arXiv subjects

Gregory B. Cook

Publications and source records attributed to Gregory B. Cook.

34 records · Page 2Linked to original sources

arXiv2005

Conformal thin-sandwich puncture initial data for boosted black holes

We apply the puncture approach to conformal thin-sandwich black-hole initial data. We solve numerically the conformal thin-sandwich puncture (CTSP) equations for a single black hole with non-zero linear momentum. We show that conformally flat solutions for a boosted black hole have the same maximum gravitational radiation content as the corresponding Bowen-York solution in the conformal transverse-traceless decomposition. We find that the physical properties of these data are independent of the free slicing parameter.

gr-qc↗

arXiv2005

Excision boundary conditions for black hole initial data

We define and extensively test a set of boundary conditions that can be applied at black hole excision surfaces when the Hamiltonian and momentum constraints of general relativity are solved within the conformal thin-sandwich formalism. These boundary conditions have been designed to result in black holes that are in quasiequilibrium and are completely general in the sense that they can be applied with any conformal three-geometry and slicing condition. Furthermore, we show that they retain precisely the freedom to specify an arbitrary spin on each black hole. Interestingly, we have been unable to find a boundary condition on the lapse that can be derived from a quasiequilibrium condition. Rather, we find evidence that the lapse boundary condition is part of the initial temporal gauge choice. To test these boundary conditions, we have extensively explored the case of a single black hole and the case of a binary system of equal-mass black holes, including the computation of quasi-circular orbits and the determination of the inner-most stable circular orbit. Our tests show that the boundary conditions work well.

gr-qc↗

arXiv2003

Can a combination of the conformal thin-sandwich and puncture methods yield binary black hole solutions in quasi-equilibrium?

We consider combining two important methods for constructing quasi-equilibrium initial data for binary black holes: the conformal thin-sandwich formalism and the puncture method. The former seeks to enforce stationarity in the conformal three-metric and the latter attempts to avoid internal boundaries, like minimal surfaces or apparent horizons. We show that these two methods make partially conflicting requirements on the boundary conditions that determine the time slices. In particular, it does not seem possible to construct slices that are quasi-stationary and avoid physical singularities and simultaneously are connected by an everywhere positive lapse function, a condition which must obtain if internal boundaries are to be avoided. Some relaxation of these conflicting requirements may yield a soluble system, but some of the advantages that were sought in combining these approaches will be lost.

gr-qc↗

arXiv2002

Comparing initial-data sets for binary black holes

We compare the results of constructing binary black hole initial data with three different decompositions of the constraint equations of general relativity. For each decomposition we compute the initial data using a superposition of two Kerr-Schild black holes to fix the freely specifiable data. We find that these initial-data sets differ significantly, with the ADM energy varying by as much as 5% of the total mass. We find that all initial-data sets currently used for evolutions might contain unphysical gravitational radiation of the order of several percent of the total mass. This is comparable to the amount of gravitational-wave energy observed during the evolved collision. More astrophysically realistic initial data will require more careful choices of the freely specifiable data and boundary conditions for both the metric and extrinsic curvature. However, we find that the choice of extrinsic curvature affects the resulting data sets more strongly than the choice of conformal metric.

gr-qc↗

arXiv2001

Corotating and irrotational binary black holes in quasi-circular orbits

A complete formalism for constructing initial data representing black-hole binaries in quasi-equilibrium is developed. Radiation reaction prohibits, in general, true equilibrium binary configurations. However, when the timescale for orbital decay is much longer than the orbital period, a binary can be considered to be in quasi-equilibrium. If each black hole is assumed to be in quasi-equilibrium, then a complete set of boundary conditions for all initial data variables can be developed. These boundary conditions are applied on the apparent horizon of each black hole, and in fact force a specified surface to be an apparent horizon. A global assumption of quasi-equilibrium is also used to fix some of the freely specifiable pieces of the initial data and to uniquely fix the asymptotic boundary conditions. This formalism should allow for the construction of completely general quasi-equilibrium black hole binary initial data.

