Schwarzschild horizon in 3+1 Eddington-Finkelstein coordinates
This Jupyter/SageMath worksheet is relative to the lectures Geometry and physics of black holes
These computations are based on SageManifolds (v0.9)
The worksheet file (ipynb format) can be downloaded from here. To run it, you must start SageMath with the Jupyter notebook, with the command sage -n jupyter
First we set up the notebook to display mathematical objects using LaTeX formatting:
Spacetime
We declare the spacetime manifold :
and the 3+1 Eddington-Finkelstein coordinates as a chart on :
The mass parameter and the metric:
Let us check that we have a solution of Einstein equation in vacuum:
The scalar field defining the horizon
Let us check that each hypersurface is a null hypersurface:
The null normal
Let us check that is a null vector everywhere:
Check of the identity :
The null normal as a pregeodesic vector field
The non-affinity parameter :
Check of the pregeodesic equation :
Value of on the horizon:
The complementary null vector field
The 2-metric
We define :
Expansion along the null normal
We compute as :
Check of the formula :
Check of the forumla :
Deformation rate tensor of the cross-sections
We compute as :
Expansion of the cross-sections along the null normal :
We compute as :
Value of at the horizon: