - 17:30 | Albert-Einstein-Institut

Numerical Simulations of General Gravitational Singularities

Numerical simulations are presented of generic singularities formed in gravitational collapse. The results support the BKL conjecture: spatial derivatives become negligible compared to time derivatives; and the dynamics at each spatial point is oscillatory in the vacuum case and non-oscillatory in the case of scalar field matter.

B. Giacomazzo

Fluids I | Wednesday, July 19 | 12:30 - 13:00 | Albert-Einstein-Institut

WhiskyMHD: A New Numerical Code for General Relativistic Magnetohydrodynamics

To study astrophysical scenarios involving compact objects and magnetic fields, such as the collapse of rotating magnetized stars to black holes, it is necessary to solve Einstein equations together with those of general relativistic MHD. I will present a new numerical code developed to solve the full set of GRMHD equations which is based on well known techniques, such as approximate Riemann Solvers and excision methods, already successfully used in General Relativistic Hydrodynamics codes like Whisky. I will also discuss about the applications of this code to the study of the collapse of magnetized neutron stars.

E. Gourgoulhon

Boundaries | Friday, July 21 | 09:45 - 10:30 | Albert-Einstein-Institut

Trapping Horizons as Inner Boundary Conditions for Black Hole Spacetimes

A new paradigm is emerging for the mathematical representation of black holes: that of trapping horizons (and the related concepts of dynamical and isolated horizons). This provides a local characterization of black holes, contrary to the standard paradigm, which is based on the notion of event horizon - a highly non-local concept. In this talk, we shall review the applications of trapping horizons to numerical relativity, especially in the treatment of black hole spacetimes via the excision technique.

J.M. Martín-García

Formulations | Monday, July 17 | 11:45 - 12:30 | Albert-Einstein-Institut

Formulations of the Einstein Equations, Live Gauge, and Well-Posedness

We show that the BSSN, NOR and KST formulations are closely related. We review hyperbolic, parabolic and elliptic gauge conditions in a 3+1 framework, and give some well-posedness results for the Einstein equations in these gauges.

M. Hannam

Initial Data | Monday, July 17 | 15:15 - 16:00 | Albert-Einstein-Institut

Beyond Bowen-York Puncture Data for Spinning Black Holes

It is well-known that Bowen-York initial data contain spurious radiation. Although this “junk” radiation has been seen to be small for non spinning black-hole binaries in circular orbit, its magnitude increases when the black holes are given spin. I will discuss this problem, and suggest a possible improvement in the spinning case.

I. Hawke

Fluids II | Thursday, July 20 | 11:00 - 11:30 | Albert-Einstein-Institut

Interfaces and Surfaces in Numerical Relativity

In the future our models of astrophysical objects such as neutron stars will become much more complex. Accurate treatment of multiple fluids, elastic matter and the transition from a magnetized “fluid” to Einstein-Maxwell without fluid are all likely to be important. Here we will consider how such interfaces can be modelled and in particular look at effects already present in current simulations, by considering numerical treatments of the surface of neutron stars.

S. Hawley

Initial Data | Monday, July 17 | 14:30 - 15:15 | Albert-Einstein-Institut

Spin Dependence in Computational Black Hole Initial Data

We have implemented a parallel multigrid solver, to solve the initial data problem for 3+1 General Relativity. This involves solution of elliptic equations derived from the Hamiltonian and the momentum constraints. We use the conformal transverse-traceless method of York and collaborators which consists of a conformal decomposition with a scalar φ that adjusts the metric, and a vector potential w^i that adjusts the longitudinal components of the extrinsic curvature. The constraint equations are then solved for these quantities φ, w^i such that the complete solution fully satisfies the constraints. We apply this technique to compare with theoretical expectations for the spin orientation- and separation-dependence in the case of spinning interacting (but not orbiting) black holes. We write out a formula for the effect of the spin-spin interaction which includes a result of Wald as well as additional effect due to the rotation of the mass quadrupole moment of a spinning black hole. A subset of these spin-spin effects are confirmed via our numerical calculations, however due to computer time limitations the full parameter space has not yet been surveyed and confirmed. New improvements to the code, such as parallel FMR and a Cactus (Carpet) interface, should be in place by the time of the conference, and allow us to say more about both nonlinear and asymptotic spin-spin effects than was previously possible.

