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Gravitational scattering of two black holes at the fourth post-Newtonian approximation

Donato Bini, Thibault Damour

TL;DR

This paper computes the gauge-invariant scattering angle $χ$ for hyperbolic encounters of two spinning black holes at the $4^{\text{PN}}$ level for orbital effects and at next-to-next-to-leading order for spin effects, using the Effective-One-Body (EOB) framework. It splits the conservative dynamics into local and tail (nonlocal) parts, develops a time-localization method that extends to unbound motion, and validates tail contributions with both time-domain and Fourier-domain calculations. The authors also include NNLO linear-in-spin terms, perform a detailed large-$j$ (small $1/j$) expansion, check the Schwarzschild limit, and analyze the eccentricity dependence of tail effects across the full hyperbolic domain, including analytic fits for the eccentricity-dependent function $C(ε)$. Additionally, they connect the tail scattering correction to a time-integrated tail Hamiltonian $W^{\rm tail}$ and relate it to the gravitational-wave energy emission $ΔE_{\rm GW}$, providing a comprehensive description of high-energy scattering in GR with potential implications for gravitational-wave physics and fundamental tests of GR.

Abstract

We compute the (center-of-mass frame) scattering angle $χ$ of hyperboliclike encounters of two spinning black holes, at the fourth post-Newtonian approximation level for orbital effects, and at the next-to-next-to-leading order for spin-dependent effects. We find it convenient to compute the gauge-invariant scattering angle (expressed as a function of energy, orbital angular momentum and spins) by using the Effective-One-Body formalism. The contribution to scattering associated with nonlocal, tail effects is computed by generalizing to the case of unbound motions the method of time-localization of the action introduced in the case of (small-eccentricity) bound motions by Damour, Jaranowski and Schäfer [Phys.\ Rev.\ D {\bf 91}, no. 8, 084024 (2015)].

Gravitational scattering of two black holes at the fourth post-Newtonian approximation

TL;DR

This paper computes the gauge-invariant scattering angle for hyperbolic encounters of two spinning black holes at the level for orbital effects and at next-to-next-to-leading order for spin effects, using the Effective-One-Body (EOB) framework. It splits the conservative dynamics into local and tail (nonlocal) parts, develops a time-localization method that extends to unbound motion, and validates tail contributions with both time-domain and Fourier-domain calculations. The authors also include NNLO linear-in-spin terms, perform a detailed large- (small ) expansion, check the Schwarzschild limit, and analyze the eccentricity dependence of tail effects across the full hyperbolic domain, including analytic fits for the eccentricity-dependent function . Additionally, they connect the tail scattering correction to a time-integrated tail Hamiltonian and relate it to the gravitational-wave energy emission , providing a comprehensive description of high-energy scattering in GR with potential implications for gravitational-wave physics and fundamental tests of GR.

Abstract

We compute the (center-of-mass frame) scattering angle of hyperboliclike encounters of two spinning black holes, at the fourth post-Newtonian approximation level for orbital effects, and at the next-to-next-to-leading order for spin-dependent effects. We find it convenient to compute the gauge-invariant scattering angle (expressed as a function of energy, orbital angular momentum and spins) by using the Effective-One-Body formalism. The contribution to scattering associated with nonlocal, tail effects is computed by generalizing to the case of unbound motions the method of time-localization of the action introduced in the case of (small-eccentricity) bound motions by Damour, Jaranowski and Schäfer [Phys.\ Rev.\ D {\bf 91}, no. 8, 084024 (2015)].

Paper Structure

This paper contains 22 sections, 254 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and brief EOB reminders
  3. Contribution of the local conservative dynamics to the scattering function of nonspinning bodies
  4. EOB-derived integral expression for $\chi_{\rm loc}(E,L)$
  5. PN-expanding $\chi_{\rm loc}(E,L ; 1/c^2)$
  6. Explicit expression of the 4PN-level value of $\frac{1}{2}\chi_{\rm loc}(\bar{E},j; 1/c^2)$
  7. Computing the large-$j$ expansion of the logarithmic integral ${\mathcal{I}}_\chi$
  8. Final result for the large-$j$ expansion of the local 4PN contribution to $\chi$
  9. Schwarzschild limit
  10. On the two approaches to the tail contribution to the conservative dynamics
  11. Link between the tail contribution $\chi^{\rm tail }$ to scattering and the time-integrated tail Hamiltonian $\int dt H^{\rm tail }$
  12. Explicit computation of $W^{\rm (tail)} = \int dt \, H^{\rm (tail)}$ and $\chi^{\rm (tail)}$ in the large eccentricity limit
  13. Time-domain computation of $W^{\rm (tail)} = \int dt \, H^{\rm (tail)}$
  14. Fourier-domain computation of $W^{\rm tail}=\int dt H_{\rm (tail)}$
  15. Computation of the tail contribution to scattering in the large-eccentricity limit
  16. ...and 7 more sections

Figures (1)

  • Figure 1: The behavior, as a function of $\varepsilon=1/e$, of the additional constant $C(\varepsilon=1/e)$ in Eq. \ref{['logomc']} is plotted using both the data points of table I, and the simple cubic fit \ref{['cubicfit']} for the normalized version \ref{['defc']} of $C(\varepsilon)$.