Spin-multipole effects in binary black holes and the test-body limit

TL;DR

The paper derives the leading-PN-order, all-orders-in-spin Hamiltonian for binary black holes and shows two deep connections to Kerr-spacetime dynamics. The even-spin sector maps to geodesic motion in a Kerr background with an effective spin a0, while the full dynamics correspond to a test black hole with spin in Kerr, enabling a compact, nonperturbative description of BBH motion. The results unify arbitrary-mass-ratio BBH dynamics with test-body Kerr limits across all spins and illuminate oblate-spheroidal geometric structure underlying the interactions. These insights have implications for effective-one-body modeling and tests of GR with strong spin effects in gravitational waves, and point to promising directions for extending the framework to higher PN orders.

Abstract

We discuss the effects of the black holes' spin-multipole structure in the orbital dynamics of binary black holes according to general relativity, focusing on the leading-post-Newtonian-order couplings at each order in an expansion in the black holes' spins. We first review previous widely confirmed results up through fourth order in spin, observe suggestive patterns therein, and discuss how the results can be extrapolated to all orders in spin with minimal information from the test-body limit. We then justify this extrapolation by providing a complete derivation within the post-Newtonian framework of a canonical Hamiltonian for a binary black hole, for generic orbits and spin orientations, which encompasses the leading post-Newtonian orders at all orders in spin. At the considered orders, the results reveal a precise equivalence between arbitrary-mass-ratio two-spinning-black-hole dynamics and the motion of a test black hole in a Kerr spacetime, as well as an intriguing relationship to geodesic motion in a Kerr spacetime.

Figures (3)

• Figure 1: Contributions to the two-body Hamiltonian in the PN-spin expansion, for arbitrary-mass-ratio binaries with spin-induced multipole moments (such as BBHs). The lower-right inset indicates the increments in the PN orders in moving right or diagonally down and left; c.f. (\ref{['schematicH']}). The first row has the nonspinning (point-mass) $S^0$ contributions, labelled by the order in the PN expansion: N for Newtonian, 1PN for relative order $\epsilon^1$, etc. For the spin contributions, the second row gives the linear-in-spin or spin-orbit (SO) parts, the third row gives the quadratic-in-spin parts, etc. LO stands for the leading-(PN-)order part (the $S^n$ contributions at the leading order in $\epsilon$), NLO stands for next-to-leading order, etc. Terms in red text are unknown. Terms in black text have been calculated, and confirmed by independent groups, as discussed at LOs below and as reviewed e.g. in Blanchet:2006 at higher orders, all except for (i) the recent NNLO S$^2$ calculations of Levi:Steinhoff:2015:3, and (ii) the underlined LO-S$^n$ terms with $n\ge 5$, which are presented here for BBHs. The recent 4PN results of Damour:2014jtaDamour:2015isaBernard:2015njpDamour:2016ablBernard:2016wrg have been recently also confirmed by independent calculations.
• Figure 2: Relationships between the coordinates (\ref{['coords']}) on flat 3-space, with $(r,\theta,\Phi)$ being oblate spheroidal coordinates with "ring-radius" $a_0=|\boldsymbol a_0|$, showing a quadrant of a plane containing the $Z$-axis. The surfaces of constant $r$ are oblate ellipsoids with foci on "the ring" $\rho^2=X^2+Y^2=a_0^2$ in the $Z=0$ plane, with a cross-section shown in red. The ring pierces this plane orthogonally at the $\otimes$ symbol, with its center at the origin. The locus $r=0$ is the disk $Z=0$, $\rho<a_0$ bounded by the ring. The surfaces of constant $\theta$ are half- one-sheeted hyperboloids with foci on the same ring, with a cross-section shown in blue; as $r\to\infty$, they asymptote to cones opening an angle $\theta$ from the +$Z$-axis. The locus $\theta=\pi/2$ is the plane $Z=0$ minus the disk $\rho<a_0$. The ring represents the "ring singularity" of an effective Kerr black hole with rescaled spin $\boldsymbol a_0$.
• Figure 3: Feynman diagrams of the leading-order gravito-electric and gravito-magnetic interactions.