Self Gravitating General Solutions to GR

[Wallace]

Hi All, I was hoping someone could help point me in the right direction. I'm interested in simulations of General Relativity with generic mass distributions that are self gravitating, i.e. that the metric of the space-time being considered is dynamically altered by the distribution of mass whose motion is being goverened by that space-time. Phew! I hope that makes some sense!

Basically what I am curious about is that the GR solutions I know of are either vacuum solutions (Schwarschild, Kerr etc) or rely on complete homegenaity and isotropy such as the FLRW solutions. I also know that analytically the 2 body problem is not completely solved, though I think with some approximations can be determined, but certainly the N-body GR problem is far from solved analytically.

There are sophisticated Magneto-Hydro-Dynamic simulations of GR modelling such things as Neutron star collapse, but these as far as I have found all assume the 'test fluid' approximations where the underlying space-time is fixed and unaffected by the fluid.

Does anyone know of any work performed (references would be great!) on numerical simulations dealing with a full arbitrary GR model where the mass curves the space that then directs the mass to accelerate with the full feed-back between the two. Clearly such a thing would be very hard and require a lot of computation, but I can't find any examples where this possibility is even discussed. If I could even find a source that explains exactly why this could not be done that would be a great help.

Can anyone point me in the right direction?

 

[JesseM]

I'm not sure, but wouldn't something like this have had to be true of the simulated black hole collision described here? From the article:

NASA scientists have reached a breakthrough in computer modeling that allows them to simulate what gravitational waves from merging black holes look like. The three-dimensional simulations, the largest astrophysical calculations ever performed on a NASA supercomputer, provide the foundation to explore the universe in an entirely new way.

According to Einstein's math, when two massive black holes merge, all of space jiggles like a bowl of Jell-O as gravitational waves race out from the collision at light speed.

Previous simulations had been plagued by computer crashes. The necessary equations, based on Einstein's theory of general relativity, were far too complex. But scientists at NASA's Goddard Space Flight Center in Greenbelt, Md., have found a method to translate Einstein's math in a way that computers can understand.

...

Einstein's theory of general relativity employs a type of mathematics called tensor calculus, which cannot be turned into computer instructions easily. The equations need to be translated, which greatly expands them. The simplest tensor calculus equations require thousands of lines of computer code. The expansions, called formulations, can be written in many ways. Through mathematical intuition, the Goddard team found the appropriate formulations that led to suitable simulations.

 

[Wallace]

This is still a vacuum solution as the mass is confined to two infinately small points, but the spacetime is clearly being evolved dynamically. Not exactly what I was after but the PhRevD article on this might prove a usefull starting point for me.

Thanks JesseM

 

[Chris Hillman]

I think you want the notion of a "perfect fluid solution"

Hi, Wallace,

I'm entering this thread some time after it appears to have become exinct, but it looks to me like you probably want to read about perfect fluid solutions in gtr. There is a huge literature on this. In fact, there is a huge literature on just the simplest class of such solutions, the static spherically symmetric perfect fluid solutions. See the monograph Exact Solutions of Einstein's Field Equations, by Stephani et al, Second Ed., Cambridge University Press, 2001. Then you can search the arXiv; look especially for papers coauthored by Matt Visser on the static spherically symmetric perfect fluids, wherer there have been major advances since 2001.

In addition to perfect fluids, which are suitable for modeling stars, you might also try looking for models of elastic solids. You would probably also be interested in null dust solutions which can be used to model for example the influx of incoherent massless radiation into a black hole, such as the well-known Vaidya null dust.

Chris Hillman