F. Herrmann

Evolutions II | Tuesday, July 18 | 15:15 - 15:45 | Albert-Einstein-Institut

Binary Black Hole Evolutions

We report on recent studies of close-separated binary black hole systems with unequal masses. We compute energies and angular momenta radiated as well as kick velocities.

L. Kidder

Multiblocks | Thursday, July 20 | 15:15 - 16:00 | Albert-Einstein-Institut

Evolving Black Hole Spacetimes with Spectral Methods

I will describe in detail the multidomain pseudospectral collocation methods that we use to evolve spacetimes with black holes. I will discuss recent results simulating binary systems of compact objects.

L. Lindblom

Formulations | Monday, July 17 | 09:45 - 10:30 | Albert-Einstein-Institut

Solving Einstein's Equation With Generalized Harmonic Gauge Conditions

A first order symmetric hyperbolic representation of the Einstein evolution system will be presented that uses the generalized harmonic method of specifying the coordinates. This new system has constraint satisfying solutions that are identical to those of the Einstein system, and includes terms that dynamically suppress the growth of small constraint violations. Simple numerical tests of this system will be presented that demonstrate its constraint suppressing qualities.

C. Lousto

Evolutions I | Tuesday, July 18 | 09:00 - 09:45 | Albert-Einstein-Institut

Gravitational Radiation from Spinning-Black-Hole Binaries

We study the dynamics of spinning-black-hole binaries by numerically solving the full nonlinear field equations of General Relativity. We compute trajectories, merger times, and radiation waveforms. We find that the last stages of the orbital motion of black-hole binaries are profoundly affected by the individual spins. In order to cleanly display its effects, we consider two equal mass holes with individual spin parameters S/m2=0.75, both aligned and anti-aligned with the orbital angular momentum. We choose initial data corresponding to quasicircular orbits with a period of 125 M for both cases. The computed merger time for the aligned spin case is approximately 225 M, performing nearly three orbits before merger, while for the anti-aligned case the merger time is approximately 105 M, performing just less than one orbit before merger. The total energy radiated for the former case is approximately 6% while for the latter it is only approximately 2%. The final Kerr hole remnants have rotation parameters a/M=0.9 and a/M=0.44 respectively, showing the difficulty of creating a maximally rotating black hole out of the merger of two spinning holes.

G. Manca

Fluids I | Wednesday, July 19 | 11:45 - 12:30 | Albert-Einstein-Institut

Dynamical Bar-Mode instability in General Relativity

We present new results for the critical value of the parameter β=T/W for the onset of the instability and on mode dynamics of the dynamical bar-mode instability of differentially rotating compact stars in full general relativity. We present new estimations of the thresholds for the onset of the bar-mode instability obtained as the extrapolation of the value of instability parameter β for which there is an infinite growth-time of the (m=2)-mode in series of constant “baryonic” mass stars. Moreover, we discuss the importance of mode coupling and of the compactness of the star for the evolution of the bar-mode.

P. Marronetti

Evolutions I | Tuesday, July 18 | 12:30 - 13:00 | Albert-Einstein-Institut

New Techniques for the Evolution of Compact-Object Binaries

The past year has seen the introduction of new algorithms in numerical relativity that made possible the stable evolution of black hole binaries in circular orbits. Our group at Florida Atlantic University, in collaboration with groups at University of Jena and The University of Texas at Brownsville, is currently working on such simulations using the technique known as “moving punctures” that permits the evolution of black hole spacetimes without using grid excision. I will review some of our latest results and also comment on some of the other projects in numerical astrophysics being developed at FAU.

