Coalescence of two spinning black holes: An effective one-body approach

Abstract

We generalize to the case of spinning black holes a recently introduced ``effective one-body'' approach to the general relativistic dynamics of binary systems. We show how to approximately map the conservative part of the third post-Newtonian (3PN) dynamics of two spinning black holes of masses ${m}_{1},$ ${m}_{2}$ and spins ${\mathit{S}}_{1},$ ${\mathit{S}}_{2}$ onto the dynamics of a non-spinning particle of mass $\ensuremath{\mu}\ensuremath{\equiv}{m}_{1}{m}_{2}{/(m}_{1}{+m}_{2})$ in a certain effective metric ${g}_{\ensuremath{\mu}\ensuremath{\nu}}^{\mathrm{eff}}{(x}^{\ensuremath{\lambda}};M,\ensuremath{\nu},\mathit{a})$ which can be viewed either as a spin deformation [with the deformation parameter $\mathit{a}\ensuremath{\equiv}{\mathit{S}}_{\mathrm{eff}}/M]$ of the recently constructed 3PN effective metric ${g}_{\ensuremath{\mu}\ensuremath{\nu}}^{\mathrm{eff}}{(x}^{\ensuremath{\lambda}};M,\ensuremath{\nu}),$ or as a $\ensuremath{\nu}$ deformation [with the comparable-mass deformation parameter $\ensuremath{\nu}\ensuremath{\equiv}{m}_{1}{m}_{2}{/(m}_{1}{+m}_{2}{)}^{2}]$ of a Kerr metric of mass $M\ensuremath{\equiv}{m}_{1}{+m}_{2}$ and (effective) spin ${\mathit{S}}_{\mathrm{eff}}\ensuremath{\equiv}[{1+3m}_{2}{/(4m}_{1})]{\mathit{S}}_{1}+[{1+3m}_{1}{/(4m}_{2})]{\mathit{S}}_{2}.$ The combination of the effective one-body approach, and of a Pad\'e definition of the crucial effective radial functions, is shown to define a dynamics with much improved post-Newtonian convergence properties, even for black hole separations of the order of $6 {GM/c}^{2}.$ The complete (conservative) phase-space evolution equations of binary spinning black hole systems are written down and their exact and approximate first integrals are discussed. This leads to the approximate existence of a two-parameter family of ``spherical orbits'' (with constant radius), and of a corresponding one-parameter family of ``last stable spherical orbits'' (LSSO). These orbits are of special interest for forthcoming LIGO-VIRGO-GEO gravitational wave observations. The binding energy and total angular momentum of LSSO's are studied in some detail. It is argued that for most (but not all) of the parameter space of two spinning holes the approximate (leading-order) effective one-body approach introduced here gives a reliable analytical tool for describing the dynamics of the last orbits before coalescence. This tool predicts, in a quantitative way, how certain spin orientations increase the binding energy of the LSSO. This leads to a detection bias, in LIGO-VIRGO-GEO observations, favoring spinning black hole systems, and makes it urgent to complete the conservative effective one-body dynamics given here by adding (resummed) radiation reaction effects, and by constructing gravitational waveform templates that include spin effects. Finally, our approach predicts that the spin of the final hole formed by the coalescence of two arbitrarily spinning holes never approaches extremality.