Kerr

##Low Level Tools for Black Hole Quasi-normal Modes

• Kerr QNM frequencies
• Kerr separation constants
• Spheroidal Harmonics
• Leaver's continued fraction method
• Solution space mapping
• Fit equations for QNM excitations
• Fit equations for frequencies and separation constants

How to add to your system's PYTHONPATH

If you have cloned the repository to

/Users/home/kerr_public/

Then you will want to add the following line to your ~/.bash_profile (or .bashrc or equivalent)

export PYTHONPATH="${PYTHONPATH}:/Users/home/kerr_public/"

You will then want to source the related file

source ~/.bash_profile

Quick Examples

Get Kerr QNM Frequencies and Damping times using tabulated results of Leaver's Method

from kerr import leaver
l,m,n = 3,3,0
jf = 0.99
# NOTE that jf<0 accesses the retrograde frequencies 
untiless_qnm_frequency, separation_constant = leaver( jf, l, m, n )

The result is untiless_qnm_frequency = 1.3230831096974336-0.029402669080879983j

Get Kerr QNM Frequencies and Damping times using fitting formulas

from kerr.formula.ksm2_cw import CW as cwfit
l,m,n = 3,3,0
jf = 0.99
# NOTE that there is a convention difference between the fit and the results of kerr.leaver
untiless_qnm_frequency = cwfit[l, m, n](jf)

The result is untiless_qnm_frequency = 1.32307097094+0.0294057641931j

A note on the extremal kerr limit (jf -> 0)

The above example has been chosen to demonstrate that at near the extremal kerr limit, many QNM frequencies aspmtote to m/2. In the case of (l,m)=(3,3), the extremal frequency is 1.5 (M=c=1). While the tabulated evaluations of leaver's method have not been carried out exactly at the extremal limit, the fits enforce this behavior and evaluations of Leaver's method very near the kerr limit have been performed as a check.

A few working notes

• kerr_public/notes/ns/mmrd.pdf

Useful ipython notebooks

• Test Spheroidal Harmonic implementation kerr_public/notes/ns/notebooks/test_slm.ipynb
• Query how well the QNM fits satisfy Leaver's equations kerr_public/notes/ns/notebooks/test_python_fit_equations.ipynb
• Explore the QNM solution space using Leaver's continued fraction method kerr_public/examples/leaver_example.ipynb