Kerr spacetime in 3+1 Kerr coordinates
This Jupyter/SageMath notebook is relative to the lectures Geometry and physics of black holes.
The corresponding tools have been developed within the SageManifolds project.
NB: a version of SageMath at least equal to 9.1 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 :
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 :
The -dual 1-form (i.e. the 1-form of components ) is obtained via the method down()
:
We apply extra trigonometric simplifications on the components and factor them, via the method apply_map
:
The scalar product :
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 :