Binary Black Hole Problem Research Articles

Binary black hole simulation with an adaptive finite element method: Solving the Einstein constraint equations

In this paper we start a systematic investigation of applying an adaptive finite element method to the Einstein equations, especially binary compact object simulations. To our knowledge, this is the first study on this topic. Puncture type initial data are solved with the adaptive finite element method. The numerical scheme proposed in the current work can be straightforwardly extended to the general type of initial data of the Einstein equations. The Parallel Hierarchical Grid library and the existing numerical relativity code AMSS-NCKU are used to develop the adaptive finite element Einstein solver. In the unsmooth toy model problem, the adaptive mesh refinement operation can catch the unsmooth region efficiently. The numerical solution deviates the exact solution by an error less than $1{0}^{\ensuremath{-}5}$. In the binary black hole problem, our solution is consistent with the one gotten by the TwoPuncture code which uses a pseudospectral method. As we expected, the solution gotten by the finite element method is less accurate than that gotten by the spectral method. But the relative error distributes almost uniformly. The adaptive mesh refinement method is quite efficient and it does not waste computational effort. Our finite element code is more flexible than the TwoPuncture code. It can be used to treat other general initial data problems such as the three black holes problem, besides the binary black hole problem. We test one typical three black holes problem also. In all of the test cases, our adaptive finite element code works quite well.

The overlap of numerical relativity, perturbation theory and post-Newtonian theory in the binary black hole problem

Inspiralling and coalescing binary black holes are promising sources of gravitational radiation. The orbital motion and gravitational-wave emission of such system can be modeled using a variety of approximation schemes and numerical methods in general relativity: The post-Newtonian (PN) formalism, black hole perturbation theory (BHP), numerical relativity (NR) simulations and the effective one-body (EOB) model. We review recent work at the multiple interfaces of these analytical and numerical techniques, emphasizing the use of coordinate-invariant relationships to perform meaningful comparisons. Such comparisons provide independent checks of the validity of the various calculations, they inform the development of a universal, semi-analytical model of the binary dynamics and gravitational-wave emission and they help to delineate the respective domains of validity of each approximation method. For instance, several recent comparisons suggest that perturbation theory may find applications in a broader range of physical problems than previously thought, including the radiative inspiral of intermediate mass-ratio and comparable-mass black hole binaries.

Continuum and Discrete Initial-Boundary Value Problems and Einstein's Field Equations.

Many evolution problems in physics are described by partial differential equations on an infinite domain; therefore, one is interested in the solutions to such problems for a given initial dataset. A prominent example is the binary black-hole problem within Einstein’s theory of gravitation, in which one computes the gravitational radiation emitted from the inspiral of the two black holes, merger and ringdown. Powerful mathematical tools can be used to establish qualitative statements about the solutions, such as their existence, uniqueness, continuous dependence on the initial data, or their asymptotic behavior over large time scales. However, one is often interested in computing the solution itself, and unless the partial differential equation is very simple, or the initial data possesses a high degree of symmetry, this computation requires approximation by numerical discretization. When solving such discrete problems on a machine, one is faced with a finite limit to computational resources, which leads to the replacement of the infinite continuum domain with a finite computer grid. This, in turn, leads to a discrete initial-boundary value problem. The hope is to recover, with high accuracy, the exact solution in the limit where the grid spacing converges to zero with the boundary being pushed to infinity.The goal of this article is to review some of the theory necessary to understand the continuum and discrete initial boundary-value problems arising from hyperbolic partial differential equations and to discuss its applications to numerical relativity; in particular, we present well-posed initial and initial-boundary value formulations of Einstein’s equations, and we discuss multi-domain high-order finite difference and spectral methods to solve them.

The current status of binary black hole simulations in numerical relativity

Since the breakthroughs in 2005 which have led to long-term stable solutions of the binary black hole problem in numerical relativity, much progress has been made. I present here a short summary of the state of the field, including the capabilities of numerical relativity codes, recent physical results obtained from simulations and improvements to the methods used to evolve and analyse binary black hole spacetimes.

