5.3 The outer Cauchy boundary in numerical relativity

A special issue arising in general relativity is whether the boundary conditions on an outer worldtube preserve the constraints. It is typical of hyperbolic reductions of the Einstein equations that the Hamiltonian and momentum constraints propagate in a domain of dependence dictated by the characteristics. Unless the boundary conditions enforce these constraints, they will be violated outside the domain of dependence of the initial Cauchy hypersurface. This issue of a constraint-preserving initial boundary value problem has only recently been addressed [232]. The first fully nonlinear treatment of a well-posed constraint preserving formulation of the Einstein initial-boundary value problem (IBVP) has subsequently been given by Friedrich and Nagy [98Jump To The Next Citation Point]. Their treatment is based upon a frame formulation in which the evolution variables are the tetrad, connection coefficients and Weyl curvature. 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 [63236Jump To The Next Citation Point64100128207162Jump To The Next Citation Point]. See [213] for a review.

Well-posedness of the IBVP, in addition to constraint preservation, is a necessary condition for computational success. This is particularly cogent for dealing with waveform extraction in the simulation of black holes by BSSN or harmonic formulations. There is no well-posed outer boundary theory for the BSSN formulation and the strategy is to place the boundary out far enough so that it does no harm. The harmonic formulation has a simpler mathematical structure as a system of coupled quasilinear wave equations which is more amenable to an analytic treatment.

Standard harmonic coordinates satisfy the covariant wave equation

1 √ --- Γ α = − □x α = − √--- ∂βγαβ = 0 , γαβ = − ggαβ. (44 ) − g
(This can easily be generalized to include gauge forcing [97], whereby Γ α = fα(xβ,gβγ). For simplicity of discussion, I will set α Γ = 0, although gauge forcing is an essential tool in simulating black holes [195Jump To The Next Citation Point].)

When Γ α = 0, Einstein’s equations reduce to the 10 quasilinear wave equations

μν αβ αβ g ∂μ∂νγ = S , (45 )
where Sαβ does not enter the principle part and vanishes in the linearized approximation. Straightforward techniques can be applied to formulate a well-posed IBVP for the system (45View Equation). The catch is that Einstein’s equations are not necessarily satisfied unless the constraints are also satisfied.

In the harmonic formalism, the constraints can be reduced to the harmonic coordinate conditions (44View Equation). For the resulting IBVP to be constraint preserving, these harmonic conditions must be built into the boundary condition. Numerous early attempts to accomplish this failed because (44View Equation) contains derivatives tangent to the boundary, which do not fit into the standard methods for obtaining the necessary energy estimates. The use of pseudo-differential techniques developed for similar problems in elasticity theory has led to the first well-posed formulation for the general IBVP for the harmonic Einstein equations [162]. Subsequently, these results were also obtained using standard energy estimates by means of a novel, non-conventional choice of the energy for the harmonic system [160]. Furthermore, the allowed boundary conditions include those of the Sommerfeld type which are nonreflecting in the sense that the boundary data for μν g falls off as 4 O (1∕R ) as the boundary radius R → ∞ [161]

A Cauchy evolution code, the Abigel code, has been based upon a discretized of this well-posed harmonic IBVP [236Jump To The Next Citation Point]. The code was tested to be stable, convergent and constraint preserving in the nonlinear regime [16]. A linearized version of the Abigel code has been used to successfully carry out CCM (see Section 5.8).

In the present harmonic codes used to simulate the binary black holes, the best that can be done is to impose a constraint preserving boundary condition for which homogeneous boundary data, i.e. zero boundary values, is a good approximation. One proposal of this type [98] is a boundary condition that requires the Newman–Penrose [179] Weyl tensor component Ψ0 to vanish. In the limit that the outer boundary goes to infinity this outer boundary condition becomes exact. In the present state of the art of black hole simulations, this approach comes closest to a satisfactory treatment of the outer boundary [205].

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