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

We consider Detweiler’s redshift variable $z$ for a nonspinning mass $m_1$ in circular motion (with orbital frequency $\mathrm{\Omega }$) 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 $z(m_2\mathrm{\Omega },m_1/m_2)$ 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, $6\mathrm{PN},\dots$, contributions to the $O({\nu }^2)$ contribution to the EOB radial potential $A(u;\nu )$]. We apply our results to the second-self-force-order contribution to the frequency shift of the last stable orbit. 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.