pykerr

This package provides functions to get the frequency, damping time, and spheroidal harmonics of Kerr black holes. Solutions for the l=2 to l=7 modes are provided, including all m=[-l, ..., l] and up to 7 overtones (where n=0 is the fundamental mode), for dimensionless black hole spins up to +/- 0.9997. Currently, only spin weights of s=-2 are supported.

Installation

The easiest way to install pykerr is via pip:

pip install pykerr

or conda:

conda install -c conda-forge pykerr

You can also install from source by cloning the repository at https://github.com/cdcapano/pykerr. Required packages are listed in the requirements.txt file, which you can install by running pip install -r requirements.txt from within the source code directory, followed by pip install ..

Examples

1. Get the frequency and damping time of a Kerr black hole:

>>> import pykerr
>> pykerr.qnmfreq(200., 0.7, 2, 2, 0)
86.04823229677822
>>> pykerr.qnmtau(200., 0.7, 2, 2, 0)
0.012192884850631896

2. Get the spheroidal harmonics for the same black hole, viewed from a polar angle of pi/3 and an azimuthal angle of 0:

>>> pykerr.spheroidal(numpy.pi/3, 0.7, 2, 2, 0, phi=0.)
(0.3378406925922286-0.0007291958007333236j)

Details

pykerr uses pre-tabulated values for the Kerr QNM frequencies and angular separation constants. These are used to obtain solutions for the spheroidal harmonics using the recursion relation given by Eq. 18 in Leaver (1985) [1]. The pre-tabulated values for the QNM frequency, damping time, and angular separation constant comes from Berti et al. (2006) [2], made available as text files on the GRIT Ringdown website. Those files are repackaged into compressed hdf files that are released with this package. A cubic spline is applied to the pretabulated values to provide fast evaluation of the spheroidal harmonics and QNM frequencies at any arbitrary spin <= 0.9997. Pre-tabulated normalization constants for the s=-2 spheroidal harmonics are also provided, with a cubic spline being used to interpolate them.

pykerr does not calculate QNM frequencies and angular separation constants. For that, see the various Mathematica packages that are publicly available or the qnm package, which can be installed via pip. Interpolated values have been checked against London (2017).

Conventions

This uses the convention that the -m and +m modes are related to each other by:

f_{l-mn} = -f_{lmn}
tau_{l-mn} = tau_{lmn}
A_{l-mn} = A*_{lmn}

with dimensionless spin a/M being positive or negative. Negative spin means the perturbation is counter-rotating with respect to the black hole, while positive spin means the perturbation is co-rotating.

Custom tabulation

If you would like pykerr to use your own tabulated values, clone the code from soucre, then add or replace the hdf files stored in pykerr/data. The files should be named l{l}.hdf, where {l} is the l index. The hdf files need to have a top-level spin dataset with spins stored as 10^4 times the spin value. Each mode should be provided as a separate group, named {l}{m}{n}. In that group, there should be complex datasets called omega and alm that provides the dimensionless complex angular frequencies and angular separation constants, respectively. See the convert_to_hdf.py in the scripts directory, which is used to convert the text files from the GRIT website into the correct format, as an example. Once your files are implace, install pykerr using pip install .. That will copy your files to the install directory to be used by your code at runtime.

If you add values for spins beyond 0.9997 and would like pykerr to support them, change pykerr.qnm.MAX_SPIN appropriately. This can also be done at run time. It is recommended that at least three data points be provided beyond your spin limit to avoid boundary effects from the cubic spline.

To add pre-tabulated normalization constants to the data files use the tabulate_norms.py script, provided in the scripts directory.

References

1. E. W. Leaver, "An Analytic representation for the quasi normal modes of Kerr black holes", Proc. R. Soc. Lond. A 402 285-298 (1985).

2. E. Berti, V. Cardoso, and C. M. Will, "On gravitational-wave spectroscopy of massive black holes with the space interferometer LISA", PRD 73 064030 (2006), arXiv:0512160 [gr-qc].