SageMath notebooks associated to the Black Hole Lectures (https://luth.obspm.fr/~luthier/gourgoulhon/bh16)
Kerr-Schild coordinates on Kerr spacetime
This Jupyter/SageMath notebook is relative to the lectures Geometry and physics of black holes.
The involved computations are based on tools developed through the SageManifolds project.
NB: a version of SageMath at least equal to 8.8 is required to run this notebook:
First we set up the notebook to display mathematical objects using LaTeX formatting:
To speed up computations, we ask for running them in parallel on 8 threads:
Spacetime
We declare the spacetime manifold :
3+1 Kerr coordinates
We restrict the 3+1 Kerr patch to , in order to introduce latter the Kerr-Schild coordinates:
The Kerr parameters and :
Kerr metric
We define the metric by its components w.r.t. the 3+1 Kerr coordinates:
The inverse metric is pretty simple:
as well as the determinant w.r.t. to the 3+1 Kerr coordinates:
Ingoing principal null geodesics
Let us check that is a null vector:
Computation of :
Kerr-Schild form of the Kerr metric
Let us introduce the metric such that where :
is a flat metric:
which proves that is a Kerr-Schild metric.
Let us check that is a null vector for as well:
Kerr-Schild coordinates
Check of the identity
Minkowskian expression of in terms of Kerr-Schild coordinates:
Equivalently, we may check the following identity:
Expression of and in the Kerr-Schild frame:
Expression of the Killing vector in terms of the Kerr-Schild frame:
Plots
Numerical values of the event and Cauchy horizons:
The BH event horizon:
Plot of the domain
We use replace by in the transformation formulas, because in what follows, we consider that is positive: