SageMath notebooks associated to the Black Hole Lectures (https://luth.obspm.fr/~luthier/gourgoulhon/bh16)
Radial free fall in Schwarzschild spacetime
This Jupyter/SageMath worksheet is relative to the lectures Geometry and physics of black holes.
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
Schwarzschild-Droste coordinate as a function of
The function :
The primitive:
Since this is a complicated expression, let us try to find something simpler by taking as a variable; we have then :
This time, integrate
yields a simpler expression for :
We force some obvious simplification by
We note that, given the range of , the argument of the logarithm is negative; let us enforce as the primitve of instead of or :
We move back to , with some extra simplification enforced:
Finally we invoke trig_reduce()
to let appear :
Plot of as a function of
Plot of as a function of
Spacetime diagram based on IEF coordinates
Infall time
Proper time spent inside the black hole
We may check the above limit by asking for a Taylor expansion in terms of :