gr-qc↗

arXiv2000

Initial Data for Numerical Relativity

Initial data are the starting point for any numerical simulation. In the case of numerical relativity, Einstein's equations constrain our choices of these initial data. We will examine several of the formalisms used for specifying Cauchy initial data in the 3+1 decomposition of Einstein's equations. We will then explore how these formalisms have been used in constructing initial data for spacetimes containing black holes and neutron stars. In the topics discussed, emphasis is placed on those issues that are important for obtaining astrophysically realistic initial data for compact binary coalescence.

gr-qc↗

arXiv2000

Quasi-circular Orbits for Spinning Binary Black Holes

Using an effective potential method we examine binary black holes where the individual holes carry spin. We trace out sequences of quasi-circular orbits and locate the innermost stable circular orbit as a function of spin. At large separations, the sequences of quasi-circular orbits match well with post-Newtonian expansions, although a clear signature of the simplifying assumption of conformal flatness is seen. The position of the ISCO is found to be strongly dependent on the magnitude of the spin on each black hole. At close separations of the holes, the effective potential method breaks down. In all cases where an ISCO could be determined, we found that an apparent horizon encompassing both holes forms for separations well inside the ISCO. Nevertheless, we argue that the formation of a common horizon is still associated with the breakdown of the effective potential method.

gr-qc↗

arXiv2000

Black hole evolution by spectral methods

Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that prohibit long-term evolution. Some of these instabilities may be due to the numerical method used, traditionally finite differencing. In this paper, we explore the use of a pseudospectral collocation (PSC) method for the evolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of Einstein's equations. We demonstrate that our PSC method is able to evolve a spherically symmetric black hole spacetime forever without enforcing constraints, even if we add dynamics via a Klein-Gordon scalar field. We find that, in contrast to finite-differencing methods, black hole excision is a trivial operation using PSC applied to a hyperbolic formulation of Einstein's equations. We discuss the extension of this method to three spatial dimensions.

gr-qc↗

arXiv1998

Treating instabilities in a hyperbolic formulation of Einstein's equations

We have recently constructed a numerical code that evolves a spherically symmetric spacetime using a hyperbolic formulation of Einstein's equations. For the case of a Schwarzschild black hole, this code works well at early times, but quickly becomes inaccurate on a time scale of 10-100 M, where M is the mass of the hole. We present an analytic method that facilitates the detection of instabilities. Using this method, we identify a term in the evolution equations that leads to a rapidly-growing mode in the solution. After eliminating this term from the evolution equations by means of algebraic constraints, we can achieve free evolution for times exceeding 10000M. We discuss the implications for three-dimensional simulations.

gr-qc↗

arXiv1998

Effects of Rapid Stellar Rotation on Equation of State Constraints Derived from Quasi-Periodic Brightness Oscillations

Quasi-periodic X-ray brightness oscillations (QPOs) with frequencies around a kilohertz have now been discovered in more than a dozen neutron stars in low-mass X-ray binary systems using the Rossi X-ray Timing Explorer. There is strong evidence that the frequencies of the kilohertz oscillations are the orbital frequencies of accreting gas in nearly circular orbits around these stars. Some stars that produce kilohertz QPOs may have spin frequencies greater than 400 Hertz. For spin rates this high, first-order analytic treatments of the effects of the star's rotation on its structure and the spacetime are inaccurate. Here we use the results of a large number of fully relativistic, self-consistent numerical calculations of the stellar structure of rapidly rotating neutron stars and the interior and exterior spacetime to investigate the constraints on the properties of such stars that can be derived if stable circular orbits of various frequencies are observed. We have computed the equatorial radius of the star, the radius of the innermost stable circular orbit, and the frequency of the highest-frequency stable circular orbit as functions of the stellar spin rate, for spin rates up to the maximum possible and for several illustrative equations of state. Our calculations show that the upper bounds on the stiffness of neutron star matter implied by a given orbital frequency are typically significantly stricter for stars with spin frequencies more than 400 Hertz than for slowly rotating stars.

astro-ph↗

arXiv1997

Numerical Evolution of Black Holes with a Hyperbolic Formulation of General Relativity

