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

TL;DR

The paper develops a global spacetime framework for binary black holes in circular orbits by imposing a helical Killing vector and using a conformally flat 3-metric on a Misner-Lindquist two-sheet manifold. It reduces the problem to five Einstein equations for Ψ, N, and β, solved as coupled elliptic equations with carefully constructed boundary conditions at infinity and on the throats, and fixes the orbital velocity Ω via a relativistic virial condition linking monopole terms. A generalized Smarr relation and explicit expressions for ADM mass and angular momentum are derived, while an analysis of asymptotic behavior reveals the tension between exact helicoidal symmetry and asymptotic flatness due to a time-varying quadrupole, highlighting the quasi-stationary nature of the model. The approach provides a tractable, gauge-consistent way to generate initial data for binary black hole evolution and identifies key limitations and directions for future work, including non-corotating and irrotational configurations.

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 solving 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.

Figures (4)

• Figure 1: Construction of the spacetime manifold.
• Figure 2: Coordinate systems $(t,r_1,\theta_1,\varphi_1)$ and $(t,r_2,\theta_2,\varphi_2)$ on the spacetime manifold $\cal M$. Shown here is a $t={\rm const}$ section of $\cal M$, with the dimension in the $\theta$ direction suppressed, leaving only $(r_1,\varphi_1)$ or $(r_2,\varphi_2)$.
• Figure 3: Kruskal diagrams showing the slicing of the extended Schwarzschild spacetime by two families of maximal hypersurfaces; left: lapse function symmetric with respect to the isometry $I$; right: lapse function antisymmetric with respect to the isometry $I$. $R$ denotes the standard Schwarzschild radial coordinate and $r$ the isotropic one.
• Figure 4: Cartesian coordinate systems used for the computation of the asymptotic behavior of the shift vector.