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Equation of motion for photon $$ \Sigma \frac{dt}{d\lambda} = aL\left(1-\frac{r^2+a^2}{\Delta}\right) + \omega\left(\frac{\left(r^2+a^2\right)^2}{\Delta}-a^2 \sin ^2\theta\right)\ , $$ $$ \Sigma\frac{dr}{d\lambda} = \sqrt{R(r)}=\sqrt{\left( \omega\left(r^2+a^2\right)-aL\right)^2-K\Delta}\ , $$ $$ \Sigma\frac{d\theta}{d\lambda} = \sqrt{\Theta(\theta)} = \sqrt{K-\left(\frac{L}{\sin\theta}-a\omega\sin\theta\right)^2}\ , $$ $$ \Sigma\frac{d\phi}{d\lambda} = L\left(\frac{1}{\sin^2\theta}- \frac{a^2}{\Delta}\right)+a\omega \left(\frac{r^2+a^2}{\Delta}-1\right)\ , $$ where $K$ is the Carter constant of the motion.

Qmechanic
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Bturimov
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    Check out this blogpost: http://rantonels.github.io/starless/ It includes a description of how to do it as well as a python code. – Void Mar 05 '19 at 18:08
  • @Void: that's an answer I think (it's the answer I was going to give if I could remember enough to search for it!) –  Mar 05 '19 at 18:38
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    I'm voting to close this question as off-topic because it's about writing or obtaining a piece of code and not physics. [scicomp.se] might be better suited. – Kyle Kanos Mar 05 '19 at 20:50
  • The team of Double Negative Ltd. and Kip Thorne wrote an article explaining how the images of the wormhole (not the black hole however) were created. It is the belief of the last author that wormholes can serve to teach general relativity. The article pursues that objective. I leave the link in case it were of interest: https://aapt.scitation.org/doi/pdf/10.1119/1.4916949 – Zythos Jun 09 '20 at 18:44

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I wrote some open-source Python code for that purpose: https://github.com/bcrowell/karl . I believe the code used in the actual movie is not open source. There is a very complete discussion of the techniques by Riazuelo: https://arxiv.org/abs/1511.06025 . I don't think Riazuelo's source code is available. Mine only currently handles Schwarzschild black holes, i.e., not rotating ones like in the movie.

My video simulation of falling into a black hole: https://youtu.be/z-H-PipYCKc

This lecture https://www.youtube.com/watch?v=lCWwsg6CtL0 by Riazuelo (in French) has some nice videos.

A related question, with an answer from me that includes some still images and discussion of the physics: What will the universe look like for anyone falling into a black hole?

I hadn't known until seeing Void's comment today that there were two open-source projects for this by Riccardo Antonelli: starless and schwarzschild. Wow, nice!