RegularizationParameters

Overview

In the mode-sum approach to the self-force, the retarded field is computed numerically one spherical harmonic $\ell$-mode at a time. Each mode is finite at the particle, but their sum diverges. The regularization parameters are analytic expressions for the large-$\ell$ behaviour of the singular (Detweiler–Whiting) field. Subtracting them mode by mode leaves a sum that converges to the regular field at the particle:

where the next term is $F_{a[4]}/[(2\ell-3)(2\ell-1)(2\ell+3)(2\ell+5)]$, and so on.

The first two parameters, $F_{a[-1]}$ and $F_{a[0]}$, are enough to make the sum converge. Each higher-order parameter speeds up convergence by another two powers of $\ell$.

This collection gathers the latest regularization parameters for scalar, electromagnetic and gravitational fields in Schwarzschild and Kerr spacetimes. It also has parameters for related scalar quantities: the singular field itself, the Detweiler redshift $\tfrac12 h_{ab}u^a u^b$, and the spin precession and tidal invariants. The notation follows Heffernan, Ottewill and Wardell (2012), which writes the parameters as $F_{a[n]}$ with $n$ the order in distance from the particle. In the original notation:

There are two repositories:

• RegularizationParameters holds the parameters as Mathematica notebooks for Schwarzschild, equatorial Kerr and accelerated (non-geodesic) Schwarzschild orbits. This is the main, most complete resource.
• RegularizationParametersPackage holds Regularisation, an early-stage (v0.1.0) Mathematica package. It evaluates the scalar-field self-force parameters $F_{a[-1]}$, $F_{a[0]}$ and $F_{a[2]}$ for generic (eccentric, inclined) orbits in Kerr spacetime, from Heffernan (2022).

Available parameters

The tables below list where each set of parameters was first derived. If you use a set of parameters, please cite the corresponding papers as well as the Toolkit.

Schwarzschild spacetime

Geodesic self-force parameters

Useful scalar quantities for geodesic motion

The Schwarzschild notebook also has the regularization parameters for the gravitational spin-precession invariant $\Delta\psi$ and the tidal invariants $\Delta\lambda$ on circular orbits, as defined in S. R. Dolan, P. Nolan, A. C. Ottewill, N. Warburton and B. Wardell, Tidal invariants for compact binaries on quasi-circular orbits, Phys. Rev. D 91, 023009 (2015), arXiv:1406.4890.

Non-geodesic parameters

Kerr spacetime

Geodesic self-force parameters

Useful scalar quantities for geodesic motion

Installation

Parameter notebooks

The notebooks are not a package. Clone the repository and open them in Mathematica:

git clone https://github.com/BlackHolePerturbationToolkit/RegularizationParameters.git

Regularisation package

The Regularisation package is not on the BHPToolkit paclet server. Clone its repository and put the package’s Kernel directory on your $Path:

git clone https://github.com/BlackHolePerturbationToolkit/RegularizationParametersPackage.git

AppendTo[$Path, "/path/to/RegularizationParametersPackage/Regularisation/Kernel"];
Needs["Regularisation`"]

Put the Kernel directory itself on $Path. The main file loads its component files by bare file name, so loading the directory with PacletDirectoryLoad does not currently work. The package compiles its functions to C (CompilationTarget -> "C"), so it needs a C compiler that Mathematica can use.

Usage

Parameter notebooks

Each notebook defines the parameters as symbolic expressions in the orbital parameters. The same symbols are reused for the scalar, electromagnetic and gravitational sections, so evaluate only the section you need.

Schwarzschild. Orbits are parametrized by the energy E0, the angular momentum L, the radial four-velocity ur and the radial position r0. For example, a circular orbit of radius $10M$ has:

r0 = 10 M;
E0 = (1 - 2 M/r0) Sqrt[r0/(r0 - 3 M)];
L = r0 Sqrt[M/(r0 - 3 M)];
ur = 0;

Self-force parameters are stored as Flt[n], Flr[n] and Flϕ[n] for $n = -1, 0, 2, 4, 6$, with the $\ell$-dependence written out explicitly. The scalar field parameters are Φl[n], $\tfrac12 h_{ab}u^a u^b$ is Hl[n], the spin-precession parameters are Δψ[n], and the tidal parameters are Δλ1[n], Δλ2[n], Δλ3[n] and ΔλB[n]. The spin-precession and tidal parameters are valid only for circular orbits.

The leading-order parameter often depends on the direction from which the worldline is approached. This enters through Sign[Δr]: $+1$ if the self-force is computed from just outside $r_0$, $-1$ if from just inside.

Kerr (equatorial). Orbits are parametrized by the energy ℰ, the angular momentum ℒ, the radial four-velocity ur, the radial position r and the black hole spin a. The self-force parameters are stored as Ft[n], Fr[n] and Fϕ[n], the scalar field parameters as Φ[n], and $\tfrac12 h_{ab}u^a u^b$ as H[n]. Here the $\ell$-dependent prefactors are factored out, as in the equation in the Overview. The $m$-mode parameters (Fmt[4], Fmr[4], Fmϕ[4] and Φm[4] for the scalar field, plus an $m$-mode section that redefines H[2] and H[4] for $\tfrac12 h_{ab}u^a u^b$) include their $m$-dependence explicitly. The $m$-mode $\tfrac12 h_{ab}u^a u^b$ parameters are valid only for circular orbits.

The scalar, electromagnetic and gravitational parameters in the Kerr notebook have been checked in several ways. They reduce to the Schwarzschild values for $a \to 0$. Covariant and coordinate calculations agree. They satisfy the Barack–Ori relations between the $A_\alpha$ and $B_\alpha$ parameters for the three fields. The scalar $\ell$-mode and $m$-mode parameters correctly regularize independent numerical data.