R. Matzner

Initial Data | Monday, July 17 | 14:30 - 15:15 | Albert-Einstein-Institut

Black Holes: Data and Methods

I will discuss the physical content of some data setting methods for binary black hole spacetimes.

Y. Mizuno

Fluids II | Thursday, July 20 | 09:00 - 09:45 | Albert-Einstein-Institut

Newly-Developed 3D GRMHD Code and its Application to Jet Formation

We have developed a new three-dimensional general Relativistic magnetohydrodynamic code by using a conservative, high-resolution shock-capturing scheme. The numerical fluxes are calculated using the HLL approximate Riemann solver scheme. The flux-interpolated constrained transport scheme is used to maintain a divergence-free magnetic field. We have performed various 1-dimensional test problems in both special and general relativity by using several reconstruction methods and found that the new 3D GRMHD code shows substantial improvements over our previous model. The preliminary results show the jet formations from a geometrically thin accretion disk near a non-rotating and a rotating black hole. We will discuss the jet properties depended on the rotation of a black hole and the magnetic field strength.

A. Nagar

Boundaries | Friday, July 21 | 12:30 - 13:00 | Albert-Einstein-Institut

Transition from Inspiral to Plunge in Binary Black Hole Systems in the Extreme Mass Ratio

Using black hole perturbation theory and effective-one-body ideas for the radiation-reaction force, we completely determine the gravitational wave signature for the transition inspiral-plunge-ringdown for a binary (non-spinning) black hole system in the extreme mass ratio. We discuss the features of gravitational waveforms up to l=4 multipole, energy, angular momentum and linear momentum losses. We focus on the mechanism of excitation of the black hole quasi-normal modes and propose analytical methods to well reproduce the smooth transition between the inspiral and the ringdown phase of the gravitational wave signal.

S. Noble

Fluids II | Thursday, July 20 | 12:30 - 13:00 | Albert-Einstein-Institut

Imaging Accretion Disk Evolutions

Very Long Baseline Interferometry (VLBI) at sub-millimeter wavelengths shows promise at resolving the shadow cast by the supermassive black hole at the Galactic Center, Sgr A*, in the near future. In order to accurately test theoretical models of Sgr A* using these observations, a direct comparison of VLBI data to numerical models must be made. We will present our progress at modeling sub-millimeter images of Sgr A* using accretion disk simulation data from our fixed background GRMHD code called HARM. The particular disk evolutions imaged are chosen so as to model the advection-dominated thick disk thought to exist in the innermost regions around the central black hole. Synchrotron and bremsstrahlung radiation are considered in the optically thin limit, which allows us to solve the radiative transfer equations using only simulation and geodesic data in a post-processing step. The GRMHD simulations employ a new grid structure that simultaneously resolves the small spatial scales of the equatorial inflow and the large scale outflows ejected along the spin axis. Other new improvements to HARM will also be discussed.

J. Novak

Formulations | Monday, July 17 | 11:00 - 11:45 | Albert-Einstein-Institut

Solution of the Gravitational Wave Tensor Equation Using Spectral Methods

Far from the sources, the propagation of gravitational waves can be described by a wave equation for a transverse and traceless symmetric 3-tensor. A numerical scheme is presented, based on spectral methods, which is well adapted to solve this problem, keeping the transversality property during the evolution. In particular it is shown how it is possible to use spherical components of the tensor, together with spectral methods in spherical coordinates, transforming the 3D problem to a simple system of ordinary differential equations in term of the radia coordinate. The evolution of 3D gravitational wave spacetime is presented as a numerical example and a discussion of robust outgoing-wave conditions is given.

