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SageMath notebooks

SageMath is a modern free, open-source mathematics software system, which is based on the Python programming language.

The following SageMath notebooks are in Jupyter format and involve tools developed through the SageManifolds project.

Each notebook can be viewed in the browser by clicking on its name in the list below. It can be downloaded by clicking on the [ipynb] link. All notebooks can be downloaded in a single directory from the GitHub repository.

Even if SageMath is not installed on your computer, you can run an interactive version of the notebook by clicking on the button Execute on Binder in the top right menu of Jupyter nbviewer.

Minkowski spacetime

• Conformal completion of Minkowski spacetime [ipynb]

Schwarzschild spacetime

• Schwarzschild horizon [ipynb]
• Solving Einstein equation: Kottler solution [ipynb]
• Basic computations in Schwarzschild spacetime [ipynb] (Christoffel symbols, Riemann tensor, Kretschmann scalar)
• Radial null geodesics in Schwarzschild spacetime [ipynb]
• Kruskal-Szekeres coordinates in Schwarzschild spacetime [ipynb]
• Carter-Penrose diagram of Schwarzschild spacetime (based on Frolov-Novikov coordinates) [ipynb]
• Standard (singular) Carter-Penrose diagram of Schwarzschild spacetime [ipynb]
• Einstein-Rosen bridge in Schwarzschild spacetime [ipynb]

Kerr spacetime

• Kerr metric as a solution of Einstein equation [ipynb]
• Kerr spacetime in advanced Kerr coordinates [ipynb]
• Kerr spacetime in Kerr coordinates [ipynb]
• Plot of principal null geodesics in Kerr spacetime [ipynb]
• Kerr-Schild coordinates on Kerr spacetime [ipynb]
• ZAMO frame on Kerr spacetime [ipynb]
• Carter frame on Kerr spacetime [ipynb]
• Walker-Penrose Killing tensor on Kerr spacetime [ipynb]
• Timelike and null geodesics in Kerr spacetime [ipynb]
• Circular equatorial orbits in Kerr spacetime [ipynb]
• Zero-energy null geodesics in Kerr spacetime [ipynb]
• Existence and stability of spherical photon orbits in Kerr spacetime [ipynb]
• Plots of spherical photon orbits geodesics in Kerr spacetime [ipynb]
• Plots of null geodesics in Kerr spacetime [ipynb]
• Shadow and critical curve of a Kerr black hole [ipynb]
• Images of an accretion disk around a Kerr black hole [ipynb]
• Critical curve of a Kerr black hole onto the EHT image of M87* [ipynb]
• Extremal Kerr black hole [ipynb]
• Maximal extension of the extremal Kerr spacetime [ipynb]
• Isometric embbedings of 2D sections of the extremal Kerr spacetime [ipynb]
• Near-horizon extremal Kerr geometry [ipynb]
• NHEK spacetime [ipynb]

Black hole formation

• Lemaître-Tolman gravitational collapse [ipynb]
• Oppenheimer-Snyder collapse: spacetime diagrams [ipynb]
• Oppenheimer-Snyder collapse: curvature [ipynb]
• Black hole formation in Vaidya spacetime [ipynb]
• Solving the ODE for outgoing radial null geodesics in Vaidya spacetime [ipynb]
• Naked singularity in Vaidya collapse [ipynb]
• False trapped surface in Minkowski spacetime [ipynb]