Kerr spacetime in 3+1 Kerr coordinates
This Jupyter/SageMath worksheet is relative to the lectures Geometry and physics of black holes
These computations are based on SageManifolds (v0.9)
Click here to download the worksheet file (ipynb format). To run it, you must start SageMath with the Jupyter notebook, with the command sage -n jupyter
First we set up the notebook to display mathematical objects using LaTeX formatting:
Spacetime
We declare the spacetime manifold :
and the 3+1 Kerr coordinates as a chart on :
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:
Let us check that we are dealing with a solution of the Einstein equation in vacuum:
The Christoffel symbols w.r.t. the 3+1 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 :
Thefore the surface gravity of the Kerr black hole is nothing but the value of the non-affinity coefficient of on :