Binary black hole initial data for numerical general relativity based on post-Newtonian data

TL;DR

This work establishes a bridge between post-Newtonian theory and full numerical relativity by deriving fully relativistic initial data for binary black holes from post-2-Newtonian expressions in the ADMTT gauge. Using the York-Lichnerowicz conformal decomposition and a generalized puncture method, the authors project PN data onto the GR constraint manifold, addressing singularities near the holes and ambiguities in the mapping through an extended conformal rescaling. Numerical experiments demonstrate convergence of the elliptic solve and show that, with an extended York procedure (parameterized by $q$), the ADM mass tracks the PN energy in the regime where PN is valid and yields physically reasonable behavior near the ISCO. This approach provides a practical pathway to astrophysically realistic BBH initial data for evolutions and waveform generation, with future work aimed at including spins, higher PN orders, and globally valid TT data for improved accuracy. The method thus offers a concrete mechanism to connect early inspiral PN descriptions to nonlinear GR dynamics and gravitational-wave predictions.

Abstract

With the goal of taking a step toward the construction of astrophysically realistic initial data for numerical simulations of black holes, we for the first time derive a family of fully general relativistic initial data based on post-2-Newtonian expansions of the 3-metric and extrinsic curvature without spin. It is expected that such initial data provide a direct connection with the early inspiral phase of the binary system. We discuss a straightforward numerical implementation, which is based on a generalized puncture method. Furthermore, we suggest a method to address some of the inherent ambiguity in mapping post-Newtonian data onto a solution of the general relativistic constraints.

Figures (8)

• Figure 1: Hamiltonian constraint violation for a black hole separation of $r_{12}=8M$. The Hamiltonian constraint of pure PN data is much larger than the Hamiltonian constraint after solving (i.e. applying the the York procedure). We numerically solve for three different resolutions $h$. The inset is a blow up of the central region, which shows that our numerical scheme is second order convergent as expected.
• Figure 2: The momentum constraint for a separation of $r_{12}=8M$. We observe second order convergence in the resolution $h$ after solving. The momentum constraint violation of pure PN data is larger than after solving.
• Figure 3: The solutions of $u$ and $W^x$ along the y-axis for a black hole separation of $r_{12}=8M$. For comparison we also show $\psi_{PN}$, which diverges at $y=\pm 4$.
• Figure 4: Components of the 3-metric and extrinsic curvature for a black hole separation of $r_{12}=8M$. The data are shown before (dashed lines) and after applying the York procedure (solid lines). The components of the 3-metric change on the order of $\sim 1\%$.
• Figure 5: PN energy of Eq. (\ref{['PN_Etot']}) and ADM masses before and after solving (i.e. applying the York procedure) versus coordinate separation $r_{12}$ along the PN inspiral sequence. The data here were computed by keeping all momentum terms in $\psi_{PN}$, without consistently dropping higher order terms. In this case the ADM mass of pure PN data does not agree well with the PN energy. The ADM mass after solving (with $q=0.0$) increases on the order of $\sim 1\%$, when compared to the ADM mass of pure PN data. Furthermore the ADM mass after solving increases with decreasing separation, which is physically not acceptable. For comparison we also show the ADM mass of two puncture black holes along the PN sequence with constant bare masses, which show a similar increase in ADM mass.