Characteristic Evolution and Matching.

I review the development of numerical evolution codes for general relativity based upon the characteristic initial value problem. Progress is traced from the early stage of 1D feasibility studies to 2D axisymmetric codes that accurately simulate the oscillations and gravitational collapse of relativistic stars and to current 3D codes that provide pieces of a binary black hole spacetime. Cauchy codes have now been successful at simulating all aspects of the binary black hole problem inside an artificially constructed outer boundary. A prime application of characteristic evolution is to eliminate the role of this artificial outer boundary via Cauchy-characteristic matching, by which the radiated waveform can be computed at null infinity. Progress in this direction is discussed.

IMEX Evolution of Scalar Fields on Curved Backgrounds

Inspiral of binary black holes occurs over a time-scale of many orbits, far longer than the dynamical time-scale of the individual black holes. Explicit evolutions of a binary system therefore require excessively many time steps to capture interesting dynamics. We present a strategy to overcome the Courant-Friedrichs-Lewy condition in such evolutions, one relying on modern implicit-explicit ODE solvers and multidomain spectral methods for elliptic equations. Our analysis considers the model problem of a forced scalar field propagating on a generic curved background. Nevertheless, we encounter and address a number of issues pertinent to the binary black hole problem in full general relativity. Specializing to the Schwarzschild geometry in Kerr-Schild coordinates, we document the results of several numerical experiments testing our strategy.

Simulation of binary black hole spacetimes with a harmonic evolution scheme

A numerical solution scheme for the Einstein field equations based on generalized harmonic coordinates is described, focusing on details which are not provided before in the literature and which are of particular relevance to the binary black hole problem. This includes demonstrations of the effectiveness of constraint damping, and how the time slicing can be controlled through the use of a source function evolution equation. In addition, some results from an ongoing study of binary black hole coalescence, where the black holes are formed via scalar field collapse, are shown. Scalar fields offer a convenient route to exploring certain aspects of black hole interactions, and one interesting though tentative suggestion from this early study is that behaviour reminiscent of ‘zoom-whirl’ orbits in particle trajectories is also present in the merger of equal mass, non-spinning binaries, with appropriately fine-tuned initial conditions.

Geometry of a naked singularity created by standing waves near a Schwarzschild horizon, and its application to the binary black hole problem

The most promising way to compute the gravitational waves emitted by binary black holes (BBHs) in their last dozen orbits, where post-Newtonian techniques fail, is a quasistationary approximation introduced by Detweiler and being pursued by Price and others. In this approximation the outgoing gravitational waves at infinity and downgoing gravitational waves at the holes' horizons are replaced by standing waves so as to guarantee that the spacetime has a helical Killing vector field. Because the horizon generators will not, in general, be tidally locked to the holes' orbital motion, the standing waves will destroy the horizons, converting the black holes into naked singularities that resemble black holes down to near the horizon radius. This paper uses a spherically symmetric, scalar-field model problem to explore in detail the following BBH issues: (i) The destruction of a horizon by the standing waves. (ii) The accuracy with which the resulting naked singularity resembles a black hole. (iii) The conversion of the standing-wave spacetime (with a destroyed horizon) into a spacetime with downgoing waves by the addition of a ``radiation-reaction field''. (iv) The accuracy with which the resulting downgoing waves agree with the downgoing waves of a true black-hole spacetime (with horizon). The model problem used to study these issues consists of a Schwarzschild black hole endowed with spherical standing waves of a scalar field. It is found that the spacetime metric of the singular, standing-wave spacetime, and its radiation-reaction-field-constructed downgoing waves are quite close to those for a Schwarzschild black hole with downgoing waves -- sufficiently close to make the BBH quasistationary approximation look promising for non-tidally-locked black holes.

