We present a new class of 3D black hole initial data sets for numerical
relativity. These data sets go beyond the axisymmetric, ``gravity wave plus
rotating black hole'' single black hole data sets by creating a dynamic,
distorted hole with adjustable distortion parameters in 3D. These data sets
extend our existing test beds for 3D numerical relativity, representing the
late stages of binary black hole collisions resulting from on-axis collision or
3D spiralling coalescence, and should provide insight into the physics of such
systems. We describe the construction of these sets, the properties for a
number of example cases, and report on progress evolving them.Comment: 3 pages, 2 postscript figures, LaTeX, uses mprocl.sty (available at
  http://shemesh.fiz.huji.ac.il/MG8/submission.html) To appear in the
  proceedings of the Marcel Grossmann 8 (Jerusalem, 1997

Brandt, S. R.

Camarda, K.

Seidel, E.

English

arXiv

We present a new class of 3D black hole initial data sets for numerical relativity. These data sets go beyond the axisymmetric, ``gravity wave plus rotating black hole'' single black hole data sets by creating a dynamic, distorted hole with adjustable distortion parameters in 3D. These data sets extend our existing test beds for 3D numerical relativity, representing the late stages of binary black hole collisions resulting from on-axis collision or 3D spiralling coalescence, and should provide insight into the physics of such systems. We describe the construction of these sets, the properties for a number of example cases, and report on progress evolving them

Brandt, Steven R.

Camarda, Karen

Max Planck Society eDoc Server

Three dimensional distorted black holes: Initial data and evolution

Brandt, S.

