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A new analytical method for self-force regularization II. Testing the efficiency for circular orbits

Wataru Hikida, Sanjay Jhingan, Hiroyuki Nakano, Norichika Sago, Misao Sasaki, Takahiro Tanaka

TL;DR

The paper tests a novel analytic self-force regularization framework for a scalar charge in Schwarzschild spacetime by decomposing the retarded Green function into $\tilde{S}$ and $\tilde{R}$ parts, ensuring all singular behavior is confined to $\tilde{S}$. It demonstrates that the $(\tilde{S}-S)$-part can be computed to $18$PN with rapid convergence for circular orbits, while the $\tilde{R}$-part converges efficiently through an $\ell$-mode expansion, with $\ell \le 13$ sufficing for accurate waveforms over a one-year observation. The total regularized self-force agrees with the Detweiler–Messaritaki–Whiting result for a test case, validating the method and showing the conservative part is $O(\mu/M)$ suppressed while the dissipative part governs the inspiral. These findings support efficient, high-accuracy waveform templates and motivate extensions to Kerr backgrounds and second-order perturbation theory.

Abstract

In a previous paper, based on the black hole perturbation approach, we formulated a new analytical method for regularizing the self-force acting on a particle of small mass $μ$ orbiting a Schwarzschild black hole of mass $M$, where $μ\ll M$. In our method, we divide the self-force into the $\tilde S$-part and $\tilde R$-part. All the singular behaviors are contained in the $\tilde S$-part, and hence the $\tilde R$-part is guaranteed to be regular. In this paper, focusing on the case of a scalar-charged particle for simplicity, we investigate the precision of both the regularized $\tilde S$-part and the $\tilde R$-part required for the construction of sufficiently accurate waveforms for almost circular inspiral orbits. For the regularized $\tilde S$-part, we calculate it for circular orbits to 18 post-Newtonian (PN) order and investigate the convergence of the post-Newtonian expansion. We also study the convergence of the remaining $\tilde{R}$-part in the spherical harmonic expansion. We find that a sufficiently accurate Green function can be obtained by keeping the terms up to $\ell=13$.

A new analytical method for self-force regularization II. Testing the efficiency for circular orbits

TL;DR

The paper tests a novel analytic self-force regularization framework for a scalar charge in Schwarzschild spacetime by decomposing the retarded Green function into and parts, ensuring all singular behavior is confined to . It demonstrates that the -part can be computed to PN with rapid convergence for circular orbits, while the -part converges efficiently through an -mode expansion, with sufficing for accurate waveforms over a one-year observation. The total regularized self-force agrees with the Detweiler–Messaritaki–Whiting result for a test case, validating the method and showing the conservative part is suppressed while the dissipative part governs the inspiral. These findings support efficient, high-accuracy waveform templates and motivate extensions to Kerr backgrounds and second-order perturbation theory.

Abstract

In a previous paper, based on the black hole perturbation approach, we formulated a new analytical method for regularizing the self-force acting on a particle of small mass orbiting a Schwarzschild black hole of mass , where . In our method, we divide the self-force into the -part and -part. All the singular behaviors are contained in the -part, and hence the -part is guaranteed to be regular. In this paper, focusing on the case of a scalar-charged particle for simplicity, we investigate the precision of both the regularized -part and the -part required for the construction of sufficiently accurate waveforms for almost circular inspiral orbits. For the regularized -part, we calculate it for circular orbits to 18 post-Newtonian (PN) order and investigate the convergence of the post-Newtonian expansion. We also study the convergence of the remaining -part in the spherical harmonic expansion. We find that a sufficiently accurate Green function can be obtained by keeping the terms up to .

Paper Structure

This paper contains 10 sections, 36 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Analytic regularization scheme
  3. Convergence test of the PN expansion
  4. The ($\tilde{S}-S$)-part
  5. The $\tilde{R}$-part and the regularized self-force
  6. Comparison with Detweiler, Messaritaki and Whiting Detweiler:2002gi
  7. Errors in gravitational wave cycles
  8. Conclusion and discussion
  9. Analytic expressions for the $\tilde{R}$-part
  10. Error analysis

Figures (8)

  • Figure 1: The relative error of post-Newtonian formulas in the $r$-component of the ($\tilde{S}-S$)-force at $r_0=6M, 10M, 20M$ and $50M$. The horizontal axis is the order of the post-Newtonian expansion. The top figure displays the convergence in the Taylor expansion and the bottom figure is that in the Pade approximation.
  • Figure 2: Plots of errors in the post-Newtonian expansion of both the $r$-component (upper panel) and the $t$-component (lower panel) of the $\tilde{R}$-force.
  • Figure 3: Plots of the errors in the $\ell$ expansion of both the $r$-component (upper panel) and the $t$-component (lower panel) of the $\tilde{R}$-force. Here, $\hat{\Delta}_{\alpha}^{\tilde{R}}(\ell)$ is defined in the same way as in the case of $\Delta_{\alpha}^{\tilde{R}}(n)$, but in the $\ell$-expansion instead of the post-Newtonian expansion. The convergence is much faster than that of the naive post-Newtonian expansion.
  • Figure 4: Error in the number of cycles caused by truncating the $\ell$-expansion in the $\tilde{R}$-force as a function of the mass of the central black hole.
  • Figure 5: Plots of the error in the $\ell$-expansion of both the $r$-component (upper panel) and the $t$-component (lower panel) of $F^{1,(8,25)}$, $F^{2,(8,25)}$ and $F^{3,(8,25)}$ (denoted by $1$, $2$ and $3$, respectively) at $r_0=6M$.
  • ...and 3 more figures