Binary black-hole problem at the third post-Newtonian approximation in the orbital motion: Static part

Post-Newtonian expansions of the Brill-Lindquist and Misner-Lindquist solutions of the time-symmetric two-black-hole initial value problem are derived. The static Hamiltonians related to the expanded solutions, after identifying the bare masses in both solutions, are found to differ from each other at the third post-Newtonian approximation. By shifting the position variables of the black holes the post-Newtonian expansions of the three metrics can be made to coincide up to the fifth post-Newtonian order resulting in identical static Hamiltonians up the third post-Newtonian approximation. The calculations shed light on previously performed binary point-mass calculations at the third post-Newtonian approximation.

The Characteristic Treatment of Black Holes

The characteristic initial value problem has been successfully implemented as a robust computational algorithm (the PITT NULL CODE) to evolve 4-dimensional vacuum spacetimes. It has been applied to the calculation of gravitational waveforms emitted by black holes and to the event horizon structure in the merger of black holes. The characteristic code also has potential application to the binary black hole problem via Cauchy-characteristic matching. Because the event horizon is itself a characteristic hypersurface, it can be analyzed by characteristic techniques as a stand-alone object. We have developed an analytic conformal model of null hypersurfaces which gives new insight into the intrinsic geometry of the pairof-pants horizon found in the numerical simulation of the head-on collision of black holes and into the initially toroidal horizon found in the simulation of a collapsing, rotating cluster. Most studies of black hole formation and merger have been restricted to axisymmetry. However, axisymmetric horizons, like the Schwarzschild horizon, are non-generic. When applied to a non-axisymmetric horizon, the characteristic approach reveals substantially new features. In particular, coalescing black holes generically go through a toroidal phase before they become spherical. The conformal structure of the event horizon supplies part of the data for a simulation of the exterior space-time. This provides a new way to calculate the post-merger waveforms from a binary black hole inspiral. §1. Null cone evolution A longterm project 1) to develop the pioneering work of Bondi 2) and Penrose 3) into a computational algorithm for the characteristic initial value problem has recently culminated in a highly accurate,efficient and robust code — the PITT NULL CODE. 4) Because the evolution proceeds on a space-time foliation by null cones which are generated by the characteristic rays of the theory,this approach offers several advantages for numerical work. I will describe here the fruits of these investigations relevant to black hole physics. In null cone coordinates,Einstein’s equations reduce to propagation equations along the radial light rays,which can be integrated in hierarchical order for one variable at a time. This leads to a highly efficient characteristic marching algorithm. There is one complex evolution variable which describes the free degrees of freedom of the gravitational field and four auxiliary variables. A compactified grid,based upon Penrose’s conformal description of null infinity,removes the necessity of an artificial outgoing radiation condition and makes possible a rigorous description of geometrical quantities such as the Bondi mass and news function. The news function supplies both the true waveform and polarization of the gravitational radiation incident on a distant antenna. Furthermore the grid domain is exactly the region in which waves propagate,which is ideally efficient for the purpose of radiation studies. Since each

Exact analytic characteristic initial data for axisymmetric, non-rotating, vacuum spacetimes, with an application to the binary black hole problem

Bondi's approach to the construction of a coordinate system is used with a different choice of gauge, in accordance with which the radial coordinate r is an affine parameter, to cast the metric tensor into a form suitable for use with the Newman-Penrose null tetrad formalism. The choice of tetrad has the result that the equations and all the functions that appear in them are real-valued. A group classification of the Sachs equations in this gauge leads to a unique expression for the first of the five independent elements of the Weyl spinor, and to the corresponding exact solutions for two of the metric functions on an initial null hypersurface. A proof is presented that the result for constitutes the appropriate characteristic initial value function for all physically realistic axisymmetric, non-rotating vacuum spacetimes. Integration of the field equations on the axis of symmetry when an equatorial symmetry plane is also present, and on the equatorial plane itself when time rates of change can be neglected, shows that these data produce results consistent with Newton's laws and with the Schwarzschild solution in the appropriate limits. The solution on the axis of symmetry indicates that the Weyl curvature increases without limit between black holes as their separation decreases.