Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Experimental mathematics meets gravitational self-force

Nathan K. Johnson-McDaniel, Abhay G. Shah, Bernard F. Whiting

Abstract

It is now possible to compute linear in mass-ratio terms in the post-Newtonian (PN) expansion for compact binaries to very high orders using black hole perturbation theory applied to various invariants. For instance, a computation of the redshift invariant of a point particle in a circular orbit about a black hole in linear perturbation theory gives the linear-in-mass-ratio portion of the binding energy of a circular binary with arbitrary mass ratio. This binding energy, in turn, encodes the system's conservative dynamics. We give a method for extracting the analytic forms of these PN coefficients from high-accuracy numerical data using experimental mathematics techniques, notably an integer relation algorithm. Such methods should be particularly important when the calculations progress to the considerably more difficult case of perturbations of the Kerr metric. As an example, we apply this method to the redshift invariant in Schwarzschild. Here we obtain analytic coefficients to 12.5PN, and higher-order terms in mixed analytic-numerical form to 21.5PN, including analytic forms for the complete 13.5PN coefficient, and all the logarithmic terms at 13PN. At these high orders, an individual coefficient can have over 30 terms, including a wide variety of transcendental numbers, when written out in full. We are still able to obtain analytic forms for such coefficients from the numerical data through a careful study of the structure of the expansion. The structure we find also allows us to predict certain "leading logarithm"-type contributions to all orders. The additional terms in the expansion we obtain improve the accuracy of the PN series for the redshift observable, even in the very strong-field regime inside the innermost stable circular orbit, particularly when combined with exponential resummation.

Experimental mathematics meets gravitational self-force

Abstract

It is now possible to compute linear in mass-ratio terms in the post-Newtonian (PN) expansion for compact binaries to very high orders using black hole perturbation theory applied to various invariants. For instance, a computation of the redshift invariant of a point particle in a circular orbit about a black hole in linear perturbation theory gives the linear-in-mass-ratio portion of the binding energy of a circular binary with arbitrary mass ratio. This binding energy, in turn, encodes the system's conservative dynamics. We give a method for extracting the analytic forms of these PN coefficients from high-accuracy numerical data using experimental mathematics techniques, notably an integer relation algorithm. Such methods should be particularly important when the calculations progress to the considerably more difficult case of perturbations of the Kerr metric. As an example, we apply this method to the redshift invariant in Schwarzschild. Here we obtain analytic coefficients to 12.5PN, and higher-order terms in mixed analytic-numerical form to 21.5PN, including analytic forms for the complete 13.5PN coefficient, and all the logarithmic terms at 13PN. At these high orders, an individual coefficient can have over 30 terms, including a wide variety of transcendental numbers, when written out in full. We are still able to obtain analytic forms for such coefficients from the numerical data through a careful study of the structure of the expansion. The structure we find also allows us to predict certain "leading logarithm"-type contributions to all orders. The additional terms in the expansion we obtain improve the accuracy of the PN series for the redshift observable, even in the very strong-field regime inside the innermost stable circular orbit, particularly when combined with exponential resummation.

Paper Structure

This paper contains 15 sections, 54 equations, 3 figures.

Table of Contents

  1. Introduction and Summary
  2. Self-force calculation
  3. Obtaining analytic forms of the PN coefficients of the individual $(\ell,m)$ modes of $\Delta U$
  4. The PN expansion of the $(2,2)$ mode of $\Delta U$ and a way to simplify a general $(\ell, m)$ mode of $\Delta U$
  5. Applying PSLQ to the coefficients of the PN expansion of the modes of $\Delta U$
  6. An example: Obtaining the analytic form of the $\beta_7$ coefficient of the full $\Delta U$ from the decimal form given in SFW
  7. Combining together the values at different radii to increase the number of digits known
  8. Overview of our method for obtaining analytic forms of the coefficients of the modes of $\Delta U$
  9. Checks on the output of PSLQ
  10. Terms predicted by the simplifying factorization to arbitrarily high orders
  11. Computing the infinite sum over renormalized $\ell$ modes to obtain the final result for $\Delta U$
  12. Checking the results for $\Delta U$ by making an independent fit
  13. Convergence
  14. Conclusions and outlook
  15. Obtaining the $e^{2\bar{\nu}_{\ell m}\mathop{\mathrm{eulerlog}}\nolimits_m(R)}$ and $e^{2\bar{\nu}_{\ell m}\log(2/R)}$ contributions to the simplifications of the modes of $\Delta U$

Figures (3)

  • Figure 1: The largest prime in the denominator of the expression for $\beta_7$ returned by Mathematica's FindIntegerNullVector function applied to the vector $\{\beta_7, 1, \gamma, \log(2)\}$ for $\beta_7$ evaluated to varying numbers of digits from $10$ to $45$. The vertical dotted line marks the point at which this function returns an accurate analytic expression for $\beta_7$.
  • Figure 2: Convergence of the $21.5$PN expression for $\Delta U$ for orbits at various radii, comparing with the numerical data from Dolan et al.Dolanetal-tidal and Akcay et al.ABDS. Specifically, we show the convergence of the plain series, as well as the results of factoring out the test particle binding energy and/or performing exponential resummation on the entire series.
  • Figure 3: An estimate of the Schwarzschild radial coordinate (in $M$) of the radius of convergence of the PN series for $\Delta U/u$, obtained from $a_n^{2/n}$, where $a_n$ denotes the nonlogarithmic coefficient of $v^n$. We also show the same estimate for the test particle energy flux at infinity $(dE/dt)_\infty$, scaled by the Newtonian energy flux (from Fujita22PN), for comparison.