Gravitational spin-orbit coupling in binary systems, post-Minkowskian approximation and effective one-body theory

TL;DR

This work introduces a gauge-invariant framework based on scattering holonomy to extract spin-orbit coupling information in gravitational binaries at the first post-Minkowskian order. By relating the PM spin holonomy to the EOB spin dynamics in an anti-DJS gauge, the authors derive explicit closed-form expressions for the gyro-gravitomagnetic ratios $g_S^{1{ m PM}}$ and $g_{S_*}^{1{ m PM}}$ as functions of momentum and mass ratio, valid to all orders in $v/c$. The results reproduce known Kerr and PN limits, reveal a decay of both ratios in the ultrarelativistic regime for comparable masses, and illuminate issues related to self-force expansions. The approach sets the stage for higher-PM extensions (e.g., 2PM) and potential resummations that could improve modeling of strong-field binary dynamics and gravitational-wave generation.

Abstract

A novel approach for extracting gauge-invariant information about spin-orbit coupling in gravitationally interacting binary systems is introduced. This approach is based on the "scattering holonomy", i.e. the integration (from the infinite past to the infinite future) of the differential spin evolution along the two worldlines of a binary system in hyperboliclike motion. We apply this approach to the computation, at the first post-Minkowskian approximation (i.e. first order in $G$ and all orders in $v/c$), of the values of the two gyrogravitomagnetic ratios describing spin-orbit coupling in the Effective One-Body formalism. These gyrogravitomagnetic ratios are found to tend to zero in the ultrarelativistic limit.

Figures (3)

• Figure 1: Schematic representation of the definition of the (abstract) spatial spin vectors $S_a$ ($a=1,2)$ orthogonal to the time direction $U$ defining the c.m. frame. They are obtained from the covariant spin vectors $s_a$ (orthogonal to $p_a = m_a u_a + O(s^2)$) by the Lorentz boost that rotates $u_a$ into the local value of $U$ (drawn vertically), while leaving invariant the 2-plane orthogonal to $u_a$ and $U$. For ease of drawing, we have sketched the scattering geometry as if the interaction were repulsive rather than attractive, and we have represented orthogonality relations as if we were in an Euclidean space.
• Figure 2: Geometrical configuration (in Minkowski spacetime) underlying the computation of the scattering holonomy at the 1PM order. The 1PM metric generated by $\mathcal{L}_2$ is computed at the field point $x$ (which is then made to move all along $\mathcal{L}_1$). The point $x_2^\perp(x) \in \mathcal{L}_2$ is the foot of the perpendicular drawn from the field point $x$ to $\mathcal{L}_2$. The segment $b$ (oriented from $\mathcal{L}_2$ towards $\mathcal{L}_1$) denotes the four-vectorial impact parameter of $\mathcal{L}_1$ wrt $\mathcal{L}_2$. All the labelled angles ($\alpha, \beta,\gamma,\delta$) indicate orthogonal incidences (in the Minkowski sense).
• Figure 3: Our 1PM results, Eqs. \ref{['gsfin']}, \ref{['gssfin']}, for the EOB spin-orbit couplings $g_S^{\rm 1PM}({\mathbf p}^2,\nu)$ and $g_{S*}^{\rm 1PM}({\mathbf p}^2,\nu)$ are plotted as functions of ${\mathbf p}^2$ for the two limiting values of the symmetric mass-ratio $\nu=0$ and $\nu=\frac{1}{4}$.