Excision boundary conditions for black-hole initial data

Abstract

We define and extensively test a set of boundary conditions that can be applied at black hole excision surfaces when the Hamiltonian and momentum constraints of general relativity are solved within the conformal thin-sandwich formalism. These boundary conditions have been designed to result in black holes that are in quasiequilibrium and are completely general in the sense that they can be applied with any conformal three-geometry and slicing condition. Furthermore, we show that they retain precisely the freedom to specify an arbitrary spin on each black hole. Interestingly, we have been unable to find a boundary condition on the lapse that can be derived from a quasiequilibrium condition. Rather, we find evidence that the lapse boundary condition is part of the initial temporal gauge choice. To test these boundary conditions, we have extensively explored the case of a single black hole and the case of a binary system of equal-mass black holes, including the computation of quasi-circular orbits and the determination of the inner-most stable circular orbit. Our tests show that the boundary conditions work well.

Circular orbits and spin in black-hole initial data

“…[5] that has not been adequately verified relates to the spins of the individual black holes. In Ref.…”

“…A very effective approach for constructing numerical black-hole binary initial data has been developed and explored by two of the authors [5] (see also Refs. [6 -8]).…”

“…In Ref. [5], the authors focused attention on two specific, limiting cases of binary initial data: corotating black holes and irrotational (nonspinning) black holes. These numerical initial-data solutions were compared against previous numerical results [ 8,9] and to analytic post-Newtonian estimates [10,11] for binaries in circular orbits and appear to give the best agreement yet between numerical and analytic models of close black-hole binaries in circular orbits.…”

“…In Ref. [5], the spin of each black hole is fixed by a particular choice of boundary conditions applied at the surface of the black hole. For the case of corotating black holes, the choice of boundary conditions is unambiguous.…”

Approximate initial data for binary black holes

“…[7][8][9][10]; see also [11][12][13][14] for alternative ways of constructing binary black hole initial data.) There is general consensus that the latter formalism is better suited for the construction of quasiequilibrium data (but see [15] for a very promising alternative approach), even though, at least in terms of global quantities, both formalisms lead to very similar results for configurations outside the innermost stable circular orbit (see, e.g., [9,16]). …”