C. Ott

Fluids II | Thursday, July 20 | 12:00 - 12:30 | Albert-Einstein-Institut

Rotating Collapse of Realistic Stellar Iron Cores in General Relativity - II. 3D Full GR Results

We present the first-ever 2D and 3D simulations of the rotating collapse of realistic stellar cores in general relativity employing a finite-temperature equation of state and an approximate treatment of deleptonization. Part II: (a) We compare fully nonlinear and conformally-flat spacetime evolution methods and find that the conformally-flat treatment is sufficiently accurate for the core-collapse supernova problem. (b) We study the formation of postbounce 3D structure in our simulations and find that a number of models are unstable to a so-called low-T/|W| rotational instability with dominant m=1 character that leads to prolonged narrow-band gravitational wave emission from its quadrupole daughter mode.

C. Palenzuela

Formulations | Monday, July 17 | 12:30 - 13:00 | Albert-Einstein-Institut

Dealing with Constraint Violations in First Order Evolution Systems

It is described a simple way to convert, withouth introducing new constraint violations, a second order system (in space) into a first order system, like the BSSN or the Z4 formalism of the Einstein Equations. The procedure is described for the last evolution system and tested numerically in different scenarios.

H. Pfeiffer

Initial Data | Monday, July 17 | 17:00 - 17:45 | Albert-Einstein-Institut

Circular Orbits and Spin in Binary Black Hole Initial Data

The construction of initial data for black-hole binaries usually involves the choice of free parameters that define the spin and the orbital frequency of the binary. We discuss and compare two criteria to determine the orbital frequency, the Komar-mass ansatz and the effective potential method, and find excellent agreement. Furthermore, we implement quasi-local spin measures, and use these to construct a sequence of truly non-spinning black-hole binaries in circular orbits.

D. Pollney

Evolutions II | Tuesday, July 18 | 15:45 - 16:15 | Albert-Einstein-Institut

Binary Black Hole evolutions using BSSN

In the past year, black hole evolutions in numerical relativity have made significant strides forward in terms of the ability to evolve binaries for periods of more than one orbit. The use of mesh refinement techniques have finally allowed us to apply sufficient resolution to the problem. In addition, the previous commonly applied paradigm of co-rotating grids and excision techniques, have been replaced by moving punctures. While this has resulted in unprecedented evolution timescales, it is important to critically examine these new techniques and in particular their consistency with alternate methods. This talk describes recent results obtained with the AEI-LSU BSSN evolution code, demonstrating orbital evolutions which can be compared with previously applied fixed-BH excision methods. In the case of moving punctures, gravitational waveforms can be extracted using both Weyl scalars and Zerilli-based techniques, and show good consistency in the waveform and radiated energies. We present initial comparisons between co-rotating plus excision with current moving-puncture evolutions, and find good consistency at given resolutions, particularly at early times.

F. Pretorius

Evolutions I | Tuesday, July 18 | 11:00 - 11:30 | Albert-Einstein-Institut

Binary Black Hole Coalescence

I will present an update on the ongoing effort to simulate the merger of binary black hole systems using a technique based on generalized harmonic coordinates. Topics covered may include evolution of Cook-Pfeiffer quasi circular inspiral datasets, binaries constructed via scalar field collapse and comparisons of the resultant waveforms to those calculated with perturbative techniques.

O. Rinne

Boundaries | Friday, July 21 | 09:00 - 09:45 | Albert-Einstein-Institut

Well-Posed Radiation-Controlling Boundary Conditions for the Harmonic Einstein Equations

The initial-boundary value problem for the Einstein equations in a first-order generalized harmonic formulation is analyzed. We impose boundary conditions that preserve the constraints and control the incoming gravitational radiation. High-frequency perturbations about any given spacetime (with arbitrary subluminal shift vector at the boundary) are analyzed using the Fourier-Laplace technique. We also show that our system does not admit weak instabilities with polynomial time dependence. Numerical tests support our claim that the initial-boundary value problem is most likely to be well-posed.

O. Sarbach

Boundaries | Friday, July 21 | 11:45 - 12:30 | Albert-Einstein-Institut

Outer Boundary Conditions

I will review some recent progress on the initial-boundary value formulation of General Relativity, the construction of absorbing boundary conditions and wave extraction.

