A few years ago, I came across this browser-based app that simulates the trajectories of test particles in the Schwarzschild spacetime and shows an animation. I used it a little bit as a demonstration for teaching. One thing I didn't like was that when the trajectory hit the event horizon, the simulation just stopped abruptly. Now I'm working on a fun project to do something similar but without this limitation. (I would also like to make it a little more user-friendly.)
I imagined that one reason for the limitation was that maybe the simulation was using Schwarzschild coordinates, which misbehave at the event horizon. I've worked out the necessary Christoffel coefficients in Kruskal-Szekeres coordinates so as to sidestep this issue.
But now as I think more about the problem I think there is a whole nother issue, which is that we want some reasonable way of visualizing the motion. The obvious simple thing to do in the exterior region is to take the Schwarzschild coordinates $(r,\theta)$ and plot points in polar coordinates, letting time run in the animation according to the Schwarzschild $t$ coordinate. However, this seems to stop making any kind of sense at the event horizon, where the Schwarzschild coordinates misbehave. Inside the horizon, $t$ becomes the spacelike coordinate, so I guess we could plot $(t,\theta)$ in polar coordinates, but the transition across the horizon is obviously going to be silly.
Is there any better way to do this kind of visualization?
A method that certainly works is to plot a point crawling across a Penrose diagram, with time in the animation representing the proper time of the particle. But this seems a little esoteric for the average person playing with a simulation they come across on a web site.
Maybe one could show an optical simulation according to some other observer. This might be hard to do computationally. If the observer is outside the horizon, they never see the particle cross the horizon. If the observer is inside the horizon, the observer might hit the singularity and die before the simulation had gotten as far as we would like.
MTW have the following interesting embedding diagram (p. 837):
This provides a method of visualizing the addition regions of spacetime in the maximal extension of the Schwarzschild spacetime.