# GR and Dynamical Systems

## Main Question or Discussion Point

I'm doing a research project for a class in advanced differential geometry, and have chosen to cover dynamical systems. However, I've found that the relationship between dynamical systems is limited to exploring flows on manifolds in the most abstract sense. As a result, I was hoping on expand on my topic by including general relativity (since it seems to be strongly based on differential geometry).

Can anybody think of any dynamical systems in General Relativity that I can play around with? Even suggestions on how to formulate dynamical systems out of problems that have been solved in a purely physical sense would be helpful.

For example, I'm working on trying to analyze the dynamics of a massive particle outside of a blackhole using the Schwarzschild metric. I know how to do this physically, but am not entirely sure as how to form the corresponding dynamical system.

Any ideas?

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In fact, in particular I would be content to find a way to transform this situation into a dynamical system

A free massive particle with proper time $\tau$ is evolving in a Schwarzschild geometry produced by a massive object of mass M, located at r = 0

I'm really not sure how I should go about this though. Could I perhaps use a Hamiltonian approach with Kinetic and (Effective) potential energy?

For example, I'm working on trying to analyze the dynamics of a massive particle outside of a blackhole using the Schwarzschild metric. I know how to do this physically, but am not entirely sure as how to form the corresponding dynamical system.
Unless you are modeling a test mass you cannot use the Schwarzschild metric for a two body problem. To approach a two body problem you might want to take a look at Chris Hillman's short introduction over here: http://www.mountainman.com.au/news98_x.htm
You also might want to lookup the double Kerr solution.

Be aware though that a full analytical solution for a two body problem is impossible. One either makes an approximation or resort to numerical solutions.

For those who are interested here is a link to some visualisations derived from numerical solutions: http://jean-luc.aei.mpg.de/Movies/ [Broken]

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I don't necessarily want full solutions. I'd be more than content to analyze behaviours within local neighbourhoods.

I'm not too terribly concerned with the complexity of the problem so long as I have something I can play around with

I've realized I've been thinking about this the wrong way. Instead of modeling particles, I can just use the field equations to 'derive' a metric. That is, the field equations yield a set of coupled non-linear differential equations which I can use to analyze the behaviour of a metric in local neighbourhoods.

If only it were that simple...

Critical phenomena is general relativity is an excellent example that can be described in the terms of what you want. The review by Gundlach is a good place to start.

http://arxiv.org/abs/0711.4620