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:
'SageMath version 8.0.beta6, Release Date: 2017-05-12'
First we set up the notebook to display mathematical objects using LaTeX formatting:
We declare the spacetime manifold M:
4-dimensional differentiable manifold M
We use coordinates (t,r,θ,ϕ) analogous to the 3+1 Eddington-Finkelstein coordinates in Schwarzschild spacetime, i.e. coordinates such that the advanced time v=t+r is constant on the ingoing radial null geodesics:
A zoom on the trapping horizon in its dynamical part: notice that the "outgoing" null geodesics cross it with a vertical tangent, in agreement with the cross-sections of the trapping horizon being marginally trapped surfaces.