Checking that Kerr metric is a solution of Einstein equation
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 Boyer-Lindquist coordinates as a chart on :
Kerr metric
We define the metric by its components w.r.t. the Boyer-Lindquist coordinates:
The inverse metric:
The Christoffel symbols:
The Einstein equation
Let us check that the Ricci tensor of vanishes identically, which is equivalent to the Einstein equation in vacuum:
On the contrary, the Riemann tensor is not zero:
The Kretschmann scalar
The Kretschmann scalar is the following square of the Riemann tensor: We compute first the tensors and by respectively lowering and raising the indices of with the metric :
Then we perform the contraction:
A variant of this expression can be obtained by invoking the factor()
method on the coordinate function representing the scalar field in the manifold's default chart:
The Schwarzschild value of the Kretschmann scalar is recovered for