Effective one-body approach to general relativistic two-body dynamics

TL;DR

The paper develops an effective-one-body (EOB) framework that maps the general relativistic two-body problem onto a test-particle motion in a ν-deformed Schwarzschild metric, enabling a nonperturbative resummation of the 2PN dynamics. By matching real and effective energy levels and enforcing a physically natural energy mapping, it derives a remarkably simple effective metric with A(R)=1 - 2GM/(c^2R) + 2ν(GM/(c^2R))^3 + ... and D(R)=1 - 6ν(GM/(c^2R))^2 + ..., capturing strong-field effects through the symmetric mass ratio ν. The analysis yields ISCO predictions for comparable-mass binaries that are more tightly bound than the Schwarzschild test-mass case, with concrete real-energy and frequency estimates after appropriate mapping, and provides explicit canonical transformations linking real and effective phase-space variables. The framework offers a practical bridge between PN theory and numerical relativity, with clear paths to include radiation reaction and to extend to spins and higher PN orders, enhancing gravitational-wave modeling for detectors like LIGO/VIRGO.

Abstract

We map the general relativistic two-body problem onto that of a test particle moving in an effective external metric. This effective-one-body approach defines, in a non-perturbative manner, the late dynamical evolution of a coalescing binary system of compact objects. The transition from the adiabatic inspiral, driven by gravitational radiation damping, to an unstable plunge, induced by strong spacetime curvature, is predicted to occur for orbits more tightly bound than the innermost stable circular orbit in a Schwarzschild metric of mass M = m1 + m2. The binding energy, angular momentum and orbital frequency of the innermost stable circular orbit for the time-symmetric two-body problem are determined as a function of the mass ratio.

Figures (5)

• Figure 1: The effective radial potential $W_j(R)$ (at the 2PN level and for $\nu =1/4$) versus the dimensionless radial variable $c^2R/(GM)$ for three different values of the dimensionless angular momentum $j = c {\cal J}_{\rm real}/(GM\mu)$. Note that the effective radial potential tends to one for $R \rightarrow \infty$. The stable circular orbits are located at the minima of the effective potential and are indicated by heavy black circles. The innermost stable circular orbit corresponds to the critical value $j_*$. In the case of the $j_1$ curve the orbit of a particle with energy $E_0^R = \widehat{\cal E}_0$ is an elliptical rosette.
• Figure 2: Variation with $\nu$ (at the 2PN level) of the ISCO values of the real non-relativistic energy $E_{\rm real} \equiv \widehat{\cal E}_{\rm real}^{\rm NR} \equiv ({\cal E}_{\rm real} -M\,c^2)/\mu c^2$ (on the left) and of the real angular momentum ${j} \equiv c{\cal J}_{\rm real}/GM\mu$ (on the right), divided by the corresponding Schwarzschild values $|E_{\rm S}| \equiv |\widehat{\cal E}_{\rm S}^{\rm NR}| = 1 - \sqrt{8/9} \simeq 0.05719$ and ${j}_{\rm S} = \sqrt{12}$, respectively.
• Figure 3: ISCO values (at the 2PN level) of the quantity $z =(GM \omega_{\rm real}/c^3)^{-2/3}$, divided by the Schwarzschild value $z_S = 6$, versus $\nu$.
• Figure 4: ISCO values (for $\nu = 1/4$) of the real non-relativistic energy $E \equiv \widehat{\cal E}_{\rm real}^{\rm NR}$, divided by the corresponding Schwarzschild value $E_S \equiv \widehat{\cal E}_{\rm S}^{\rm NR}$, versus $z/z_S$. On the left we have compared our predictions at the 1PN level $({\blacksquare})$ and 2PN level $(\blacklozenge)$ with the results obtained in [21] $(\blacktriangleright)$ and [22] $(\blacktriangleleft)$. The $(\ast)$ indicates the Schwarzschild predictions. The right panel is a magnification of the part of the left one in which we analyze the robustness of our method by exhibiting the points $(\bullet)$ obtained by introducing in the effective metric reasonable 3PN and 4PN contributions: $(a'_4 , a'_5) = (\pm 4 , -4)$, $(\pm 4 , 0)$ and $(\pm 4 , +4)$ in the notation of Eq. (\ref{['eq5.31']}).
• Figure 5: Inspiraling circular orbits in $(q^\prime, p^\prime)$ coordinates including radiation reaction effects for $\nu = 0.1$ (left panel) and $\nu = 1/4$ (right panel). The location of the ISCO and of the horizon are indicated.