What are some examples of dynamical systems in General Relativity?

In summary, the conversation revolves around exploring dynamical systems in the context of advanced differential geometry and general relativity. The individual is seeking suggestions on how to approach this topic, specifically in regards to analyzing the behavior of a massive particle outside of a black hole using the Schwarzschild metric. Various approaches, such as using Hamiltonian methods or the field equations, are suggested, with the reminder that a full analytical solution for a two body problem is impossible. The conversation also references Chris Hillman's introduction and the double Kerr solution as sources for further research. In conclusion, the conversation highlights the complexity of the topic and suggests looking into critical phenomena in general relativity as a potential example.
  • #1
Kreizhn
743
1
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 black hole 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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  • #2
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 [itex] \tau [/itex] 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?
 
  • #3
Kreizhn said:
For example, I'm working on trying to analyze the dynamics of a massive particle outside of a black hole 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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  • #4
I don't necessarily want full solutions. I'd be more than content to analyze behaviours within local neighbourhoods.

Thanks for the link.
 
  • #5
I'm not too terribly concerned with the complexity of the problem so long as I have something I can play around with
 
  • #6
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.
 
  • #7
If only it were that simple...
 
  • #8
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
 

1. What is the relationship between General Relativity (GR) and Dynamical Systems?

General Relativity is a theory of gravity that describes the behavior of large-scale objects in the universe. Dynamical Systems is a field of mathematics that studies the behavior of physical systems over time. In the context of GR, Dynamical Systems is used to study the evolution of spacetime and the behavior of objects under the influence of gravity.

2. How does GR explain the curvature of spacetime?

According to GR, the presence of massive objects, such as planets and stars, causes spacetime to curve. This curvature is what we experience as gravity. The more massive an object is, the greater its curvature of spacetime and the stronger its gravitational pull.

3. Can GR be applied to both small and large-scale systems?

Yes, GR can be applied to both small and large-scale systems. It is a theory that explains the behavior of gravity on all scales, from subatomic particles to the entire universe.

4. What are some real-world applications of GR and Dynamical Systems?

GR and Dynamical Systems have many real-world applications, such as predicting the orbits of planets and satellites, studying the behavior of black holes, and understanding the expansion of the universe. They are also used in fields like astrophysics, cosmology, and aerospace engineering.

5. How does GR differ from Newton's theory of gravity?

While Newton's theory of gravity describes gravity as a force between two objects, GR explains gravity as the curvature of spacetime caused by the presence of massive objects. GR also accounts for the effects of gravity on the fabric of spacetime, such as time dilation and gravitational lensing, which are not predicted by Newton's theory.

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