Binary black holes in circular orbits. I. A global spacetime approach

Abstract

We present a new approach to the problem of binary black holes in the pre-coalescence stage, i.e. when the notion of orbit has still some meaning. Contrary to previous numerical treatments which are based on the initial value formulation of general relativity on a (3-dimensional) spacelike hypersurface, our approach deals with the full (4-dimensional) spacetime. This permits a rigorous definition of the orbital angular velocity. Neglecting the gravitational radiation reaction, we assume that the black holes move on closed circular orbits, which amounts to endowing the spacetime with a helical Killing vector. We discuss the choice of the spacetime manifold, the desired properties of the spacetime metric, as well as the choice of the rotation state for the black holes. As a simplifying assumption, the space 3-metric is approximated by a conformally flat one. The problem is then reduced to the resolution of five of the ten Einstein equations, which are derived here, as well as the boundary conditions on the black hole surfaces and at spatial infinity. We exhibit the remaining five Einstein equations and propose to use them to evaluate the error induced by the conformal flatness approximation. The orbital angular velocity of the system is computed through a requirement which reduces to the classical virial theorem at the Newtonian limit. PACS number(s): 04.70.Bw, 97.60.Lf, We dedicate this work to the memory of our dear friend Jean-Alain Marck. I. BACKGROUND AND MOTIVATIONBinary black holes have been the subject of numerous studies in the past two decades, both from the analytical and numerical point of view. These studies are motivated by the fact that the coalescence of two black holes is expected to be one of the strongest sources of gravitational waves detectable by the interferometric detectors LIGO, GEO600, TAMA300 and VIRGO, currently coming on-line [1].From the analytical point of view, the most recent works are based on the post-Newtonian formalism (see e.g. Ref.[2] for a review) or on the effective one-body approach developed by . In these works, the black holes are treated as point mass particles 1 , which is a very good approximation when the black holes are far apart. For closer configurations, one may turn instead to some numerical approach. The numerical studies can be divided in two classes: (i) the initial value problem for two black holes (see Ref.[7] for a review) and (ii) the time evolution of the initial data (see Ref.[8] for a review and Refs. [9][10][11] for recent results). One of the major problems in this respect is to get physically relevant initial data. Indeed, initial data representing two black holes have been obtained long ago by Misner [12] and Lindquist [13], as well as Brill and Lindquist [14] (see also Ref. [15] or Appendices A and B of Ref. [16]). However these solutions correspond to two momentarily static black holes and are therefore far from representing some stage in the evolution of an isolated binary black hole in our universe. Based on the seminal work of Bowen and Yo...

Circular orbits and spin in black-hole initial data

“…An answer to the first question was proposed by Gourgoulhon et al [7] where they made the ansatz that 0 be chosen so that the ADM mass [13] and the Komar mass [12] agree. We call this the ''Komar-mass ansatz,'' and when it is applied, we refer to it as the ''Komar-mass condition.''…”

“…Refs. [7,29] and references therein). These arguments show that a system in quasiequilibrium necessarily satisfies the Komar-mass ansatz.…”

“…The boundary condition on the shift is chosen so that the time coordinate, t n , is helical and tracks the rotation of the system [5][6][7]24,25]. If we wish to consider systems with one or more black holes, and if we excise the interior of the black hole to avoid difficulties with singularities, then we must also impose boundary conditions on the excision surfaces.…”

“…In order to construct quasicircular orbits (as opposed to general elliptical orbits), Gourgoulhon et al [7] proposed that the data must satisfy a simple condition: the Komar [12] and the ADM [13] masses must agree if the orbits are quasicircular. The Komar mass is only a reasonable definition of the total mass of a system if the system is stationary.…”

Approximate initial data for binary black holes

“…[3][4][5][6], see also below), while most more recent calculations solve the constraint equations in the conformal thin-sandwich formalism (e.g. [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]).…”