We describe a numerical code that solves Einstein's equations for a Schwarzschild black hole in spherical symmetry, using a hyperbolic formulation introduced by Choquet-Bruhat and York. This is the first time this formulation has been used to evolve a numerical spacetime containing a black hole. We excise the hole from the computational grid in order to avoid the central singularity. We describe in detail a causal differencing method that should allow one to stably evolve a hyperbolic system of equations in three spatial dimensions with an arbitrary shift vector, to second-order accuracy in both space and time. We demonstrate the success of this method in the spherically symmetric case.

gr-qc↗

arXiv1997

Binary Neutron Stars in Quasi-Equilibrium Circular Orbit: A Fully Relativistic Treatment

We present a numerical scheme that solves the initial value problem in full general relativity for a binary neutron star in quasi-equilibrium. While Newtonian gravity allows for a strict equilibrium, a relativistic binary system emits gravitational radiation, causing the system to lose energy and slowly spiral inwards. However, since inspiral occurs on a time scale much longer than the orbital period, we can adopt a quasi-equilibrium approximation. In this approximation, we integrate a subset of the Einstein equations coupled to the equations of relativistic hydrodynamics to solve the initial value problem for binaries of arbitrary separation, down to the innermost stable orbit.

gr-qc↗

arXiv1996

Implementing an apparent-horizon finder in three dimensions

Locating apparent horizons is not only important for a complete understanding of numerically generated spacetimes, but it may also be a crucial component of the technique for evolving black-hole spacetimes accurately. A scheme proposed by Libson et al., based on expanding the location of the apparent horizon in terms of symmetric trace-free tensors, seems very promising for use with three-dimensional numerical data sets. In this paper, we generalize this scheme and perform a number of code tests to fully calibrate its behavior in black-hole spacetimes similar to those we expect to encounter in solving the binary black-hole coalescence problem. An important aspect of the generalization is that we can compute the symmetric trace-free tensor expansion to any order. This enables us to determine how far we must carry the expansion to achieve results of a desired accuracy. To accomplish this generalization, we describe a new and very convenient set of recurrence relations which apply to symmetric trace-free tensors.

gr-qc↗

arXiv1995

Testing a Simplified Version of Einstein's Equations for Numerical Relativity

Solving dynamical problems in general relativity requires the full machinery of numerical relativity. Wilson has proposed a simpler but approximate scheme for systems near equilibrium, like binary neutron stars. We test the scheme on isolated, rapidly rotating, relativistic stars. Since these objects are in equilibrium, it is crucial that the approximation work well if we are to believe its predictions for more complicated systems like binaries. Our results are very encouraging.

gr-qc↗

arXiv1994

Collisions of boosted black holes: perturbation theory prediction of gravitational radiation

We consider general relativistic Cauchy data representing two nonspinning, equal-mass black holes boosted toward each other. When the black holes are close enough to each other and their momentum is sufficiently high, an encompassing apparent horizon is present so the system can be viewed as a single, perturbed black hole. We employ gauge-invariant perturbation theory, and integrate the Zerilli equation to analyze these time-asymmetric data sets and compute gravitational wave forms and emitted energies. When coupled with a simple Newtonian analysis of the infall trajectory, we find striking agreement between the perturbation calculation of emitted energies and the results of fully general relativistic numerical simulations of time-symmetric initial data.

gr-qc↗

arXiv1994

Three-dimensional initial data for the collision of two black holes II: Quasi-circular orbits for equal-mass black holes

The construction of initial-data sets representing binary black-hole configurations in quasi-circular orbits is studied in the context of the conformal-imaging formalism. An effective-potential approach for locating quasi-circular orbits is outlined for the general case of two holes of arbitrary size and with arbitrary spins. Such orbits are explicitly determined for the case of two equal-sized nonrotating holes, and the innermost stable quasi-circular orbit is located. The characteristics of this innermost orbit are compared to previous estimates for it, and the entire sequence of quasi-circular orbits is compared to results from the post-Newtonian approximation. Some aspects of the numerical evolution of such data sets are explored.

gr-qc↗