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Approximate Analytical Solutions to the Initial Data Problem of Black Hole Binary Systems
General Relativity and Quantum Cosmology · Submitted 24 Jan 2000
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The authors' main claim Thus, these data sets are suitable for use in evolution codes.
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We present approximate analytical solutions to the Hamiltonian and momentum constraint equations, corresponding to systems composed of two black holes with arbitrary linear and angular momentum. The analytical nature of these initial data solutions makes them easier to implement in numerical evolutions than the traditional numerical approach of solving the elliptic equations derived from the Einstein constraints. Although in general the problem of setting up initial conditions for black hole binary simulations is complicated by the presence of singularities, we show that the methods presented in this work provide initial data with $l_1$ and $l_\infty$ norms of violation of the constraint equations falling below those of the truncation error (residual error due to discretization) present in finite difference codes for the range of grid resolutions currently used. Thus, these data sets are suitable for use in evolution codes. Detailed results are presented for the case of a head-on collision of two equal-mass M black holes with specific angular momentum 0.5M at an initial separation of 10M. A straightforward superposition method yields data adequate for resolutions of $h=M/4$, and an "attenuated" superposition yields data usable to resolutions at least as fine as $h=M/8$. In addition, the attenuated approximate data may be more tractable in a full (computational) exact solution to the initial value problem.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We present approximate analytical solutions to the Hamiltonian and momentum constraint equations, corresponding to systems composed of two black holes with arbitrary linear and angular momentum. | contribution 1.00 | 0.96 | 0.25 | asserted 1.00 |
| s1 | The analytical nature of these initial data solutions makes them easier to implement in numerical evolutions than the traditional numerical approach of solving the elliptic equations derived from the Einstein constraints. | interpretation 0.86 | 0.35 | 0.12 | asserted 0.99 |
| s2 | Although in general the problem of setting up initial conditions for black hole binary simulations is complicated by the presence of singularities, we show that the methods presented in this work provide initial data with $l_1$ and $l_\infty$ norms of violation of the constraint equations falling below those of the truncation error (residual error due to discretization) present in finite difference codes for the range of grid resolutions currently used. | result 0.99 | 0.94 | 0.29 | asserted 1.00 |
| s3 | Thus, these data sets are suitable for use in evolution codes. | interpretation 0.97 | 0.70 | 0.39 | asserted 0.99 |
| s4 | Detailed results are presented for the case of a head-on collision of two equal-mass M black holes with specific angular momentum 0.5M at an initial separation of 10M. | result 0.61 | 0.91 | 0.07 | asserted 0.98 |
| s5 | A straightforward superposition method yields data adequate for resolutions of $h=M/4$, and an "attenuated" superposition yields data usable to resolutions at least as fine as $h=M/8$. | result 0.86 | 0.91 | 0.10 | asserted 0.99 |
| s6 | In addition, the attenuated approximate data may be more tractable in a full (computational) exact solution to the initial value problem. | interpretation 0.59 | 0.32 | 0.07 | hedged possibility 1.00 |
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Original abstract
We present approximate analytical solutions to the Hamiltonian and momentum constraint equations, corresponding to systems composed of two black holes with arbitrary linear and angular momentum. The analytical nature of these initial data solutions makes them easier to implement in numerical evolutions than the traditional numerical approach of solving the elliptic equations derived from the Einstein constraints. Although in general the problem of setting up initial conditions for black hole binary simulations is complicated by the presence of singularities, we show that the methods presented in this work provide initial data with $l_1$ and $l_\infty$ norms of violation of the constraint equations falling below those of the truncation error (residual error due to discretization) present in finite difference codes for the range of grid resolutions currently used. Thus, these data sets are suitable for use in evolution codes. Detailed results are presented for the case of a head-on collision of two equal-mass M black holes with specific angular momentum 0.5M at an initial separation of 10M. A straightforward superposition method yields data adequate for resolutions of $h=M/4$, and an "attenuated" superposition yields data usable to resolutions at least as fine as $h=M/8$. In addition, the attenuated approximate data may be more tractable in a full (computational) exact solution to the initial value problem.
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3 arXiv papers cite this in 4 sentences
Papers citing it · 1 sentence without a confident label
Uses its method or tool 1 paper
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In the case of binary black hole or neutron star initial data, it is common practice to remove the metric variables of one star at the location of the other (analogous to approaches introduced in Refs. [ref] ), achieved by non-trivial attenuation functions $f_(A)$.
Generic initial data for binary boson stars · Modified superposition · 0.86
Compares with it 1 paper
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Note that the superposed metrics in this paper and in Ref. [ref] are similar to the superposed-Kerr-Schild conformal metrics used in Refs. [ref] ; however, the conformal metrics used here and in Ref. [ref] are flat everywhere except near each black hole.
Reducing spurious gravitational radiation in binary-black-hole simulations by using conformally curved initial data · Introduction · 0.93
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