Test black holes, scattering amplitudes and perturbations of Kerr spacetime

TL;DR

The paper interrogates the proposed link between quantum higher-spin amplitudes and classical spinning black-hole dynamics, testing whether the aligned-spin two-body problem at $igO(G^2)$ can be reconstructed from a spinning test-black-hole limit via a simple mapping. It develops a minimal effective test-BH framework with a dynamical mass function and tidal couplings, expresses the conservative dynamics through a canonical Hamiltonian, and translates scattering data into gauge-invariant circular-orbit observables. By combining the first-law relations for spinning binaries with first-order self-force results in Kerr spacetimes, the authors confront the conjectured $igO(G^2 a^4)$ contributions and find that the linear-in-mass-ratio self-force data fixes the three Wilson coefficients to the GOV values, confirming the amplitude-based predictions through linear order in the mass ratio. The work thereby strengthens the bridge between on-shell amplitude methods and classical GR, providing high-order spin-dependent constraints at 4.5PN and 5PN, and outlining how perturbations of Kerr spacetime can further refine the structure of spinning-test-BH couplings. These results have implications for precision gravitational-wave modeling in extreme-mass-ratio regimes and for validating amplitude-derived effective descriptions of spinning binary dynamics.

Abstract

It has been suggested that amplitudes for quantum higher-spin massive particles exchanging gravitons lead, via a classical limit, to results for scattering of spinning black holes in general relativity, when the massive particles are in a certain way minimally coupled to gravity. Such limits of such amplitudes suggest, at least at lower orders in spin, up to second order in the gravitational constant $G$, that the classical aligned-spin scattering function for an arbitrary-mass-ratio two-spinning-black-hole system can be obtained by a simple kinematical mapping from that for a spinning test black hole scattering off a stationary background Kerr black hole. Here we test these suggestions, at orders beyond the reach of the post-Newtonian and post-Minkowskian results used in their initial partial verifications, by confronting them with results from general-relativistic "self-force" calculations of the linear perturbations of a Kerr spacetime sourced by a small orbiting body, here considering only results for circular orbits in the equatorial plane. We translate between scattering and circular-orbit results by assuming the existence of a local-in-time canonical Hamiltonian governing the conservative dynamics of generic (bound and unbound) aligned-spin orbits, while employing the associated first law of spinning binary mechanics. We confirm, through linear order in the mass ratio, some previous conjectures which would begin to fill in the spin-dependent parts of the conservative dynamics for arbitrary-mass-ratio aligned-spin binary black holes at the fourth-and-a-half and fifth post-Newtonian orders.

Figures (1)

• Figure 1: The tree- and 1-loop-level structure of the massive spin-s particle-graviton on-shell scattering amplitude $\mathcal{A}$, where $q$ is the momentum transfer. When moving from the amplitude $\mathcal{A}$ to a classical scattering-angle function $\chi$, the momentum transfer $q$ of $\mathcal{A}$ translates into the impact parameter $b$ of $\chi$. For higher-spin particles the amplitude structure $\mathcal{A}=\mathcal{A}^\text{tree}+\mathcal{A}^\text{1-loop}+\mathcal{O}(G^3)$ suggests the mapping \ref{['EOBmap']}. To see this, we follow the arguments in Vines:2018gqi, and point out that the structure $\mathcal{A}^\text{1-loop}=\mathcal{A}^\text{1-loop}_\triangleleft+\mathcal{A}^\text{1-loop}_\triangleright$ yields the following decomposition of the scattering-angle function: $\chi[\mathcal{A}]=\chi[\mathcal{A}^\text{tree}]+\chi[\mathcal{A}^\text{1-loop}_\triangleleft]+\chi[\mathcal{A}^\text{1-loop}_\triangleright]+\mathcal{O}(G^3)$, where $\chi[\mathcal{A}^\text{1-loop}_\triangleleft]=E m_2 f_\triangleleft$ and $\chi[\mathcal{A}^\text{1-loop}_\triangleright]=E m_1 f_\triangleright$. Crucially, $f_\triangleright\equiv f_\triangleleft$, and $f_\triangleright$ is independent of both $m_1$ and $m_2$. Therefore, even in the extreme-mass-ratio limit (i.e., $m_1\rightarrow 0$), the full arbitrary mass-ratio information can be recovered. The validity of this decomposition is discussed in the text, and in Vines:2018gqi.