SageMath notebooks associated to the Black Hole Lectures (https://luth.obspm.fr/~luthier/gourgoulhon/bh16)
Standard Carter-Penrose diagram of Schwarzschild spacetime
This Jupyter/SageMath notebook is relative to the lectures Geometry and physics of black holes
The involved computations make use of tools developed through the SageManifolds project.
NB: a version of SageMath at least equal to 7.5 is required to run this notebook:
First we set up the notebook to display mathematical objects using LaTeX formatting:
Spacetime
We declare the spacetime manifold :
The Schwarzschild-Droste domain
The domain of Schwarzschild-Droste coordinates is :
The Schwarzschild-Droste coordinates :
Kruskal-Szekeres coordinates
Plot of Schwarzschild-Droste grid on in terms of KS coordinates
Adding the Schwarzschild horizon to the plot:
Adding the curvature singularity to the plot:
Extension to and
The second Schwarzschild-Droste domain:
Definition of regions and :
Schwarzschild-Droste coordinates in and :
Standard compactified coordinates
The coordinates associated with the conformal compactification of the Schwarzschild spacetime are
The chart of compactified coordinates plotted in terms of itself:
The transition map from Kruskal-Szekeres coordinates to the compactified ones:
Transition map between the Schwarzschild-Droste chart and the chart of compactified coordinates
The transition map is obtained by composition of previously defined ones:
Carter-Penrose diagram
The diagram is obtained by plotting the curves of constant Schwarzschild-Droste coordinates with respect to the compactified chart.