SageMath notebooks associated to the Black Hole Lectures (https://luth.obspm.fr/~luthier/gourgoulhon/bh16)
Einstein-Rosen bridge
This Jupyter/SageMath notebook is relative to the lectures Geometry and physics of black holes.
The involved computations make use of tools developed through the SageManifolds project.
NB: a version of SageMath at least equal to 7.5 is required to run this notebook:
First we set up the notebook to display mathematical objects using LaTeX formatting:
The rescaled Lambert function:
The minimal value of the Kruskal-Szekeres coordinate on a hypersurface :
Plot of as a function of on a hypersurface :
Plot of hypersurfaces in the Kruskal diagram
We generate first plots for the curvature singularity and the bifurcate horizon:
We add the inner limits of the isometring embeddings for :
The minimal value of the Kruskal-Szekeres coordinate on an embedded surface of constant :
The plot of the hypersurfaces for selected values of :
Isometric embeddings
The function :
The exact integral is not known in the general case:
The minimal value of on an embedded surface of constant :
The integration of to get is performed numerically, using an algorithm for an adaptive integration with (integrable) singularities (qags
):
Test that it works:
Function for some selected values of :
3D plots of the embeddings
Zoom out of the surface (Flamm paraboloid):