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 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;v) [where v = 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, ..., contributions to the O(v(2)) contribution to the EOB radial potential A(u;v)]. 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.