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High-order post-Newtonian contributions to the two-body gravitational interaction potential from analytical gravitational self-force calculations

Donato Bini, Thibault Damour

TL;DR

This work advances analytical gravitational self-force methods to 6PN order for conservative two-body dynamics by computing Detweiler's gauge-invariant redshift h_{kk}^R(u) along circular orbits via Regge-Wheeler-Zerilli perturbation theory enhanced with Mano–Suzuki–Takasugi techniques. From h_{kk}^R(u), the authors derive the first-order GSF contribution a(u) to the EOB radial potential A(u;ν) and, subsequently, the binding-energy function E_B(x;ν), identifying new 5PN non-log, 5.5PN, and 6PN contributions (including logs) and revealing remarkable cancellations in ν-dependence. A key conceptual result is the emergence of a conservative 5.5PN term linked to second-order tail effects, interpreted through hereditary tail machinery and energy balance, with analytic confirmation and numerical cross-checks against Shah et al. These results sharpen the bridge between self-force calculations and PN-EOB descriptions, providing high-precision, gauge-invariant coefficients for strong-field binary dynamics and clarifying the origin of tail-driven conservative corrections. The findings underscore the utility of MST techniques in capturing near-zone/tail physics and illuminate PN convergence behavior near the light ring.

Abstract

We extend the analytical determination of the main radial potential describing (within the effective one-body formalism) the gravitational interaction of two bodies beyond the 4th post-Newtonian approximation recently obtained by us. This extension is done to linear order in the mass ratio by applying analytical gravitational self-force theory (for a particle in circular orbit around a Schwarzschild black hole) to Detweiler's gauge-invariant redshift variable. By using the version of black hole perturbation theory developed by Mano, Suzuki and Takasugi, we have pushed the analytical determination of the (linear in mass ratio) radial potential to the 6th post-Newtonian order (passing through 5 and 5.5 post-Newtonian terms). In principle, our analytical method can be extended to arbitrarily high post-Newtonian orders.

High-order post-Newtonian contributions to the two-body gravitational interaction potential from analytical gravitational self-force calculations

TL;DR

This work advances analytical gravitational self-force methods to 6PN order for conservative two-body dynamics by computing Detweiler's gauge-invariant redshift h_{kk}^R(u) along circular orbits via Regge-Wheeler-Zerilli perturbation theory enhanced with Mano–Suzuki–Takasugi techniques. From h_{kk}^R(u), the authors derive the first-order GSF contribution a(u) to the EOB radial potential A(u;ν) and, subsequently, the binding-energy function E_B(x;ν), identifying new 5PN non-log, 5.5PN, and 6PN contributions (including logs) and revealing remarkable cancellations in ν-dependence. A key conceptual result is the emergence of a conservative 5.5PN term linked to second-order tail effects, interpreted through hereditary tail machinery and energy balance, with analytic confirmation and numerical cross-checks against Shah et al. These results sharpen the bridge between self-force calculations and PN-EOB descriptions, providing high-precision, gauge-invariant coefficients for strong-field binary dynamics and clarifying the origin of tail-driven conservative corrections. The findings underscore the utility of MST techniques in capturing near-zone/tail physics and illuminate PN convergence behavior near the light ring.

Abstract

We extend the analytical determination of the main radial potential describing (within the effective one-body formalism) the gravitational interaction of two bodies beyond the 4th post-Newtonian approximation recently obtained by us. This extension is done to linear order in the mass ratio by applying analytical gravitational self-force theory (for a particle in circular orbit around a Schwarzschild black hole) to Detweiler's gauge-invariant redshift variable. By using the version of black hole perturbation theory developed by Mano, Suzuki and Takasugi, we have pushed the analytical determination of the (linear in mass ratio) radial potential to the 6th post-Newtonian order (passing through 5 and 5.5 post-Newtonian terms). In principle, our analytical method can be extended to arbitrarily high post-Newtonian orders.

Paper Structure

This paper contains 15 sections, 101 equations.

Table of Contents

  1. Introduction and summary of new results
  2. Analytical computation of conservative GSF effects along circular orbits
  3. General strategy for computing $h_{kk}^R$
  4. Solving in terms of their sources the radial functions $R_{lm\omega }^{\rm (odd)}$, $R_{lm\omega }^{\rm (even)}$ and $Z_{lm\omega} (r)$ entering the Regge-Wheeler-Zerilli metric perturbation
  5. Evaluating the even and odd multipolar pieces $h_{kk,lm}^{\rm (even/odd)}$ of $h_{kk}$ in terms of homogeneous solutions $X_{l\omega}^{\rm (in)}$, $\ X_{l\omega}^{\rm (up)}$ of the Regge-Wheeler equation
  6. Strategy for analytically computing the PN expansion of $h_{kk}^R(u)$
  7. Expanding the hypergeometric-series "in" and "up" homogeneous solutions in powers of $\eta := 1/c$
  8. Deriving sufficiently accurate homogeneous PN-expanded solutions for general multipole degree $l$
  9. Summing over $m$
  10. Subtracting $\tilde{D}_0$, and summing the series over the multipolar degree $l$
  11. 6PN-accurate computation of the EOB radial potential $A(u;\nu)$ at linear order in $\nu$
  12. Binding energy of binary systems in terms of the orbital frequency
  13. Possible effect of the second-order tail integral in the inner metric and value of the $5.5$ PN coefficient $a_{6.5}(\nu)$
  14. Conclusions
  15. Relation $\nu$ vs $l$ at higher orders in $\epsilon$