Kruskal-Szekeres coordinates in 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 9.4 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 ingoing Eddington-Finkelstein domain
The domain of ingoing Eddington-Finkelstein coordinates :
The Schwarzschild-Droste domain
The domain of Schwarzschild-Droste coordinates is :
The Schwarzschild-Droste coordinates :
Eddington-Finkelstein coordinates
The ingoing Eddington-Finkelstein chart:
Kruskal-Szekeres coordinates
Plot of the IEF grid in terms of KS coordinates:
Adding the Schwarzschild horizon to the plot:
Adding the curvature singularity to the plot:
Zoom showing the regular sclicing of the hypersurface by the hypersurfaces:
Plot of Schwarzschild-Droste grid on in terms of KS coordinates
Radial null geodesics
The outgoing family:
The ingoing family:
Extension to and
The second Schwarzschild-Droste domain:
Definition of regions and :
Schwarzschild-Droste coordinates in and :