Constants of motion in gravitational self-force theory

TL;DR

This work develops and applies a complete set of constants of motion for gravitational self-force theory in the nonspinning Schwarzschild setting, defining energy $E$, azimuthal angular momentum $L$, and radial/polar actions $J_r$, $J_\theta$ at first order in the mass ratio. Using a multiscale SF framework and a fixed-frequency gauge, the authors compute 1SF corrections to $(E,L,J_r)$ and to the fundamental frequencies, obtaining analytic 9PN results and numerical data across a wide bound-orbit domain, with consistency against known PN results up to 4PN. They derive PN expansions for the geodesic relations, identify and analyze special surfaces in parameter space (circular orbits, separatrix, ISCO), and demonstrate the utility of these constants for streamlined waveform generation via flux-balance laws at 1PA order. The results enable efficient hybridization between SF, PN, and PM approaches, inform EOB models, and provide a robust foundation for 1PA EMRI waveform modeling and cross-formalism validation, including explicit data and high-order PN expressions for practical use.

Abstract

Synergies between self-force theory and other approaches to the gravitational two-body problem have traditionally relied on calculations of gauge-invariant observables as functions of orbital frequencies. However, in self-force theory one can also define a complete set of constants of motion: energy, azimuthal angular momentum, and radial and polar actions. Here we outline how directly utilizing these constants allows for more straightforward comparisons and hybridizations across the parameter space, as well as more streamlined waveform generation through flux-balance laws. Restricting to the case of nonspinning binaries and first order in self-force, we compute the constants of motion and the corrections to fundamental frequencies numerically as well as analytically (to 9PN in a post-Newtonian expansion), establishing consistency with the highest-order (4PN) results available from post-Newtonian theory. We also apply the results to identify the perturbed locations of special curves in the parameter space: circular orbits and the separatrix between bound and plunging orbits.

Figures (5)

• Figure 1: Comparison of the analytical 10PN expression for $\langle z_{1} \rangle$, expanded in terms of $p$ and $e$, against our numerical results. On the horizontal axis we use the semi-latus rectum distance from the separatrix at $p_{\star}=6+2e$. The irregularities on the right side of the plot are due to numerical errors; in the large-$p$ region, we expect the PN expansion to be more accurate than the numerical results.
• Figure 2: Comparison between our analytical (9PN) and numerical results for the 1SF corrections to the energy (top), angular momentum (middle), and radial action (bottom).
• Figure 3: The 1SF correction to the energy for $e=0.31974$ as a function of $p-p_{\star}$. We see that $\hat{E}_{(1)}$ diverges when it crosses the isofrequency curve at $p-p_\star = p_{\rm iso}-p_\star\approx 0.0023852$ (vertical dashed line). If we instead divide by the pole $(p - p_{\rm iso})^{-1}$, we get a smooth result.
• Figure 4: Plot of $\delta e_{\odot}^2(p)$ using various schemes. The blue dots and error bars are obtained by evaluating Eq. \ref{['eq:delta_e2_inTermsOf_Jr1']} using our numerical results for $\hat{J}_{r(1)}^\odot(p)$. The green dot-dashed line is obtained by replacing $\rho(x)$ by its 9PN expression in Eq. \ref{['eq:e2circ_rho_p']}, whereas the orange dashed line is obtained by using a non-resummed 9PN expansion \ref{['eq:delta_e2_PN']} for $\delta e_{\odot}^2(p)$. The vertical grey dotted lines correspond, from left to right, to $p=6$, $p=p_\text{iso}\approx 6.38$ and $p = p_\text{sign} \approx 6.95$. Thus, for $p_\text{iso}<p<p_\text{sign}$, we have $\delta e_{\odot}^2 (p)<0$ and $e_{\odot}$ is an imaginary number.
• Figure 5: A plot of the separatrix curve $\hat{E}^\star_{(1)}(\hat{L})$ in terms of both the angular momentum, $\hat{L}$, and the eccentricity which is geodesically linked to the angular momentum, $e_{(0)}^\star(\hat{L})$; see \ref{['seq:e0starL']} for its explicit definition. Error bars are included based on the estimated numerical uncertainty in the data.