Time-symmetric initial data for binary black holes in numerical relativity

Abstract

We look for physically realistic initial data in numerical relativity which are in agreement with post-Newtonian approximations. We propose a particular solution of the time-symmetric constraint equation, appropriate to two momentarily static black holes, in the form of a conformal decomposition of the spatial metric. This solution is isometric to the post-Newtonian metric up to the 2PN order. It represents a non-linear deformation of the solution of Brill and Lindquist, i.e. an asymptotically flat region is connected to two asymptotically flat (in a certain weak sense)sheets, that are the images of the two singularities through appropriate inversion transformations.The total ADM mass M as well as the individual masses m 1 and m 2 (when they exist) are computed by surface integrals performed at infinity. Using second order perturbation theory on the Brill-Lindquist background, we prove that the binary's interacting mass-energy M − m 1 − m 2 is well-defined at the 2PN order and in agreement with the known post-Newtonian result. 04.80.Nn, 97.60.Jd, 97.60.Lf * Electronic address: blanchet@iap.fr 1 I. MOTIVATION AND RELATION TO OTHER WORKSThe numerical computation of the collision of two black holes is of paramount importance for the observation of gravitational waves by the network of laser-interferometric detectors.When investigating this problem the ten Einstein field equations are separated into: (i) four constraint equations that are to be satisfied by some initial data given on an initial threedimensional Cauchy hypersurface; (ii) six hyperbolic-like equations describing the dynamical evolution of the gravitational field on neighbouring hypersurfaces. The Bianchi identities guarantee that the constraint equations are satisfied on neighbouring hypersurfaces if they are on the initial hypersurface. There are infinitely many ways that the initial data can be chosen to represent the starting state of the evolution of black holes. It is widely admitted that the problem of choosing physically realistic initial conditions for the collision of two black holes has not yet been solved. There has been a lot of concern in the litterature [1] for knowing what would really motivate physically a particular choice of initial data.Let us consider the problem of time-symmetric initial data, which are physically appropriate to two momentarily static black holes, i.e. with zero initial velocities. The dynamical evolution of time-symmetric data describes the subsequent head-on collision of the two black holes. In this situation the constraint equations reduce to the Hamiltonian or scalar contraint equation R = 0 (in vacuum -the case appropriate to black holes), with R being the three-dimensional scalar curvature. Considering as usual [2,3,4] a conformal decomposition of the spatial metric (spatial indices i, j, · · · = 1, 2, 3),where γ ij is the physical metric and γ ij denotes the conformal (unphysical) metric, we obtain the Lichnerowicz [2] equation, which is an elliptic-type equation to be satisfied by the conformal fa...

Post-Newtonian freely specifiable initial data for binary black holes in numerical relativity

“…In contrast to the above methods, Blanchet [1] presents an alternative set of PN moti-vated free data for the time-symmetric constraints, corresponding to two Schwarzschild black holes momentarily at rest 5 . This work proposes a particular solution to the time-symmetric constraints in the form of the conformal metric,γ ij .…”

“…where we refer to the notation used in [1,56] Similarly, the transverse-traceless velocity-dependent terms, …”

“…see Section III A in [1]. Therefore, we do not allow the addition of such terms, here and henceforth, to our solutions for Poisson-type equations.…”

“…(3.7) in [1]. In contrast, the linear-order terms in (4.23) in both the perturbation metric, h ij (4.19), and Brill-Lindquistlike conformal factor, ψ (4.13), include terms due to the velocity of the black holes.…”

“…respectively, and refer to the results given in Section III B in [1] for solving the timesymmetric components in (4.23). Substitution of (4.19) and (4.20) into (4.23), allows us to determine the exact solution for κ (1) , (1) .…”