The Bowen-York family of spinning black hole initial data depends essentially on one, positive, free parameter. The extreme limit corresponds to making this parameter equal to zero. This choice represents a singular limit for the constraint equations. We prove that in this limit a new solution of the constraint equations is obtained. These initial data have similar properties to the extreme Kerr and Reissner-Nordstrom black hole initial data. In particular, in this limit one of the asymptotic ends changes from asymptotically flat to cylindrical. The existence proof is constructive, we actually show that a sequence of Bowen-York data converges to the extreme solution.
The Bowen-York initial value data typically used in numerical relativity to represent spinning black hole are not those of a constant-time slice of the Kerr spacetime. If Bowen-York initial data are used for each black hole in a collision, the emitted radiation will be partially due to the ``relaxation'' of the individual holes to Kerr form. We compute this radiation by treating the geometry for a single hole as a perturbation of a Schwarzschild black hole, and by using second order perturbation theory. We discuss the extent to which Bowen-York data can be expected accurately to represent Kerr…
We prove the existence of initial data sets which possess an asymptotically flat and an asymptotically cylindrical end. Such geometries are known as trumpets in the community of numerical relativists.
A class of vacuum initial-data sets is described which are based on certain expressions for the extrinsic curvature first studied and employed by Bowen and York. These expressions play a role for the momentum constraint of general relativity which is analogous to the role played by the Coulomb solution for the Gauss-law constraint of electromagnetism.
The most popular method to construct initial data for black-hole-binary simulations is the puncture method, in which compactified wormholes are given linear and angular momentum via the Bowen-York extrinsic curvature. When these data are evolved, they quickly approach a ``trumpet'' topology, suggesting that it would be preferable to use data that are in trumpet form from the outset. To achieve this, we extend the puncture method to allow the construction of Bowen-York trumpets, including an outline of an existence and uniqueness proof of the solutions. We construct boosted, spinning and binary…
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. It is possible to reduce the spurious radiation by applying the puncture approach to multiple Kerr black holes, as we demonstrate for examples of head-on collisions of equal-mass black-hole binaries.
There is derived, for a conformally flat three-space, a family of linear second-order partial differential operators which send vectors into tracefree, symmetric two-tensors. These maps, which are parametrized by conformal Killing vectors on the three-space, are such that the divergence of the resulting tensor field depends only on the divergence of the original vector field. In particular these maps send source-free electric fields into TT-tensors. Moreover, if the original vector field is the Coulomb field on $\mathbb{R}^3\backslash \lbrace0\rbrace$, the resulting tensor fields on $\mathbb{R…
A generalization of the Bowen-York initial data to the case with a positive cosmological constant is investigated. We follow the construction presented recently by Bizoń, Pletka and Simon, and solve numerically the Lichnerowicz equation on a compactified domain $\mathbb S^1 \times \mathbb S^2$. In addition to two branches of solutions depending on the polar variable on $\mathbb S^2$ that were already known, we find branches of solutions depending on two variables: the polar variable on $\mathbb S^2$ and the coordinate on $\mathbb S^1$. Using Vanderbauwhede's results concerning bifurcations fro…
We study the tidal deformations of the shape of a spinning black hole horizon due to a binary companion in the Bowen-York initial data set. We use the framework of quasi-local horizons and identify a black hole by marginally outer trapped surfaces. The intrinsic horizon geometry is specified by a set of mass and angular-momentum multipole moments $\mathcal{M}_n$ and $\mathcal{J}_n$ respectively. The tidal deformations are described by the change in these multipole moments caused by an external perturbation. This leads us to define two sets of dimensionless numbers, the tidal coefficients for $…
A new approach to construct initial data for binary systems with neutron star components is introduced. The approach is a generalization of the puncture initial data method for binary black holes based on Bowen-York solutions to the momentum constraint. As with binary black holes, the method allows setting orbital configurations with direct input from post-Newtonian approximations and involves solving only the Hamiltonian constraint. The effectiveness of the method is demonstrated with evolutions of double neutron star and black hole -- neutron star binaries in quasi-circular orbits.