M. Scheel

Evolutions I | Tuesday, July 18 | 11:30 - 12:00 | Albert-Einstein-Institut

Binary Black Hole Evolutions Using Spectral Methods

I discuss recent progress made by the Caltech-Cornell group on the evolution of binary black holes. I describe results from binary inspiral simulations that achieve high accuracy by the use of multidomain spectral methods.

E. Schnetter

Evolutions II | Tuesday, July 18 | 18:00 - 18:30 | Albert-Einstein-Institut

Instant Excision

We present a technique for dealing with singularities during the numerical simulation of black hole spacetimes. Simply stated, the singular initial data is replaced by some appropriate smooth function in some small region within the horizon, after which the spacetime is evolved over the entire grid without imposing an excision boundary condition, and without any special treatment of this region. The replacement function is not necessarily physical or constraint preserving. This approach is very similar in spirit though more general than the recently published “moving punctures”. We demonstrate through a number of examples, including single Kerr-Schild black holes, punctures, and quasi-circular binary black holes, that errors due to the non-physical interior do not propagate to the exterior grid, so that physical measures such as horizon mass and spin are not affected by the interior black hole evolution. The method is at least not worse than the commonly used extrapolation and “simple excision” boundary conditions.

E. Snajdr

Evolutions II | Tuesday, July 18 | 17:30 - 18:00 | Albert-Einstein-Institut

Critical Phenomena and Naked Singularity Formation in a Gravitational Collapse of an Ultrarelativistic Fluid

I will discuss the results of a study of a spherically Symmetric Gravitational collapse of an ultrarelativistic fluid (P = k * rho) in the limit of k -> 0. The critical exponents are found both semi-analytically (using a self-similar ansatz) and from numerical calculations. The limiting value (k -> 0) is calculated. We also tested the hypothesis of Harada and Maeda that naked singularities are formed in a generic gravitational collapse of ultrarelativistic fluids with k < 0.01. Our calculations confirm that in this regime black hole is not an attractor for the supercritical colla pse, instead, the solution converges to a general relativistic Larson-Penston solution (GRLP). It is known that GRLP contains a naked singularity for k < 0.01.

B. Stephens

Fluids I | Wednesday, July 19 | 09:00 - 09:45 | Albert-Einstein-Institut

Magnetohydrodynamics in Numerical Relativity: Collapse and Black Hole Formation in Magnetized, Differentially Rotating Neutron Stars

The capacity to model magnetohydrodynamical (MHD) flows in dynamical, strongly curved spacetimes significantly extends the reach of numerical relativity in addressing many problems at the forefront of theoretical astrophysics. We have developed an evolution code for the coupled Einstein-Maxwell-MHD equations which combines a BSSN solver with a simple (yet accurate) high resolution shock capturing scheme due to Harten, Lax, and van Leer (HLL). We have subjected this code to an extensive suite of test problems, including magnetized shock tube tests, magnetized spherical accretion, and the excitation of MHD waves in a plasma by linearized gravitational waves. As one application, we evolve magnetized, differentially rotating neutron stars under the influence of a small seed magnetic field. Of particular significance is the behavior found for hypermassive neutron stars (HMNSs), which have rest masses greater the mass limit for uniform rotation for a given equation of state. The remnant of a binary neutron star following merger is likely to be a HMNS. We find that magnetic braking and the magnetorotational instability lead to the collapse of HMNSs and the formation of rotating black holes surrounded by massive, hot accretion tori and collimated magnetic field lines. Such tori radiate strongly in neutrinos, and the resulting neutrino-antineutrino annihilation (possibly in concert with energy extraction by MHD effects) could provide enough energy to power short-hard gamma-ray bursts. To explore the range of outcomes, we also evolve differentially rotating neutron stars with lower masses and angular momenta than the HMNS models. Instead of collapsing, the non-hypermassive models form roughly uniformly rotating central objects which, in cases with significant angular momentum, are surrounded by massive tori.

