Absorbing boundary conditions for Einstein's field equations

Sarbach, Olivier

doi:10.1088/1742-6596/91/1/012005

Cited by 22 publications

(36 citation statements)

References 72 publications

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“…The importance of numerical simulations in general relativity has spurred a large number of works which have established many of the necessary ingredients for a well-posed IBVP. For a review, see [7]. The first complete well-posed formulation was given by Friedrich and Nagy [8] for a version of Einstein's equations in which the curvature tensor, i.e.…”

Section: Discussionmentioning

confidence: 99%

Well-posed initial-boundary value problem for the harmonic Einstein equations using energy estimates

Kreiss

Reula

Sarbach

et al. 2007

Class. Quantum Grav.

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In recent work, we used pseudo-differential theory to establish conditions that the initial-boundary value problem for second-order systems of wave equations be strongly well-posed in a generalized sense. The applications included the harmonic version of the Einstein equations. Here we show that these results can also be obtained via standard energy estimates, thus establishing strong well-posedness of the harmonic Einstein problem in the classical sense.

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Section: Discussionmentioning

confidence: 99%

Well-posed initial-boundary value problem for the harmonic Einstein equations using energy estimates

Kreiss

Reula

Sarbach

et al. 2007

Class. Quantum Grav.

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“…Typically one would impose conditions such as no incoming radiation or absorbing boundary conditions, cf. [20] for example. However, due partly to the general covariance of GR, such boundary conditions are notoriously difficult to identify and implement in practice.…”

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confidence: 99%

Quasilocal Hamiltonians in general relativity

Anderson¹

2010

Phys. Rev. D

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Abstract. We analyse the definition of quasi-local energy in GR based on a Hamiltonian analysis of the Einstein-Hilbert action initiated by Brown-York. The role of the constraint equations, in particular the Hamiltonian constraint on the timelike boundary, neglected in previous studies, is emphasized here. We argue that a consistent definition of quasi-local energy in GR requires, at a minimum, a framework based on the (currently unknown) geometric well-posedness of the initial boundary value problem for the Einstein equations.The analysis of the gravitational field by Arnowitt-Deser-Misner [1] has led to a clear and welldefined construction of the Hamiltonian, and resulting definitions of energy, linear and angular momentum in the context of asymptotically flat spacetimes. These concepts are obviously of basic importance in understanding the physics of such (infinite) isolated gravitating systems. Nevertheless, infinite systems are idealizations of more realistic physical situations, and it is desirable to have available a similar analysis in the case of physical systems of finite extent.However the understanding of this issue for domains of finite extent is much less satisfactory. Despite numerous proposals, from a number of different viewpoints, a consensus has not yet been reached on a suitable definition of the Hamiltonian or energy of a finite system, i.e. a quasi-local Hamiltonian; cf.[2] for an excellent detailed survey of the current state of the art.In this paper, we first examine and comment on the approach to the definition of energy of a finite region of spacetime based on the Hamiltonian formulation of GR. This is essentially based on a localization of the approach taken by ADM [1] and Regge-Teitelboim [3], keeping careful track of the boundary terms that arise in a Hamiltonian or Hamilton-Jacobi analysis. This approach was initiated and pioneered by Brown-York (BY) [4]. To keep the discussion focused on the central issue, we only consider the gravitational field, (so other matter fields are set to zero); in addition, we consider only the energy and not related concepts such as linear and angular momentum, although this could be done without undue difficulty. Finally most all of the discussion below applies also to more recent modifications of the BY approach by several authors, cf.[5]-[8]; however again for clarity and simplicity we focus on the Brown-York Hamiltonian and leave it to the reader to extend the analysis to the more recent alternatives.We first recall the set-up. Let M be a spacetime region, topologically of the form I × Σ, with I = [0, 1] parametrizing time and Σ a compact 3-manifold with boundary S; typically S = ∂Σ is a 2-sphere and Σ a 3-ball. The boundary ∂M of M is a union of two spatial hypersurfaces Σ 0 ∪ Σ 1 and the timelike boundary T = I × ∂S = ∪S t . These boundaries meet at the seams or corners S 0 and S 1 . The Einstein-Hilbert action is then given by (setting 8πG = 1),where g is a smooth Lorentz metric on M . The Hamiltonian in GR plays two important but a priori distin...

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“…Although this system has not yet been implemented computationally, it has spurred the investigation of simpler treatments of Einstein equations which give rise to a constraint preserving IBVP under various restrictions [63, 236, 64, 100, 128, 207, 162]. See [213] for a review.…”

Section: Cauchy-characteristic Matchingmentioning

confidence: 99%

“…In anticipation of this, I will not attempt to keep this subject up to date except for material of direct relevance to CCM. See [213] for an independent review of boundary conditions that have been used in numerical relativity.…”

Section: Introductionmentioning

confidence: 99%

Characteristic Evolution and Matching

Winicour

2009

Living Rev. Relativ.

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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.

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Absorbing boundary conditions for Einstein's field equations

Cited by 22 publications

References 72 publications

Well-posed initial-boundary value problem for the harmonic Einstein equations using energy estimates

Well-posed initial-boundary value problem for the harmonic Einstein equations using energy estimates

Quasilocal Hamiltonians in general relativity

Characteristic Evolution and Matching

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