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):

enter image description here

This provides a method of visualizing the addition regions of spacetime in the maximal extension of the Schwarzschild spacetime.

  • $\begingroup$ As fluctuations in the surface? Ideally of a hypersphere.. :P "the insight was that the informational content of all the objects that have fallen into the hole might be entirely contained in surface fluctuations of the event horizon" - en.m.wikipedia.org/wiki/Holographic_principle $\endgroup$
    – CriglCragl
    May 25, 2018 at 14:52
  • $\begingroup$ "'t Hooft showed that this field makes a logarithmic tent-pole shaped bump on the horizon of a black hole, and like a shadow, the bump is an alternative description of the particle's location and mass" en.m.wikipedia.org/wiki/… $\endgroup$
    – CriglCragl
    May 25, 2018 at 14:57

1 Answer 1


that is my little app ;)

Stopping at the horizon was an aesthetic decision, the effective potential allows horizon crossing no problem. It is easy to change this in the code but it looks a bit silly to my eyes. If you want you can have it come right out into another universe, but that wasn't the intention (I was trying to compare Newtonian and GR orbits the best I could).

Incidentally, I wrote it after I saw http://www.fourmilab.ch/gravitation/orbits/indexj.html and thought I could do better ;)

I am on my work account ATM, I can say more when I get back home later.The source is here.

  • $\begingroup$ Nice to make contact with you :-) The point of my question is to ask for suggestions on appropriate visualizations. $\endgroup$
    – user4552
    May 25, 2018 at 14:13
  • $\begingroup$ The real me this time, I actually have a 4D Kerr-de-Sitter geodesic prog which I pipe to visual python for orbit visualisation: github.com/m4r35n357/BlackHole4dVala.I have also attempted to interface with this relativistic raytracer, but not successfully yet. madore.org/~david/math/kerr.html $\endgroup$
    – m4r35n357
    May 25, 2018 at 18:37

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