MPG.PuRe

arXiv:gr-qc/9711016 v2   11 Mar 1998Three Dimensional Distorted Black Holes: Initial Data and EvolutionS. R. Brandt, K. Camarda, E. SeidelMax-Planck-Institut für GravitationsphysikSchlaatzweg 1, 14473 Potsdam, GermanyWe present a new class of 3D black hole initial data sets for numerical relativity. Thesedata sets go beyond the axisymmetric, “gravity wave plus rotating black hole” singleblack hole data sets by creating a dynamic, distorted hole with adjustable distortion pa-rameters in 3D. These data sets extend our existing test beds for 3D numerical relativity,representing the late stages of binary black hole collisions resulting from on-axis collisionor 3D spiralling coalescence, and should provide insight into the physics of such systems.We describe the construction of these sets, the properties for a number of example cases,and report on progress evolving them.1 IntroductionIn all that follows, the standard ADM 3+1 decomposition is used1 along with thestandard York decomposition of the initial value problem2. This work extendsprevious studies of axisymmetric black holes3,4 to allow for a dependence on theazimuthal coordinate. The basic idea is to combine the pure gravitational waveconstruction due to Brill and combine it with a black hole to form adjustable,highly distorted black hole initial data sets.2 Initial Value Problem I: Non-Axisymmetric Distorted Black HolesThe first type of solution we consider for a non-axisymmetric gravity wave is theBrill wave form superimposed on an Einstein-Rosen bridge, used by 2D simulationsat NCSA3. In these time-symmetric spacetimes, the three metric is given byds2 = ψ4(e2q[dη2 + dθ2]+ sin2 θ dφ2). (1)In this equation, q may be chosen freely (so long as the appropriate boundaryconditions are respected), but the form we use here is similar to that used byprevious NCSA work3q = Q0 sinn θ(1 + c cos2 φ)g̃ (2)g̃ = e−(η−η0)2/σ2 + e−(η+η0)2/σ2 . (3)In the above n is an even integer, constrained such that n ≥ 2, which specifies theangular dependence of the Brill wave. The parameter σ specifies its width, and η0its radial position. The parameter c specifies the non-axisymmetric aspect of thewave, and when c = 0 we recover the axisymmetric case3. Once these parametersare supplied, we numerically solve the Hamiltonian constraint equation for ψ, theconformal factor. The initial value code was tested against the results of the 2Dcode for accuracy, and convergence studies show that it converges to second order.13 Initial Value Problem II: Non-Axisymmetric Distorted Black Holeswith Angular MomentumAnother form of distorted black hole initial data, one which allows us to have both anon-axisymmetric gravity wave and angular momentum, is based on a conformallyflat space (q = 0). Here, the gravity wave is constructed by specifying the conformalextrinsic curvature. Its parameters are an angular momentum J , a φ-dependencegiven by m (normally we use either m = 4 so that the IVP is isometric under theoperation φ→ φ±π/2, or we usem = 0 to facilitate comparison with 2D codes), anda function of η that is typically chosen to be the same as g̃ above, and is arbitraryso long as it vanishes as η → ∞ and is symmetric at η = 0. Here we sketch theprocedure of constructing the solution to the constraints: writing Kij = ψ−2K̂ij wetakeK̂ij = m sin(mφ)h0 h1 0h1 hx − h0 00 0 −hx sin2 θ +0 0 K̂ηφ0 0 K̂θφK̂ηφ K̂θφ 0 (4)K̂ηφ = sin2 θ(3J + cos (mφ)1sin3 θ∂θ(sin4 θg))(5)K̂θφ = cos (mφ) sin θ(m2v + sin2 θ∂ηg). (6)If we now make the substitutions:h0 = ∂ηΩ −1sin θ∂θ (sin θΛ) (7)h1 = ∂ηΛ +1sin θ∂θ (sin θΩ) (8)then the η- and θ-momentum constraints become:(∂2η − 1 + ∂2θ + 2 cot θ∂θ)Ω = − (4 cos θ + sin θ∂θ) ∂φg (9)(∂2η − 1 + ∂2θ + 2 cot θ∂θ)Λ = − csc θ∂φv − ∂θhx + sin θ∂η∂φg (10)Carefully choosing the forms and boundary values of the functions on the right handside will then allow us to solve the constraints analytically in terms of the functiong̃ and its derivatives. Details will be published in a future paper.4 Evolutions of type I initial dataTo evolve this data, we first interpolate it from the spheroidal (η, θ, φ) coordinatesystem to a Cartesian coordinate system used by our evolution codes. For thepresent we limit our evolutions to a grid with an outer boundary placed at 18M ,so the results we obtain should not be taken as more than a feasibility study.Higher resolution runs, on much larger grids, are in progress and will be published5,6elsewhere. Nevertheless, we are able to evolve both axisymmetric data (c = 0), andnon-axisymmetric data (c 6= 0) and obtain qualitatively similar results. The metricelements evolve in the same fashion, with the same qualitative features we are2familiar with from 2D codes. Below we show two figures. In Figure I we see thegrowth of the metric function for a run parameterized by (Q0 = −0.1, η0 = 0.0, σ =1.0, n = 4, c = 0.5). The growth of the metric function here is familiar from all 1Dand 2D studies performed by the group. In Figure II we see the beginnings of awaveform extracted at several radii and can watch its progress as it travels outward.The wavelength matches well with the value predicted from linear theory, despitethe nearness of the outer boundary.0.0 2.0 4.0 6.0 8.0 10.0x (MADM)0.010.020.030.040.050.0g xxGrowth of Metric Function gxxt=0.00 M  4.53 M  9.27 M 13.91 M 18.54 M 23.18 M 27.81 M 32.45 M 37.09 M0.0 20.0 40.0Time (MADM)−0.4Zerilli Function AmplitudeGravity Wave Extractionr =  9.27 Mr = 11.59 Mr = 13.91 MFigure 1: Metric function gxx evaluatedalong the x-axis and plotted at severaltimes.Figure 2: The ℓ = 2 Zerilli function ex-tracted at three different radii and plot-ted as a function of time. The wavefunction is clearly propagating outward.5 Future WorkAs of this writing, much of this work has already been extended5. Furthermore,in the near future we plan to publish full descriptions of these data sets, theirimplementations, and to describe in detail comparisons with 2D code results (wherethe outer boundary may be placed arbitrarily far away), and convergence tests.References1. R. Arnowitt, S. Deser, and C. W. Misner in Gravitation: An Introduction toCurrent Research, ed. L Whitten (John Wiley, New York, 1962).2. James York in Sources of Gravitational Radiation, ed. L Smarr (CambridgeUniversity Press, Cambridge, England, 1979).3. D. Bernstein, D. Hobill, E. Seidel, and L. Smarr Phys. Rev. D 50, 3760 (1994)4. S. Brandt and E. Seidel Phys. Rev. D 52, 856 (1995); 52, 870(1995); 54,1403(1996)5. K. Camarda A Numerical Study of 3D Black Hole Spacetimes, Ph.D. thesis(University of Illinois at Urbana-Champaign, 1998).6. K. Camarda and E. Seidel Phys. Rev. D, Rapid Communications submitted(1997).3

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Three Dimensional Distorted Black Holes: Initial Data and Evolution

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