Scenarios of large extra dimensions have enhanced the importance for the study of black holes in higher dimensions. In this paper, we analyze an axisymmetric system of two black holes. Specifically, the Bowen-York method is generalized for higher dimensions in order to calculate the initial data for head-on collision of two equal-mass black holes. Then, the initial data are evolved adopting the close-slow approximation to study gravitational waves emitted during the collision. We derive an empirical formula for radiation efficiency, which depends weakly on the dimensionality. Possible implicat…
Black hole initial data is usually produced using Bowen-York type puncture initial data or by applying an excision boundary condition. The benefits of the Bowen-York initial data are the ability to specify the spin and momentum of the system as parameters of the initial data. In an attempt to extend these benefits to other formulations of the Einstein constraints, the puncture method is reformulated using distributions as source terms. It is shown how the Bowen-York puncture black hole initial data and the trumpet variation is generated by distributional sources. A heuristic argument is presen…
We evolve equal-mass, equal-spin black-hole binaries with specific spins of a/mH 0.925, the highest spins simulated thus far and nearly the largest possible for Bowen-York black holes, in a set of configurations with the spins counter-aligned and pointing in the orbital plane, which maximizes the recoil velocities of the merger remnant, as well as a configuration where the two spins point in the same direction as the orbital angular momentum, which maximizes the orbital hang-up effect and remnant spin. The coordinate radii of the individual apparent horizons in these cases are very small and t…
Initial value problem in General Relativity is often solved numerically; with only a few exceptions one of which is the "model" solution of Bowen and York where an analytical form of the solution is available. The solution describes a dynamical, time-asymmetric, gravitating system with mass and linear momentum. Here we revisit this solution and correct an error which turns out to be important for identifying the energy-content of the solution. Depending on the linear momentum, the ratio of the non-stationary part of the initial energy to the total ADM energy takes values between $[0, 0.592)$. …
A Differentiable Parametric Model of Binary-Black-Hole Initial Data: I. Conformally flat Bowen-York punctures
KU Leuven（荷语鲁汶大学）；Institute for Theoretical Physics, KU Leuven（荷语鲁汶大学理论物理研究所）；Leuven Gravity Institute, KU Leuven（荷语鲁汶大学列日引力研究所）；STADIUS Center for Dynamical Systems, Signal Processing and Data Analytics, KU Leuven（荷语鲁汶大学STADIUS动力学系统、信号处理与数据分析中心）
Initial data for black hole collisions are commonly generated using the Bowen-York approach based on conformally flat 3-geometries. The standard (constant Boyer-Lindquist time) spatial slices of the Kerr spacetime are not conformally flat, so that use of the Bowen-York approach is limited in dealing with rotating holes. We investigate here whether there exist foliations of the Kerr spacetime that are conformally flat. We limit our considerations to foliations that are axisymmetric and that smoothly reduce in the Schwarzschild limit to slices of constant Schwarzschild time. With these restricti…
We obtain an explicit solution of the momentum constraint for conformally flat, maximal slicing, initial data which gives an alternative to the purely longitudinal extrinsic curvature of Bowen and York. The new solution is related, in a precise form, with the extrinsic curvature of a Kerr slice. We study these new initial data representing spinning black holes by numerically solving the Hamiltonian constraint. They have the following features: i) Contain less radiation, for all allowed values of the rotation parameter, than the corresponding single spinning Bowen-York black hole. ii) The maxim…
We solve the Hamiltonian and momentum constraints of general relativity for two black holes with nearly extremal spins and relativistic boosts in the puncture formalism. We use a non-conformally-flat ansatz with an attenuated superposition of two Lorentz-boosted, conformally Kerr or conformally Schwarzschild 3-metrics and their corresponding extrinsic curvatures. We compare evolutions of these data with the standard Bowen-York conformally flat ansatz (technically limited to intrinsic spins $χ=S/M^2_{\text{ADM}}=0.928$ and boosts $P/M_{\text{ADM}}=0.897$), finding, typically, an order of magnit…
Initial data for boosted Kerr black hole are constructed in an axially symmetric case. Momentum and hamiltonian constraints are solved numerically using finite element method (FEM) algorithms. Both Bowen-York and puncture boundary conditions are adopted and appropriate results are compared. Past and future apparent horizons are also found numerically and the Penrose inequality is tested in detail.
