Kernel: SageMath 9.3.rc2
Plot of principal null geodesics in Kerr spacetime
This Jupyter/SageMath notebook is relative to the lectures Geometry and physics of black holes.
The computations make use of tools developed through the SageManifolds project.
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'SageMath version 9.3.rc2, Release Date: 2021-04-06'
First we set up the notebook to display mathematical objects using LaTeX rendering:
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Spacetime manifold
We declare the Kerr spacetime as a 4-dimensional Lorentzian manifold :
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4-dimensional Lorentzian manifold M
and introduce the (3+1 version of) Kerr coordinates as a chart KC
on , via the method chart()
. The argument of the latter is a string (delimited by r"..."
because of the backslash symbols) expressing the coordinates names, their ranges (the default is ) and their LaTeX symbols:
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Chart (M, (tt, r, th, tph))
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Plot of the principal null geodesics in the plane
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Adding the vectors and to the plot
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We add the vectors and at the intersection of the ingoing geodesic with the outgoing one:
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Tangent vector k at Point p_0 on the 4-dimensional Lorentzian manifold M
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Tangent vector el at Point p_0 on the 4-dimensional Lorentzian manifold M
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Plot of and as functions of
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