Binary black holes in circular orbits. II. Numerical methods and first results

TL;DR

This work develops a novel numerical scheme to model corotating binary black holes in circular orbits by exploiting a helical Killing symmetry and a conformally flat spatial metric. The five coupled elliptic equations for $N$, $\Psi$, and $\vec{\beta}$ are solved using a two-throat, multi-domain spectral method, with $\Omega$ fixed by enforcing $M_{\rm ADM}=M_{\rm Komar}$ at infinity (virial condition). The approach accurately recovers known solutions (Schwarzschild, Kerr, Misner-Lindquist) and yields a sequence of equal-mass corotating binaries; the ISCO is located and found to align well with third-order post-Newtonian predictions, suggesting strong potential for generating accurate initial data and guiding gravitational-wave modeling. The results indicate that conformal flatness is not the sole source of discrepancy with PN results, highlighting the importance of incorporating the full set of Einstein equations (via the virial constraint) in determining orbital dynamics. Overall, this method provides a rigorous, testable pathway to build physically consistent initial data for binary black-hole evolutions and GW waveform modeling.

Abstract

We present the first results from a new method for computing spacetimes representing corotating binary black holes in circular orbits. The method is based on the assumption of exact equilibrium. It uses the standard 3+1 decomposition of Einstein equations and conformal flatness approximation for the 3-metric. Contrary to previous numerical approaches to this problem, we do not solve only the constraint equations but rather a set of five equations for the lapse function, the conformal factor and the shift vector. The orbital velocity is unambiguously determined by imposing that, at infinity, the metric behaves like the Schwarzschild one, a requirement which is equivalent to the virial theorem. The numerical scheme has been implemented using multi-domain spectral methods and passed numerous tests. A sequence of corotating black holes of equal mass is calculated. Defining the sequence by requiring that the ADM mass decrease is equal to the angular momentum decrease multiplied by the orbital angular velocity, it is found that the area of the apparent horizons is constant along the sequence. We also find a turning point in the ADM mass and angular momentum curves, which may be interpreted as an innermost stable circular orbit (ISCO). The values of the global quantities at the ISCO, especially the orbital velocity, are in much better agreement with those from third post-Newtonian calculations than with those resulting from previous numerical approaches.

Figures (20)

• Figure 1: Relative difference between the calculated and the analytical lapse $N$ with respect to the number of radial spectral coefficients for the Schwarzschild black hole. The circles denote the error in the innermost shell, the squares that in the other shell and the diamonds that in the external domain.
• Figure 2: Same as Fig. \ref{['f:erreur_n']} for the conformal factor $\Psi$.
• Figure 3: Relative norm of the regularization function given by Eq. (\ref{['e:regul']}) with respect to the Kerr parameter $J/M^2$, for various numbers $N_r\times N_{\theta}\times N_{\varphi}$ of collocation points.
• Figure 4: Same as Fig. \ref{['f:regul_kerr']} for the relative difference between the angular momentum calculated by means of Eq. (\ref{['e:J_inf_seul']}) and that by means of Eq. (\ref{['e:J_hor_seul']}).
• Figure 5: Same as Fig. \ref{['f:regul_kerr']} for the relative error on the virial theorem.