We describe conformally flat initial data, with explicitly given analytic extrinsic curvature solving the vacuum momentum constraints. They follow from a solution of Dain and Friedrich discovered in 2001. The cylindrically symmetric subcase of the Bowen-York solution is a subclass of this general configuration.
There is a significant possibility that astrophysical black holes with nearly-extremal spins exist. Numerical simulations of such systems require suitable initial data. In this paper, we examine three methods of constructing binary-black-hole initial data, focusing on their ability to generate black holes with nearly-extremal spins: (i) Bowen-York initial data, including standard puncture data (based on conformal flatness and Bowen-York extrinsic curvature), (ii) standard quasi-equilibrium initial data (based on the extended-conformal-thin-sandwich equations, conformal flatness, and maximal sl…
We give a complete analytical proof of existence and uniqueness of extreme-like black hole initial data for Einstein equations, which possess a cilindrical end, analogous to extreme Kerr, extreme Reissner Nordstrom, and extreme Bowen-York's initial data. This extends and refines a previous result \cite{dain-gabach09} to a general case of conformally flat, maximal initial data with angular momentum, linear momentum and matter.
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.
We generalize Bowen-York black hole initial data to hyperboloidal constant mean curvature slices which extend to future null infinity. We solve this initial value problem numerically for several cases, including unequal mass binary black holes with spins and boosts. The singularity at null infinity in the Hamiltonian constraint associated with a constant mean curvature hypersurface does not pose any particular difficulties. The inner boundaries of our slices are minimal surfaces. Trumpet configurations are explored both analytically and numerically.
I construct initial data for equal-mass irrotational binary black holes using the conformal thin-sandwich puncture (CTSP) approach. I locate quasi-circular orbits using the effective-potential method, and estimate the location of the innermost stable circular orbit (ISCO). The ISCO prediction is consistent with results for conformal thin-sandwich data produced using excision techniques. These results also show that the ISCOs predicted by the effective-potential and ADM-Komar mass-comparison methods agree for conformal thin-sandwich data, just as they did for Bowen-York data.
This work consists of two distinct parts. In the first part we present a new method for solving the initial value problem of general relativity. Given any spatial metric with a surface orthogonal Killing field and two freely specified components of the extrinsic curvature we solve for extrinsic curvature's remaining components. For the second part, after noting that initial data for the Kerr spacetime can be derived within our formalism we construct data for axisymmetric configurations of spinning black holes. Though our method is limited to axisymmetry, it offers an advantage over the Bowen-Y…
We study conformally-flat initial data for an arbitrary number of spinning black holes with exact analytic solutions to the momentum constraints constructed from a linear combination of the classical Bowen-York and conformal Kerr extrinsic curvatures. The solution leading to the largest intrinsic spin, relative to the ADM mass of the spacetime epsilon_S=S/M^2_{ADM}, is a superposition with relative weights of Lambda=0.783 for conformal Kerr and (1-Lambda)=0.217 for Bowen-York. In addition, we measure the spin relative to the initial horizon mass M_{H_0}, and find that the quantity chi=S/M_{H_0…
We present a new computational framework for the Galerkin-collocation method for double domain in the context of ADM 3+1 approach in numerical relativity. This work enables us to perform high resolution calculations for initial sets of two arbitrary black holes. We use the Bowen-York method for binary systems and the puncture method to solve the Hamiltonian constraint. The nonlinear numerical code solves the set of equations for the spectral modes using the standard Newton-Raphson method, LU decomposition and Gaussian quadratures. We show convergence of our code for the conformal factor and th…
In this paper we study a new family of black hole initial data sets corresponding to distorted ``Kerr'' black holes with moderate rotation parameters, and distorted Schwarzschild black holes with even- and odd-parity radiation. These data sets build on the earlier rotating black holes of Bowen and York and the distorted Brill wave plus black hole data sets. We describe the construction of this large family of rotating black holes. We present a systematic study of important properties of these data sets, such as the size and shape of their apparent horizons, and the maximum amount of radiation …
