SecondOrderRicci

Overview

SecondOrderRicci computes the source for the second-order Lorenz-gauge field equations in Schwarzschild spacetime. That source is the second-order Ricci tensor $\delta^2 R_{\mu\nu}[h^{(1)}, h^{(1)}]$, which is quadratic in the first-order metric perturbation.

The code reads tensor-spherical-harmonic modes of a first-order field $h^{(1)}_{\mu\nu}$ for a particle on a circular orbit of radius $r_0$. It then computes the $(i, \ell, m)$ modes of $\delta^2 R_{\mu\nu}$ on the same radial grid. The modes are in the Barack–Lousto–Sago basis of tensor spherical harmonics, $i = 1, \dots, 10$ (see Appendix A of Wardell and Warburton (2015)). Each output mode $(\ell_3, m_3)$ is a sum over products of input modes $(\ell_1, m_1)$ and $(\ell_2, m_3 - m_1)$. The angular coupling uses Wigner 3j symbols from the GNU Scientific Library. The loop over output modes is parallelized with OpenMP.

The code is part of the Toolkit’s second-order self-force chain:

• h1Lorenz computes the first-order Lorenz-gauge field;
• Punctures gives the singular and regular parts of that field and the second-order puncture;
• SecondOrderRicci builds the quadratic source from the first-order field.

For background on the second-order scheme, see Miller and Pound (2021).

Installation

SecondOrderRicci is built from source with SCons. It needs:

• the GNU Scientific Library (GSL), for Wigner 3j symbols;
• HDF5, used for all input and output;
• a C++11 compiler with OpenMP support;
• Boost, for its multi_array type;
• SCons, the build system.

The README lists the versions it was tested with: GSL 1.16, HDF5 1.8.13, GCC 4.9.0, Boost 1.55.0 and SCons 2.3.1. The build compiles with -DH5_USE_110_API.

git clone https://github.com/BlackHolePerturbationToolkit/SecondOrderRicci.git
cd SecondOrderRicci
scons

This builds three executables, Ricci, Ricci_M and Ricci_S, in the top-level directory. If your libraries are in non-standard locations, edit the “Build options” section of src/SConscript (LIBPATH, CPPPATH, CXX and so on). The defaults point at /usr/local and the Debian/Ubuntu HDF5 locations. No installation step is needed. Run scons VERBOSE=1 to see the full compiler commands.

Usage

Input data

The first-order field must be in a directory of its own, with one HDF5 file per $(\ell, m)$ mode, named h1-l<l>m<m>.h5 (for $m \ge 0$), and no other files. Each file contains:

• grid: the radial grid. The worldline radius $r_0$ is shared between the two datasets below, so it appears once in each.
• inhom_left and inhom_right: the field to the left (including $r_0$) and right (from $r_0$) of the particle. For each field component there are six columns: the real and imaginary parts of the Barack–Lousto–Sago field $\bar h^{(i)\ell m}$, of its first radial derivative, and of its second radial derivative.

The components appear in the order $\{1,3,5,6,7,2,4\}$ for even $\ell+m$ with $\ell \ge 2$, and $\{9,10,8\}$ for odd $\ell+m$ with $\ell \ge 2$. The low multipoles use $\{1,3,6,2\}$ ($\ell = 0$), $\{1,3,5,6,2,4\}$ ($\ell = 1$, $m = 1$) and $\{9,8\}$ ($\ell = 1$, $m = 0$). Negative-$m$ modes are reconstructed by symmetry.

h1Lorenz writes files with the same names and datasets, but with only the field and its first derivative (four columns per component). The second derivatives must be added before running SecondOrderRicci. The Punctures repository has notebooks that generate the retarded, singular and regular first-order fields, including second derivatives, in this format.

Running

# Quadratic source from a single first-order field, using all available modes
./Ricci <dir>

# Source bilinear in two first-order fields A and B
./Ricci <dirA> <dirB> <delta_lmax> <lmax>

In the second form, lmax is the largest output $\ell$. For each output mode the sum over input modes is restricted to $\ell_3 - \Delta\ell_{\max} \le \ell_1, \ell_2 \le \ell_3 + \Delta\ell_{\max}$. A negative delta_lmax lowers the maximum input $\ell$ used. Both the single-field and two-field forms evaluate the quadratic terms with the first field in the first slot and the second field in the second slot.

Two variants couple the supplied field to an analytically known first-order perturbation built into the code:

./Ricci_M <dir> [<lmax>]   # coupling to the l = 0 mass perturbation
./Ricci_S <dir> [<lmax>]   # coupling to the l = 1, m = 0 odd-parity angular-momentum perturbation

These variants evaluate both orderings of the two fields.

The number of threads is set by the usual OpenMP environment variable, for example OMP_NUM_THREADS=16 ./Ricci data/h1R.

Output

The result is written to the current directory as src.h5 (src_M.h5 for Ricci_M, src_S.h5 for Ricci_S). It contains:

• r: the radial grid;
• src i=<i> l=<l> m=<m>: one dataset for each $(i, \ell, m)$ mode with $m \ge 0$. Each is an $N \times 2$ array of the real and imaginary parts of the mode on the grid.

Examples

notebooks/LoadSecondOrderRicciData.nb shows how to read the output in Mathematica. It uses SimulationTools for data handling. The core of it is:

rGrid = Import["src.h5", {"Datasets", "/r"}];
Loadd2R[i_, l_, m_] := d2R[i, l, m] = ToDataTable[Transpose[{rGrid,
     Complex @@@ Import["src.h5", {"Datasets", "/src i=" <> ToString[i] <> " l=" <> ToString[l] <> " m=" <> ToString[m]}]}]];

Do[Loadd2R[i, 2, 2], {i, 1, 7}]

The notebook also shows how to transform the modes to retarded or advanced time slicing by multiplying by $e^{\mp i m \Omega r_*}$, where $r_*$ is the tortoise coordinate.

Authors and contributors

Barry Wardell

Citing SecondOrderRicci

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

SecondOrderRicci also requests the following citation:

• Spiers, Pound and Wardell, Second-order perturbations of the Schwarzschild spacetime: Practical, covariant, and gauge-invariant formalisms, Phys. Rev. D 110, 064030 (2024) arXiv Inspire
  BibTeX

  @article{Spiers:2023mor,
      author = "Spiers, Andrew and Pound, Adam and Wardell, Barry",
      title = "{Second-order perturbations of the Schwarzschild spacetime: Practical, covariant, and gauge-invariant formalisms}",
      eprint = "2306.17847",
      archivePrefix = "arXiv",
      primaryClass = "gr-qc",
      doi = "10.1103/PhysRevD.110.064030",
      journal = "Phys. Rev. D",
      volume = "110",
      number = "6",
      pages = "064030",
      year = "2024"
  }

See how to cite for further guidance.