Towards absorbing outer boundaries in general relativity

Abstract

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.

Outer boundary conditions for Einstein's field equations in harmonic coordinates

“…As a consequence of the results of [11], the reflection coefficient can be approximated by forming the ratio of the NewmanPenrose scalars 0 and 4 at the outer boundary,…”

Approaching the Post-Newtonian Regime with Numerical Relativity: A Compact-Object Binary Simulation Spanning 350 Gravitational-Wave Cycles

“…Such a coupling might arise through enhanced reflections of the outgoing GW at the outer boundary. Our outgoing-wave boundary conditions [40] have the smallest reflection coefficient for sphericalharmonic modes with small l and a reflection coefficient of order unity when kR=l ≳ 1 [40], where k is the radial wave vector. With increasing jc c:m: j, the emitted GW will have increasing high-l content when decomposed on the outer boundary.…”

Testing outer boundary treatments for the Einstein equations

“…By evaluating 0 and 4 at the extraction radius of our test, we find that the ratio 0 / 4 agrees with their predicted ρ to leading order in 1/(kR). We note that the tetrad we use for wave extraction (appendix A.5) does not agree exactly with that of [18]. However, the tetrads do agree for the unperturbed Schwarzschild solution, so that the errors introduced into 0 and 4 due to our different choice of tetrad are second-order small in perturbation theory and hence the comparison with [18] is consistent.…”

“…We note that the tetrad we use for wave extraction (appendix A.5) does not agree exactly with that of [18]. However, the tetrads do agree for the unperturbed Schwarzschild solution, so that the errors introduced into 0 and 4 due to our different choice of tetrad are second-order small in perturbation theory and hence the comparison with [18] is consistent. For a numerical solution using our new CPBCs, we evaluate the Newman-Penrose scalars 0 (t) and 4 (t) on extraction spheres located 1.9M inside the outer boundary.…”