We present a formula that relates the variations of the area of extreme throat initial data with the variation of an appropriate defined mass functional. From this expression we deduce that the first variation, with fixed angular momentum, of the area is zero and the second variation is positive definite evaluated at the extreme Kerr throat initial data. This indicates that the area of the extreme Kerr throat initial data is a minimum among this class of data. And hence the area of generic throat initial data is bounded from below by the angular momentum. Also, this result strongly suggests th…
We present a computational framework (Rio) in the ADM 3+1 approach for numerical relativity. This work enables us to carry out high resolution calculations for initial data of two arbitrary black holes. We use the transverse conformal treatment, the Bowen-York and the puncture methods. For the numerical solution of the Hamiltonian constraint we use the domain decomposition and the spectral decomposition of Galerkin-Collocation. The nonlinear numerical code solves the set of equations for the spectral modes using the standard Newton-Raphson method, LU decomposition and Gaussian quadratures. We …
We discuss the initial value problem of general relativity in its recently unified Lagrangian and Hamiltonian pictures and present a multi-domain pseudo-spectral collocation method to solve the resulting coupled nonlinear partial differential equations. Using this code, we explore several approaches to construct initial data sets containing one or two black holes: We compute quasi-circular orbits for spinning equal mass black holes and unequal mass (nonspinning) black holes using the effective potential method with Bowen-York extrinsic curvature. We compare initial data sets resulting from dif…
We revisit the construction of maximal initial data on compact manifolds in vacuum with positive cosmological constant via the conformal method. We discuss, extend and apply recent results of Hebey et al. [19] and Premoselli [31] which yield existence, non-existence, (non-)uniqueness and (linearisation-) stability of solutions of the Lichnerowicz equation, depending on its coefficients. We then focus on so-called $(t,φ)$-symmetric data as "seed manifolds", and in particular on Bowen-York data on the round hypertorus $\mathbb{S}^2 \times \mathbb{S}$ (a slice of Nariai) and on Kerr-deSitter. In …
We study identical mass black hole binaries with spins perpendicular to the binary's orbital plane. These binaries have individual spins ranging from $s/m^2=-0.90$ to 0.90, ($s_1 = s_2$ in all cases) which is near the limit possible with standard Bowen-York puncture initial data. The extreme cases correspond to the largest initial spin simulations to date. Our results expand the parameter space covered by Rezzolla {\it et al.} and, when combining both data sets, we obtain estimations for the minimum and maximum values for the intrinsic angular momenta of the remnant of binary black hole merger…
We describe a procedure for constructing initial data for boosted black holes in the moving-punctures approach to numerical relativity that endows the initial time slice from the outset with trumpet geometry within the black hole interiors. We then demonstrate the procedure in numerical simulations using an evolution code from the Einstein Toolkit that employs 1+log slicing. The Lorentz boost of a single black hole can be precisely specified and multiple, widely separated black holes can be treated approximately by superposition of single hole data. There is room within the scheme for later im…
Sequences of initial-data sets representing binary black holes in quasi-circular orbits have been used to calculate what may be interpreted as the innermost stable circular orbit. These sequences have been computed with two approaches. One method is based on the traditional conformal-transverse-traceless decomposition and locates quasi-circular orbits from the turning points in an effective potential. The second method uses a conformal-thin-sandwich decomposition and determines quasi-circular orbits by requiring the existence of an approximate helical Killing vector. Although the parameters de…
The Lazarus project was designed to make the most of limited 3D binary black-hole simulations, through the identification of perturbations at late times, and subsequent evolution of the Weyl scalar $Ψ_4$ via the Teukolsky formulation. Here we report on new developments, employing the concept of the ``quasi-Kinnersley'' (transverse) frame, valid in the full nonlinear regime, to analyze late-time numerical spacetimes that should differ only slightly from Kerr. This allows us to extract the essential information about the background Kerr solution, and through this, to identify the radiation prese…
