Radial null geodesics in Schwarzschild spacetime
This Jupyter/SageMath worksheet is relative to the lectures Geometry and physics of black holes
These computations are based on SageManifolds (version 1.0, as included in SageMath 7.5 and higher versions)
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
NB: a version of SageMath at least equal to 7.5 is required to run this worksheet:
First we set up the notebook to display mathematical objects using LaTeX formatting:
Spacetime
We declare the spacetime manifold :
The Schwarzschild-Droste coordinates :
The Schwarzschild metric:
Radial null geodesics
The outgoing family:
The ingoing family:
Plot of some light cones:
Eddington-Finkelstein coordinates
The ingoing Eddington-Finkelstein chart:
Plot of the radial null geodesics in terms of the ingoing Eddington-Finkelstein coordinates:
Plot of the hypersurfaces in terms of the ingoing Eddington-Finkelstein coordinates: