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I am not entirely sure how to solve your issue. Indeed If I remove your chart transition, the geodesic declaration (M.integrated_geodesic(...) no longer hangs. My guess is that it is trying to express the metric in the second frame, and fails to invert/simplify something. It does not even start integrating.

But I am wondering what you are trying to achieve with this second chart. None of the systems of coordinates you are using are regular on the horizon. However, Kerr's original coordinates are (see Eq. (3) of this review). Using a coordinate system that is regular on the horizon, it is sufficient to impose $r>0.1$ (or anything small) to avoid the central singularity (no change of variable can save you, since it is a true, physical singularity). solve_across_charts should automatically stop the integration if it exits the chart domain.

If you are interested, you can check this notebook, where I implement basic black-hole ray-tracing step by step. Extending it to the Kerr metric should be straightforward (but very computationally intense, since I make use of the spherical symmetry of Schwarzschild to drastically reduce the computation time).