B. Szilágyi

Evolutions I | Tuesday, July 18 | 12:00 - 12:30 | Albert-Einstein-Institut

An Overview of the AEI Harmonic Code

The talk will present the formulation, numerical algorithm and some results of the AEI Harmonic code. The effort involves D. Pollney, L. Rezzolla, J. Thornburg and myself. Given the existence of other numerical relativity codes using a similar set of equations, the talk will relate our work to other similar efforts by highlighting differences (and similarities) with those. In addition, I also intend to compare the AEI Harmonic and BSSN codes for testbeds such as an isolated Kerr BH space-time or the head-on collision problem.

W. Tichy

Initial Data | Monday, July 17 | 16:30 - 17:00 | Albert-Einstein-Institut

Approximate Binary Black Hole Initial Data from Matched Asymptotic Expansions

We construct approximate initial data for non-spinning black hole binary systems by asymptotically matching the 4-metrics of two tidally perturbed Schwarzschild solutions in isotropic coordinates to a resummed post-Newtonian 4-metric in ADMTT coordinates. Since isotropic coordinates are very similar to ADMTT coordinates, matching yields better results than in the previous calculations. In particular, matching occurs not only in the buffer zone, but due to the similarity between ADMTT and isotropic coordinates the two metrics are also close to each other near the black hole horizons. With the help of a transition function we also obtain a global smooth 4-metric which has errors on the order of the error introduced by the more accurate of the two approximations we match. We present explicit results for the 3-metric, extrinsic curvature, lapse, and shift.

J. Thornburg

Multiblocks | Thursday, July 20 | 16:30 - 17:15 | Albert-Einstein-Institut

Multi-Patch Finite Difference Evolutions of Dynamic Black Hole Spacetimes

When using black hole excision to numerically evolve a black hole spacetime with no continuous symmetries, most 3+1 finite differencing codes use a Cartesian grid. It's difficult to do excision on such a grid, because the natural r=constant excision surface must be approximated either by a very different shape such as a contained cube, or by an irregular and non-smooth “LEGO™ sphere” which may introduce numerical instabilities into the evolution. In this paper I describe an alternate scheme, which uses multiple r*(angular coordinates) grid patches, each patch using a different (nonsingular) choice of angular coordinates. This allows excision on a smooth r=constant 2-sphere. I discuss the key design choices in such a multiple-patch scheme, including the number and shape of the patches (I use a 6-patch “inflated-cube” scheme), the way in which the patch ghost zones are “synchronized” by interpolation from neighboring patches, the tensor basis for the Einstein equations in each patch, and the handling of non-tensor field variables such as the BSSN Gamma^i tilde (I use a scheme which requires ghost zones which are twice as wide for the BSSN conformal factor φ as for Gamma^i tilde and the other BSSN field variables). Finally, I present sample numerical results from a prototype implementation of this scheme. This code simulates the time evolution of the (asymptotically flat) spacetime around a single (excised) black hole.

J. York

Public lecture | Thursday, July 20 | 19:00 - 21:00 | Schloßtheater im Neuen Palais

Dynamical Principles of General Relativity

General Relativity will be viewed as a dynamical theory. We will begin with the Action Principle in Lagrangian and in Hamiltonian forms. From these principles, we will take a leisurely stroll through variation of the action in which we obtain the equations of motion and the boundary terms. We will pay particular attention to the interpretation of the boundary terms, the constraint equations, and the Bianchi identities. This overall viewpoint gave impetus, I believe, to many of the programs of numerical relativity discussed in this meeting. A small interesting problem in quantum gravity will cap the talk.

Note: The public lecture will not take place at the conference location but at  Schloßtheater im Neuen Palais, Am Neuen Palais, D-14471 Potsdam. The lecture will be followed by a reception. A shuttle service will bring the participants to the lecture hall and then back to their hotels.