Hyperboloidal data and evolution

Husa, S.; Schneemann, Carsten; Vogel, Tilman; Zenginoğlu, Anıl

doi:10.1063/1.2218186

Cited by 22 publications

(41 citation statements)

References 15 publications

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“…For example, using maximal slicing or a slicing which insures that the mean curvature rapidly decays to zero as one approaches the outer boundary might justify criterion (iii), while forcing the normal component (with respect to the outer boundary) of the shift vector to be zero at the outer boundary guarantees (iv). On the other hand, these criteria are not justified if hyperboloidal slices are used [37,[67][68][69], where the mean curvature asymptotically approaches a constant, nonzero value. It should not be difficult to generalize our analysis to more general foliations of Minkowski spacetime using the 2 + 1 split discussed in section 2.2.…”

Section: Discussionmentioning

confidence: 99%

“…Two other approaches presented in the literature for constructing absorbing boundary conditions are: matching the nonlinear Cauchy code to a nonlinear characteristic code (see [30] for a review and [31] for recent work) and matching an incoming characteristic formulation to an outgoing one at a timelike cylinder [32]. Finally, methods that avoid introducing an artificial outer boundary altogether compactify spatial infinity [33,34], or make use of hyperboloidal slices and compactify null infinity (see, for instance, [35][36][37]). …”

Section: Introductionmentioning

confidence: 99%

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Towards absorbing outer boundaries in general relativity

Buchman¹,

Sarbach²

2006

Class. Quantum Grav.

155

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We construct exact solutions to the Bianchi equations on a flat spacetime background. When the constraints are satisfied, these solutions represent in-and outgoing linearized gravitational radiation. We then consider the Bianchi equations on a subset of flat spacetime of the form [0, T ] × B R , where B R is a ball of radius R, and analyse different kinds of boundary conditions on ∂B R . Our main results are as follows. (i) We give an explicit analytic example showing that boundary conditions obtained from freezing the incoming characteristic fields to their initial values are not compatible with the constraints. (ii) With the help of the exact solutions constructed, we determine the amount of artificial reflection of gravitational radiation from constraintpreserving boundary conditions which freeze the Weyl scalar 0 to its initial value. For monochromatic radiation with wave number k and arbitrary angular momentum number 2, the amount of reflection decays as (kR) −4 for large kR. (iii) For each L 2, we construct new local constraint-preserving boundary conditions which perfectly absorb linearized radiation with L. (iv) We generalize our analysis to a weakly curved background of mass M and compute first-order corrections in M/R to the reflection coefficients for quadrupolar odd-parity radiation. For our new boundary condition with L = 2, the reflection coefficient is smaller than that for the freezing 0 boundary condition by a factor of M/R for kR > 1.04. Implications of these results for numerical simulations of binary black holes on finite domains are discussed.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Towards absorbing outer boundaries in general relativity

Buchman¹,

Sarbach²

2006

Class. Quantum Grav.

155

View full text Add to dashboard Cite

show abstract

“…Although the correct solution is a flat spacetime, nonlinear effects and the nontrivial geometry of the time slices can easily trigger continuum instabilities in the equations. For simple examples of such effects see [21] for a nonlinear wave equation on flat space, designed to model problems arising in this testbed, and [22] for a linear example of how nontrivial geometry of the slicing can trigger instabilities already for the Maxwell equations.…”

Section: Gauge Wave Testmentioning

confidence: 99%

“…The Gowdy test is run in both future and past time directions because analytical studies [30] and numerical experiments [22,31] indicate that the sign of the extrinsic curvature may have important consequences for constraint violation. The subsidiary system governing constraint propagation can lead to unstable departure from the constraint hypersurface.…”

Section: Gowdy Wave Testmentioning

confidence: 99%

Implementation of standard testbeds for numerical relativity

Babiuc

Husa

Alic

et al. 2008

Class. Quantum Grav.

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We discuss results that have been obtained from the implementation of the initial round of testbeds for numerical relativity which was proposed in the first paper of the Apples with Apples Alliance. We present benchmark results for various codes which provide templates for analyzing the testbeds and to draw conclusions about various features of the codes. This allows us to sharpen the initial test specifications, design a new test and add theoretical insight.

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“…Fig. 2 is taken from [13] and is the result of a numerical calculation where κ has been chosen such that I + is a straight line in the corresponding conformal Gauss gauge. Illustrated is the "upper right part" of the Penrose diagram for the Schwarzschild-Kruskal spacetime.…”

Section: 31mentioning

confidence: 99%

Numerical calculations near spatial infinity

Zenginoğlu

2007

J. Phys.: Conf. Ser.

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Abstract. After describing in short some problems and methods regarding the smoothness of null infinity for isolated systems, I present numerical calculations in which both spatial and null infinity can be studied. The reduced conformal field equations based on the conformal Gauss gauge allow us in spherical symmetry to calculate numerically the entire Schwarzschild-Kruskal spacetime in a smooth way including spacelike, null and timelike infinity and the domain close to the singularity. MotivationThe main purpose of this article is to study the feasibility of the conformal Gauss gauge in numerical applications. The importance of this gauge lies in its usefulness in the analysis of the gravitational field in a neighbourhood of spatial infinity, as we discuss in the following. For review articles see [12,6]. Isolated SystemsThe motivation to study the regularity of null infinity for asymptotically flat spacetimes arises from interest in the following question: How can we describe the gravitational field of isolated systems in a rigorous and efficient way?Isolated systems in general relativity are self-gravitating models of astrophysical sources. Depending on the context, these can be planets, stars, black holes and even galaxy clusters. It is not the source that makes a system isolated, but the asymptotic region. The gravitational field of isolated systems becomes weak as we move away from the source. As nothing is completely isolated from the rest of the universe and as we can not yet test our assumptions by experiment, details of this idealization need to be decided upon in accordance with our physical intuition, expectation and most importantly, in accordance with the equations describing the model. Isolated systems are modeled in general relativity by asymptotically flat spacetimes in which the metric on the complement of a spatially compact set approaches near infinity the flat metric in a specified way.Many physical concepts like mass, linear and angular momentum are defined in the asymptotic region with respect to infinity. The use of coordinate dependent limits to infinity for calculating these quantities can deliver misleading results if one is not careful enough. A geometric, coordinate independent way to deal with isolated systems has been proposed by Penrose [15] by use of conformally rescaled spacetimes with a smooth conformal boundary. The main idea is to adjoin infinity to the physical spacetime by a conformal rescaling of the metric, as is done in complex analysis, where the Riemann sphere is constructed by adjoining infinity to the complex

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Hyperboloidal data and evolution

Cited by 22 publications

References 15 publications

Towards absorbing outer boundaries in general relativity

Towards absorbing outer boundaries in general relativity

Implementation of standard testbeds for numerical relativity

Numerical calculations near spatial infinity

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