Conservative second-order gravitational self-force on circular orbits and the effective one-body formalism

TL;DR

This paper derives an exact relation for the redshift variable in binary systems using EOB theory, incorporating high-order PN knowledge to decompose second-order self-force effects and analyze their behavior near the light ring.

Contribution

It combines EOB theory with the first law of binary dynamics to express the redshift in terms of the EOB potential and decomposes second-order self-force effects using recent high-PN results.

Findings

Decomposition of second-self-force-order contributions into known and unknown parts.

Identification of singular behaviors near the light ring.

Framework for extracting dynamical information from second-self-force computations.

Abstract

We consider Detweiler's redshift variable $z$ for a nonspinning mass $m_1$ in circular motion (with orbital frequency $\Omega$) around a nonspinning mass $m_2$. We show how the combination of effective-one-body (EOB) theory with the first law of binary dynamics allows one to derive a simple, exact expression for the functional dependence of $z$ on the (gauge-invariant) EOB gravitational potential $u=(m_1+m_2)/R$. We then use the recently obtained high-post-Newtonian(PN)-order knowledge of the main EOB radial potential $A(u ; \nu)$ [where $\nu= m_1 m_2/(m_1+m_2)^2$] to decompose the second-self-force-order contribution to the function $z(m_2 \Omega, m_1/m_2)$ into a known part (which goes beyond the 4PN level in including the 5PN logarithmic term, and the 5.5PN contribution), and an unknown one [depending on the yet unknown, 5PN, 6PN, $\ldots$, contributions to the $O(\nu^2)$ contribution to the EOB radial potential $A(u ; \nu)$]. We indicate the expected singular behaviors, near the lightring, of the second-self-force-order contributions to both the redshift and the EOB $A$ potential. Our results should help both in extracting information of direct dynamical significance from ongoing second-self-force-order computations, and in parametrizing their global strong-field behaviors. We also advocate computing second-self-force-order conservative quantities by iterating the time-symmetric Green-function in the background spacetime.