Punctures

Overview

The Punctures repository contains the puncture fields for second-order self-force calculations, specialised to a point mass on a quasicircular orbit of radius $r_0$ about a Schwarzschild black hole in the Lorenz gauge. In a puncture (or effective-source) scheme the singular part of the field near the particle is approximated analytically by a puncture $h^{\mathcal{P}}_{\mu\nu}$. Subtracting it from the field equations gives an effective source, from which a field that is finite at the particle is computed numerically. The details are in Effective source for second-order self-force calculations: quasicircular orbits in Schwarzschild spacetime (Upton, Wardell, Pound, Warburton and Barack). The covariant second-order puncture they build on was derived by Pound and Miller (2014).

The punctures are given as modes in the Barack–Lousto–Sago basis of tensor spherical harmonics, labelled by $i = 1, \dots, 10$ and $(\ell, m)$, as functions of $r_0$, the black hole mass $M$ and the radial distance from the particle, $\Delta r = r - r_0$. Mode decompositions are first done in a rotated frame where the particle sits at the pole, where only a few azimuthal numbers $m^\prime$ contribute. They are then rotated back to the physical frame with Wigner $D$-matrices.

The repository has three parts:

• First-order puncture $h^{1\mathcal{P}}_{\mu\nu}$, including terms up to order $\epsilon^2$ in the distance from the particle, together with the scripts that produce numerical mode data for the first-order singular, regular and retarded fields.
• Second-order puncture $h^{2\mathcal{P}}_{\mu\nu}$, split into the pieces $h^{\rm SS}_{\mu\nu}$, $h^{\rm SR}_{\mu\nu}$, $h^{\delta m}_{\mu\nu}$, $h^{\delta z}_{\mu\nu}$ and $h^{\rm ms}_{\mu\nu}$ defined in the paper above. These are provided as covariant expressions and as $\ell m$ modes.
• First-order regular field $h^{\mathcal{R}1}_{\mu\nu}$ and its derivatives at the particle, computed by mode-sum regularization from numerical first-order data.

Punctures is not a loadable package. It is a collection of Mathematica notebooks, scripts and precomputed expressions, intended to be used alongside the first-order Lorenz-gauge solver h1Lorenz and the second-order source code SecondOrderRicci.

Installation

Punctures is not on the BHPToolkit paclet server. Clone the repository (about 30 MB, mostly large expression files):

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

The notebooks load their companion files with NotebookDirectory[], so open them from inside the cloned repository. Nothing needs to be added to $Path.

Repository contents

Usage

Second-order punctures

Open SecondOrder/LoadSecondOrderPunctures.nb from the cloned repository and evaluate it. This loads the mode expressions in h2Pilm.m and defines h2P[piece, i, l, m]. That function returns the physical-frame $(i, \ell, m)$ mode of a piece of the second-order puncture as an expression in $r_0$, $M$, $\Omega$ and $\Delta r$. Evaluate it numerically by fixing those symbols, for example the $i=1$, $\ell=m=2$ mode of $h^{\rm SR}$:

Block[{M = 1, r0 = 7.6, Δr = 0.4, Ω = 7.6^(-3/2)}, h2P["SR", 1, 2, 2]]

Throughout the repository, Δr (\[CapitalDelta]r) is the symbol for $\Delta r$ and Ω (\[CapitalOmega]) is the symbol for the orbital frequency $\Omega$.

The notebook also plots the $h^{\rm SS}$ modes against $\Delta r$:

plothSS[i_, l_, m_][r1_] := Block[{M = 1, Ω = r1^(-3/2), r0 = r1, hSSEval},
  hSSEval = h2P["SS", i, l, m];
  Plot[Evaluate[ReIm[hSSEval]], {Δr, -2, 2}, PlotTheme -> "Detailed", FrameLabel -> {"Δr"}]]

Table[plothSS[i, 2, 2][7.6], {i, 1, 7}]
Table[plothSS[i, 2, 1][7.6], {i, 8, 10}]

Only $h^{\rm SS}$ is fully determined by the orbit. The other pieces need first-order data that you supply yourself: the first-order regular field $h^{\mathcal{R}1}_{ab}$ and its covariant derivatives (see hR1/), the first-order self-force $F^r_1$, the inspiral rate $\dot r_0$, and so on.

You can also load the rotated-frame mode expressions on their own:

Get[FileNameJoin[{"/path/to/Punctures", "SecondOrder", "h2Pilm.m"}]];
hSSlm[1][2, 0] /. {M -> 1, r0 -> 10, Δr -> 1/10} // N

First-order punctures and fields

The first-order pipeline is described in First Order/README.md:

1. Edit the “Parameters” section of h1P-exact-order-eps2.wl, which sets the working precision, lmax, the maximum rotated-frame $m^\prime$, the $\Delta r$ range and the output directory.

2. Run the script with r0 set. It writes one file per field component $i$, plus separate files for the first and second derivatives. On a cluster, First Order/README.md gives a SLURM script that runs:

   WolframKernel -run "r0=1990/100; Get[\"../h1P-exact-order-eps2.wl\"]; Quit[];"
   

3. Run process[r0] in Process-h1P.nb to combine the output into a single file.
4. Run h1S-exact-order-eps2.nb to produce HDF5 files for $h^{1\mathcal{S}}$, $h^{1\mathcal{R}}$ and $h^{1,\rm ret}$, including second derivatives.

Examples

• SecondOrder/LoadSecondOrderPunctures.nb: loads and evaluates every second-order puncture piece and plots the $h^{\rm SS}$ modes.
• hR1/hR.nb: builds the coordinate components of $h^{\mathcal{R}1}_{\mu\nu}$ and its derivatives at the particle from $(\ell, m)$ modes by mode-sum regularization.
• SecondOrder/CovariantSecondOrderPunctureMultiscale.nb: the covariant puncture expressions before mode decomposition.

Authors and contributors

Jeremy Miller, Adam Pound, Samuel D. Upton, Barry Wardell

Citing Punctures

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

Punctures also requests the following citations:

• Upton, Wardell, Pound, Warburton and Barack, Effective source for second-order self-force calculations: Quasicircular orbits in Schwarzschild spacetime, Phys. Rev. D 113, 064013 (2026) arXiv Inspire
  BibTeX

  @article{Upton:2025bja,
      author = "Upton, Samuel D. and Wardell, Barry and Pound, Adam and Warburton, Niels and Barack, Leor",
      title = "{Effective source for second-order self-force calculations: Quasicircular orbits in Schwarzschild spacetime}",
      eprint = "2508.00087",
      archivePrefix = "arXiv",
      primaryClass = "gr-qc",
      doi = "10.1103/f9j2-k64r",
      journal = "Phys. Rev. D",
      volume = "113",
      number = "6",
      pages = "064013",
      year = "2026"
  }

• Pound and Miller, Practical, covariant puncture for second-order self-force calculations, Phys. Rev. D 89, 104020 (2014) arXiv Inspire
  BibTeX

  @article{Pound:2014xva,
      author = "Pound, Adam and Miller, Jeremy",
      title = "{Practical, covariant puncture for second-order self-force calculations}",
      eprint = "1403.1843",
      archivePrefix = "arXiv",
      primaryClass = "gr-qc",
      doi = "10.1103/PhysRevD.89.104020",
      journal = "Phys. Rev. D",
      volume = "89",
      number = "10",
      pages = "104020",
      year = "2014"
  }

See how to cite for further guidance.