Regularisation package

RPscalarA, RPscalarB and RPscalarD return $F_{a[-1]}$, $F_{a[0]}$ and $F_{a[2]}$ for the scalar self-force as a list of the four components $\{t, r, \theta, \phi\}$. Each takes ten arguments: the energy, angular momentum, Carter constant, black hole spin, black hole mass, the particle’s radius and polar angle $\theta$, the sign of $\Delta r$, the sign of $u^r$ and the sign of $u^\theta$.

RPscalarA[En, Lz, Q, a, M, r, θ, sΔr, sur, suθ]
RPscalarB[En, Lz, Q, a, M, r, θ, sΔr, sur, suθ]
RPscalarD[En, Lz, Q, a, M, r, θ, sΔr, sur, suθ]

ElFactor[k, l] gives the $\ell$-dependent prefactor multiplying $F_{a[k]}$: $2\ell+1$ for $k=-1$, $1$ for $k=0$, and $1/((2\ell-1)(2\ell+3))$ for $k=2$. The wrapper RegParam[field, function, order, parameters] selects the parameter by field (currently only "Scalar") and order ($-1$, $0$, $1$ or $2$). cite returns BibTeX entries for the references appropriate to the orbit you used.

Examples

KerrRegularizationParameters.nb ends with a worked example. It takes the numerically computed $\ell$-modes of the radial scalar self-force for an eccentric equatorial orbit ($a = 0.5M$, $p = 10$, $e = 0.2$, evaluated at periastron from just inside the particle) and subtracts the regularization parameters order by order:

Frl[-1] = Table[N[(2 l + 1) Fr[-1]], {l, 0, 18}];
Frl[0] = Table[N[Fr[0]], {l, 0, 18}];
Frl[2] = Table[N[Fr[2]/((2 l - 1) (2 l + 3))], {l, 0, 18}];
Frl[4] = Table[N[Fr[4]/((2 l - 3) (2 l - 1) (2 l + 3) (2 l + 5))], {l, 0, 18}];

ListLogLogPlot[Abs[{FrRet, FrRet - Frl[-1], FrRet - Frl[-1] - Frl[0],
    FrRet - Frl[-1] - Frl[0] - Frl[2], FrRet - Frl[-1] - Frl[0] - Frl[2] - Frl[4]}],
  PlotRange -> {10^-10, 1}, DataRange -> {0, 18}, Frame -> True]

The plot shows the faster fall-off in $\ell$ as each higher-order parameter is subtracted. The notebook supplies FrRet (the retarded $\ell$-modes) and the orbital constants.

Authors and contributors

Anna Heffernan, Adrian Ottewill, Barry Wardell, Niels Warburton

Citing RegularizationParameters

If you use RegularizationParameters in your research, please acknowledge the Toolkit:

RegularizationParameters also requests the following citations:

• Heffernan, Ottewill and Wardell, High-order expansions of the Detweiler-Whiting singular field in Schwarzschild spacetime, Phys. Rev. D 86, 104023 (2012) arXiv Inspire
  BibTeX

  @article{Heffernan:2012su,
      author = "Heffernan, Anna and Ottewill, Adrian and Wardell, Barry",
      title = "{High-order expansions of the Detweiler-Whiting singular field in Schwarzschild spacetime}",
      eprint = "1204.0794",
      archivePrefix = "arXiv",
      primaryClass = "gr-qc",
      doi = "10.1103/PhysRevD.86.104023",
      journal = "Phys. Rev. D",
      volume = "86",
      pages = "104023",
      year = "2012"
  }

• Heffernan, Ottewill and Wardell, High-order expansions of the Detweiler-Whiting singular field in Kerr spacetime, Phys. Rev. D 89, 024030 (2014) arXiv Inspire
  BibTeX

  @article{Heffernan:2012vj,
      author = "Heffernan, Anna and Ottewill, Adrian and Wardell, Barry",
      title = "{High-order expansions of the Detweiler-Whiting singular field in Kerr spacetime}",
      eprint = "1211.6446",
      archivePrefix = "arXiv",
      primaryClass = "gr-qc",
      doi = "10.1103/PhysRevD.89.024030",
      journal = "Phys. Rev. D",
      volume = "89",
      number = "2",
      pages = "024030",
      year = "2014"
  }

• For the Regularisation package (scalar field, generic Kerr orbits):
  Heffernan, Regularization of a scalar charged particle for generic orbits in Kerr spacetime, Phys. Rev. D 106, 064031 (2022) arXiv Inspire
  BibTeX

  @article{Heffernan:2021olv,
      author = "Heffernan, Anna",
      title = "{Regularization of a scalar charged particle for generic orbits in Kerr spacetime}",
      eprint = "2107.14750",
      archivePrefix = "arXiv",
      primaryClass = "gr-qc",
      doi = "10.1103/PhysRevD.106.064031",
      journal = "Phys. Rev. D",
      volume = "106",
      number = "6",
      pages = "064031",
      year = "2022"
  }

• Heffernan, Self-Force Regularisation Parameters Package (Zenodo, 2022)
  BibTeX

  @software{Heffernan:2022zenodo,
      author    = {Heffernan, Anna},
      title     = {{Self-Force Regularisation Parameters Package}},
      year      = 2022,
      publisher = {Zenodo},
      version   = {1.0},
      doi       = {10.5281/zenodo.6282572},
      url       = {https://doi.org/10.5281/zenodo.6282572}
  }

See how to cite for further guidance.