Measuring Observables in 3+1 Formalism

[tendor]

Hello,

lets say I have Hamiltonian aproximation $H(\vec{x}_a,\vec{p}_a)$ of gravitational interaction that can be used for n-body simulation of mass particles and photons. Spacetime curved by simulated particles would be asymptoticly flat. But I don't have a metric etc. All I have is evolution of particles based on coordinate time, so I have masses, momenta and coordinates of particles.

My problem is I can't find materials that would told me how to use test particles and photons in order to extract observables etc. from such simulation...

 

[Muphrid]

You have particles, and so you have the stress energy tensor. Solving the Einstein equations for a given stress energy tensor is typically what's done anyway--we don't start off with the metric, we solve for it.

I'm not sure what observables you're planning on figuring out, though. Could you elaborate on what information it is you want from this system?

 

[tendor]

I'm sorry for poor description of the problem. When I've said I don't have a metric I have meant that I can't solve E.e. analyticly because configuration is too complex. If anything, I would have to get information about the metric numerically from the simulation itself.

Lets say I would like to measure a proper time of test particle located in studied area. Simulation is "ticking" with coordinate time which can be seen as a proper time of distant observer.

 

[Muphrid]

I see. Well, if you already have this simulation working, you can drop the test particle in, and at each time step, you say that $\Delta \tau = \Delta t/\gamma$ and just look at the particle's coordinate velocity at every time step to figure out $\gamma$.

 

[tendor]

Thank you Muphrid. :) That would cover time dilation due to the relative velocity. I just wonder if its really so simple in the presence of gravitation field.

I'll think about that in some better hour when my mind will working...

 

[Muphrid]

Oh, of course, silly me, your problem is that you don't know the metric, so the metric that would be involved in the GR case is at issue.

I mean, this is a general problem for numerical relativity--finding the initial data (metric, etc.) that corresponds to a starting matter distribution. You talk about a Hamiltonian approximation--is this something you already have? Are you familiar with the ADM formalism?

 

[tendor]

I'm playing with http://arxiv.org/abs/1003.0561 It works fine for things like perihelion precession, bending of light etc.

I've read about ADM formalism in general but practical usage without examples or practice and experience is another thing.