We study the collision of two slowly rotating, initially non boosted, black holes in the close limit. A ``punctures'' modification of the Bowen - York method is used to construct conformally flat initial data appropriate to the problem. We keep only the lowest nontrivial orders capable of giving rise to radiation of both gravitational energy and angular momentum. We show that even with these simplifications an extension to higher orders of the linear Regge-Wheeler-Zerilli black hole perturbation theory, is required to deal with the evolution equations of the leading contributing multipoles. Th…
Construction of astrophysically realistic initial data remains a central problem when modelling the merger and eventual coalescence of binary black holes in numerical relativity. The objective of this paper is to provide astrophysically realistic freely specifiable initial data for binary black hole systems in numerical relativity, which are in agreement with post-Newtonian results. Following the approach taken by Blanchet, we propose a particular solution to the time-asymmetric constraint equations, which represent a system of two moving black holes, in the form of the standard conformal deco…
We study the constraint equations for a class of scalar-tensor effective field theories of gravity, including the operators up to $4$ derivatives in the action ($4\partial$ST). We extend the conformal transverse traceless and conformal thin sandwich methods of General Relativity to rewrite the constraint equations of the scalar-tensor theory as a set of elliptic partial differential equations. It is shown that, at weak coupling, a unique solution exists to the corresponding elliptic boundary value problems on asymptotically Euclidean initial slices under similar conditions as in the case of Ge…
Astrophysical black holes could be nearly extremal (that is, rotating nearly as fast as possible); therefore, nearly extremal black holes could be among the binaries that current and future gravitational-wave observatories will detect. Predicting the gravitational waves emitted by merging black holes requires numerical-relativity simulations, but these simulations are especially challenging when one or both holes have mass $m$ and spin $S$ exceeding the Bowen-York limit of $S/m^2=0.93$. We present improved methods that enable us to simulate merging, nearly extremal black holes more robustly an…
The Conformal Einstein equations and the representation of spatial infinity as a cylinder introduced by Friedrich are used to analyse the behaviour of the gravitational field near null and spatial infinity for the development of data which are asymptotically Euclidean, conformally flat and time asymmetric. Our analysis allows for initial data whose second fundamental form is more general than the one given by the standard Bowen-York Ansatz. The Conformal Einstein equations imply upon evaluation on the cylinder at spatial infinity a hierarchy of transport equations which can be used to calculat…
The increase in the sensitivity of gravitational wave interferometers will bring additional detections of binary black hole and double neutron star mergers. It will also very likely add many merger events of black hole - neutron star binaries. Distinguishing mixed binaries from binary black holes mergers for high mass ratios could be challenging because in this situation the neutron star coalesces with the black hole without experiencing significant disruption. To investigate the transition of mixed binary mergers into those behaving more like binary black hole coalescences, we present results…
A general relativistic, stationary and axisymmetric black hole in a four-dimensional asymptotically-flat spacetime is fully determined by its mass, angular momentum and electric charge. The expectation that astrophysically relevant black holes do not posses charge has resulted in a limited number of investigations of moving and charged black holes in the dynamical, strong-field gravitational (and electromagnetic) regime, where numerical studies are necessary. Apart from having a theoretical interest, the advent of multimessenger astronomy with gravitational waves offers new ways to think about…
We present the topological solutions of Einstein-dilaton gravity in the presence of a non-Abelian Yang-Mills field. In 4 dimensions, we consider the $So(3)$ and $So(2,1)$ semisimple group as the Yang-Mills gauge group, and introduce the black hole solutions with spherical and hyperbolic horizons, respectively. The solution in the absence of dilaton potential is asymptotically flat and exists only with spherical horizon. Contrary to the non-extreme Reissner-Nordstrom black hole, which has two horizons with a timelike and avoidable singularity, here the solution may present a black hole with a n…
