Last Orbits of Binary Black Holes * the Isco (for Innermost Stable Circular Orbit). 2. Numerical Solution

Received (Day Month Year) Binary black hole systems in the pre-coalescence stage are numerically constructed by demanding that the associated spacetime admits a helical Killing vector. Comparison with third order post-Newtonian calculations indicates a rather good agreement until the innermost stable circular orbit. 1. Quasi-equilibrium configurations of binary black holes 1.1. Helical Killing vector Binary black holes are considered as one of the most promising sources for the gravitational wave interferometric detectors LIGO, VIRGO, GEO600 and TAMA currently under construction, as well as for the space interferometer LISA in project. From the theoretical point of view, computing the final inspiral and merger of binary black holes has proved to be a very difficult task. However, when the black holes are far enough so that the radiation reaction timescale is large as compared to the dynamical timescale, the orbits can be approximated as closed ones. Moreover, due to the accumulated effects of the reaction to gravitational radiation, these orbits can be considered as circular. In such a case (closed circular orbits), the spacetime possesses a one-parameter symmetry group, whose integral curves are helices. The associated Killing vector is called the helical Killing vector. In full general relativity, requiring the helical symmetry for a binary system imply equal amounts of outgoing and incoming grav-* This talk is dedicated to the memory of our friend and colleague Jean-Alain Marck. itational radiation. 1 The spacetime cannot therefore not be asymptotically flat. 2 However, there are two interesting cases where helical symmetry and asymptotic flatness are compatible: (i) in the post-Newtonian (PN) approximation, up to the second order and (ii) the Isenberg-Wilson-Mathews (IWM) approximation of general relativity (see e.g. Sec. IV.C of Ref. 3), which assumes that the spacetime can be foliated by a family of conformally flat spacelike hypersurfaces and reduces the Einstein equations to five elliptic equations. Both approximations (i) and (ii) do not allow for gravitational radiation, which explains why they permit asymptoti-cally flat spacetimes. They are valid approximations to general relativity when the gravitational radiation content of spacetime does not play any important dynamical role. Note that the IWM formulation contains the first order PN one. In our first study of the binary black hole problem, 4,5 we have used the helical symmetry along with the IWM approximation 1.2. Rigid rotation Regarding the rotation state of the black holes, we have chosen each black hole to be corotating (synchronized binary), so …