Schwarzschild horizon in Eddington-Finkelstein coordinates
This Jupyter/SageMath notebook is relative to the lectures Geometry and physics of black holes.
The involved computations are based on tools developed through the SageManifolds project.
NB: a version of SageMath at least equal to 9.2 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 :
and the Eddington-Finkelstein coordinates as a chart on :
The mass parameter and the metric tensor:
Let us check that we are dealing with a solution of Einstein's 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: