5.3 The outer Cauchy boundary in numerical relativity

A special issue arising in general relativity is whether the boundary conditions on an artificial 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 [282]. 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 [118]. 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 [74, 287Jump To The Next Citation Point, 75, 122, 150, 254Jump To The Next Citation Point, 192Jump To The Next Citation Point]. See [260, 250] for reviews.

The successful implementation of CCM for Einstein’s equations requires a well-posed initial-boundary value problem for the artificial outer boundary of the Cauchy evolution. This is particularly cogent for dealing with waveform extraction in the simulation of black holes by BSSN 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. (64 ) − g
This can easily be generalized to include gauge forcing [117], 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 [235Jump To The Next Citation Point].

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

gμν∂μ∂νγ αβ = Sαβ, (65 )
where αβ S 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 (65View 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 (64View 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 Equation (64View 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 of the IBVP for the harmonic Einstein equations [192Jump To The Next Citation Point]. Subsequently, well-posedness was also obtained using energy estimates by means of a novel, non-conventional choice of the energy for the harmonic system [189Jump To The Next Citation Point]. A Cauchy evolution code, the Abigel code, based upon a discretized version of these energy estimates was found to be stable, convergent and constraint preserving in nonlinear boundary tests [14]. These results were confirmed using an independent harmonic code developed at the Albert Einstein Institute [266]. A linearized version of the Abigel code has been used to successfully carry out CCM (see Section 5.8).

Given a well-posed IBVP, there is the additional complication of the correct specification of boundary data. Ideally, this data would be supplied by matching to a solution extending to infinity, e.g., by CCM. In the formulations of [192] and [189], the boundary conditions are of the Sommerfeld type for which homogeneous boundary data , i.e., zero boundary values, is a good approximation in the sense that the reflection coefficients for gravitational waves fall off as O (1∕R3) as the boundary radius R → ∞ [190]. A second differential order boundary condition based upon requiring the Newman–Penrose [216] Weyl tensor component ψ0 = 0 has also been shown to be well-posed by means of pseudo-differential techniques [254]. For this ψ0 condition, the reflection coefficients fall off at an addition power of 1 ∕R. In the present state of the art of black-hole simulations, the ψ0 condition comes closest to a satisfactory treatment of the outer boundary [252].

  Go to previous page Go up Go to next page