h1Lorenz

Overview

h1Lorenz computes the first-order metric perturbation, in the mass ratio, sourced by a point particle on a circular orbit about a Schwarzschild black hole, in the Lorenz gauge. It works in the frequency domain. It returns the tensor-spherical-harmonic modes of the perturbation and their first radial derivatives on a radial grid you supply. The modes are in the Barack–Lousto–Sago basis, $\bar h^{(i)\ell m}$ with $i = 1, \dots, 10$.

For each $(\ell, m)$ mode the code integrates the coupled radial ODEs for the independent components. It builds the remaining “gauge” components from the Lorenz gauge condition, and matches homogeneous solutions at the particle to obtain the inhomogeneous (retarded) field. For some modes it also computes asymptotic amplitudes at infinity and on the horizon. The static modes (monopole and even-parity $m = 0$ modes) are handled separately, as are the odd and even dipoles. The work over $(\ell, m)$ modes is distributed with MPI.

The code is a modified version of the one developed for Akcay, Warburton and Barack (2013). Its output is the starting point of the second-order chain in the Toolkit. The Punctures repository regularizes it to get the first-order regular field, and SecondOrderRicci uses it to build the second-order source.

Installation

h1Lorenz is built from source with SCons. It needs the GNU Scientific Library, HDF5 including the high-level hdf5_hl library, and an MPI implementation that provides mpicc (for example OpenMPI).

git clone https://github.com/BlackHolePerturbationToolkit/h1Lorenz.git
cd h1Lorenz
scons

This produces the h1Lorenz executable in the top-level directory. HDF5 include and library paths are set in src/SConscript (CPPPATH, LIBPATH). The defaults are the Debian/Ubuntu serial-HDF5 locations, so edit them for other systems.

Usage

1. Generate a radial grid

The code evaluates the field on a radial grid read from an HDF5 file. The file has a dataset r with the grid points and a dataset ImportantIndexes with three (1-based) indices: the start of the region around the particle, the particle radius $r_0$, and the end of the region around the particle. notebooks/ComputeRadialGrid.nb builds a suitable grid and writes it to input/radial_grid_r<r0>.h5. The grid has 500 points uniform in tortoise coordinate between $r = 2M + 10^{-5}M$ and $2.5M$. From there it has a spacing of $0.01M$ out to $r_0 + \Delta r_0$, where $\Delta r_0 = 2M$ ($4M$ for $r_0 > 20M$). The spacing then grows to $0.1M$, $0.5M$ and $4M$ out to $r = 100M$, $1000M$ and $10^4 M$ respectively. The same construction is in the script notebooks/ComputeRadialGrid.wl, which expects r0 and gridFile to be set before it is loaded:

wolframscript -code 'r0 = 8.1; gridFile = "input/radial_grid_r8.1.h5"; Get["notebooks/ComputeRadialGrid.wl"]'

2. Run the code

mpirun -n <numprocs> ./h1Lorenz <r0> <lmax> <gridfile> <outdir>

• r0: the orbital radius, in Schwarzschild coordinates.
• lmax: the maximum $\ell$ to compute.
• gridfile: the grid file from step 1.
• outdir: the output directory.

Use at least two processes, because one process hands out work to the others. The README notes that runs with more than two processes have not been tested recently. For example, to compute all modes up to $\ell = 15$ for $r_0 = 8.1M$:

mpirun -n 2 ./h1Lorenz 8.1 15 input/radial_grid_r8.1.h5 data/fields_r8.1

Known limitation: the even-$\ell$, $m = 0$ modes are generally not computed accurately, or at all, because the homogeneous solutions must be integrated over long regions. Replace them with the results of another calculation.

Output

Each $(\ell, m)$ mode with $m \ge 0$ is written to <outdir>/h1-l<l>m<m>.h5, which contains:

In inhom_left and inhom_right, each row is a grid point. Each field component has four columns: the real and imaginary parts of $\bar h^{(i)\ell m}$ and of its first 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 $\{1,3,6,2\}$ ($\ell=0$), $\{1,3,5,6,2,4\}$ ($\ell = m = 1$) and $\{9,8\}$ ($\ell = 1$, $m = 0$). The odd dipole includes a pure-gauge piece that makes it regular at the horizon while keeping the correct angular momentum.

Examples

notebooks/Loadh1LorenzData.nb defines Loadh1Data[r0, l, m], which reads a mode file into SimulationTools DataTables h1[r0][i, l, m] and drh1[r0][i, l, m], together with the asymptotic amplitudes CInfinity[r0][i, l, m] and CH[r0][i, l, m]. Its example run loads and plots the $\ell = 2$ modes for $r_0 = 6M$:

Loadh1Data[6, 2, 2]
Loadh1Data[6, 2, 1]

ListLogLinearPlot[ReIm[Shifted[Slab[h1[6][7, 2, 2], 2 ;; 200], -2]], Joined -> True, PlotRange -> All]

It then checks the result against the Teukolsky package. It computes the $\ell = 2$ energy flux at infinity from the asymptotic amplitudes, using Eq. (75) of arXiv:1012.5860, and compares it with TeukolskyPointParticleMode.

Related papers

• Lorenz-gauge decomposition into tensor spherical harmonics: arXiv:gr-qc/0510019
• Lorenz-gauge circular orbits in the frequency domain: arXiv:1012.5860
• Lorenz-gauge eccentric orbits in the frequency domain: arXiv:1308.5223, arXiv:1409.4419
• Lorenz-gauge circular orbits in the time domain: arXiv:gr-qc/0701069
• Lorenz-gauge eccentric orbits in the time domain: arXiv:1002.2386

Authors and contributors

Niels Warburton, Sarp Akcay

Citing h1Lorenz

If you use h1Lorenz in your research, please acknowledge the Toolkit:

h1Lorenz also requests the following citation:

• Akcay, Warburton and Barack, Frequency-domain algorithm for the Lorenz-gauge gravitational self-force, Phys. Rev. D 88, 104009 (2013) arXiv Inspire
  BibTeX

  @article{Akcay:2013wfa,
      author = "Akcay, Sarp and Warburton, Niels and Barack, Leor",
      title = "{Frequency-domain algorithm for the Lorenz-gauge gravitational self-force}",
      eprint = "1308.5223",
      archivePrefix = "arXiv",
      primaryClass = "gr-qc",
      doi = "10.1103/PhysRevD.88.104009",
      journal = "Phys. Rev. D",
      volume = "88",
      number = "10",
      pages = "104009",
      year = "2013"
  }

See how to cite for further guidance.