SageMath notebooks associated to the Black Hole Lectures (https://luth.obspm.fr/~luthier/gourgoulhon/bh16)
Kerr spacetime in Kerr coordinates
This Jupyter/SageMath worksheet is relative to the lectures Geometry and physics of black holes
These computations are based on tools developed through the SageManifolds project.
NB: a version of SageMath at least equal to 8.2 is required to run this worksheet:
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 :
and the Kerr coordinates as a chart on :
The Kerr parameters and :
Kerr metric
We define the metric by its components w.r.t. the Kerr coordinates:
The inverse metric is pretty simple:
as well as the determinant w.r.t. to the Kerr coordinates:
Let us check that we are dealing with a solution of the Einstein equation in vacuum:
The Christoffel symbols w.r.t. the Kerr coordinates:
Vector normal to the hypersurfaces
Ingoing principal null geodesics
Let us check that is a null vector:
Computation of :
Outgoing principal null geodesics
Let us check that is a null vector:
Computation of :
We check that :
Hence we may write :
Surface gravity
On , coincides with the Killing vector :
Therefore the surface gravity of the Kerr black hole is nothing but the value of the non-affinity coefficient of on :