GR simulation - is it so difficult?

[Dmitry67]

I mean, why there are so many speculations? It must be 'shut up and press enter' :)

For example, some people say that the second horizon in the Kerr black hole is not dangerous. Others claim that due to infinite concentration of energy there there is a real singularity there so everything will be torn apart.

We even don't have a model of a real kerr black hole. Real - means that there is an incoming mass and also light of the residual matter, orbiting and falling inside the ring, attempting to escape but captured by the Cauchy horizon

Why can't we put it into a parralel supercomputer and get an aswer? calulations for QM are much more complicated! We need to track the state of many volume 'voxels'... Isnt it the same (oreven easire then) done in the millenium simulation? http://www.mpa-garching.mpg.de/galform/millennium/

 

[shoehorn]

I can't make any sense of most of what you're saying. However, this bit:

Why can't we put it into a parralel supercomputer and get an aswer? calulations for QM are much more complicated!

is clearly false. Numerical studies of quantum mechanical systems, particularly many-body systems, are nowhere near as complex as numerical simulation of the Einstein equations.

There are many, many excellent papers on numerical analyses of the Einstein equations (and the various decomposition schemes) in various regimes on the ArXiv which you might be interested in.

 

[Dmitry67]

I can't make any sense of most of what you're saying. However, this bit:

is clearly false. Numerical studies of quantum mechanical systems, particularly many-body systems, are nowhere near as complex as numerical simulation of the Einstein equations.

As far as I know, simulation even of few quark systems in quantum chromodynamics is still far from what our modern supercomputers can do. Even for few quarks, not talking about the nuclei.

Probably you mix it with the simulations based on the simplifications of the QFT, which dramatically reduce the number of calculations (different models of the nuclei, lattice approaches etc).

I will be happy if you prove that I am wrong providing a link of a direct simulation (without lattice and models)

 

[Dmitry67]

Just found:
http://arxiv.org/PS_cache/nucl-th/pdf/0109/0109007v1.pdf

Monte Carlo calculations of Light Nuclei
Nuclei present a spin-isospin nightmare that make the cost of direct Monte Carlo
calculations grow exponentially with A.

So the problem is that when you simulate the gravitational interaction of N bodies, the complexity of each step is about N**2 (each body versus all others) so it is a polinomical task.

In QM it is an exponent. Check P=NP problem.

 

[shoehorn]

Typically, the simulation of the Einstein equations is highly scheme dependent in the sense that the particular strategy of decomposition chosen has significant effects on the possible run time of the simulation. For standard conformal or conformal transverse-traceless decompositions, only a limited number of spacetimes can actually be simulated without your solver blowing up the conformal factor. This is a general consequence of attempting to simulate such highly coupled elliptic systems and is the reason why I'm of the opinion that simulating the Einstein equations is significantly more involved than typical QM or QFT simulations.

 

[Hurkyl]

'Simulating' QM and GR both involve the numerical solution to a differential equation. The difficulty depends on the level of detail needed and the numerical stability of the solution, differential equation, and numerical technique. I don't think such simplistic analyses can faithfully describe the relative difficulty of particular simulations.

 

[cam875]

even the best supercomputers can't perform calculations that quick nor are they that parallel. But if quantum computers ever become good and practical than theyll be able to handle all those calculations, kind of ironic (a computer built around QM doing QM calculations)

 

[shoehorn]

'Simulating' QM and GR both involve the numerical solution to a differential equation.

Obviously.

The difficulty depends on the level of detail needed and the numerical stability of the solution, differential equation, and numerical technique.

In the case of solutions to the Einstein equations without the imposed existence of static Killing vectors, the difficulty of a numerical simulation depends principally on the fact that the underlying dynamical equations are particularly unpleasant coupled elliptic PDEs. These are typically much, much harder to examine analytically in comparison with the standard fare one encounters in (lattice or otherwise) QFTs. In numerical relativity, computational issues such as stability of the solutions are almost entirely secondary to the fact that the underlying equations are horribly intricate.

Unfortunately, despite some reasonably cute approaches to the problem coming to light over the past decade, numerical GR still relies largely on the conformal transverse-traceless decompositions of the three-metric first hinted at by York back in the early seventies. While this allows one to determine a fair amount about the qualitative nature of solutions via familiar elliptic estimates, we still know very little about the quantitative approach. Hence my claim that numerical GR is hard.

I don't think such simplistic analyses can faithfully describe the relative difficulty of particular simulations.

Of course. But to do the question justice we'd have to start a discussion of precisely why coupled elliptic PDEs are so much more intractable than the equations found in QM. Frankly, life is too short to attempt that on a message board.

 

[Dmitry67]

Yes, I understand how hard GR simulation might be (I designed few programs for N-body simulation - but with Newtonian gravity...), but as far as I know QM is harder - not in general, but QFT for non-abelian charges. Even if only 2 quarks interact, how many particles are inolved? 2 quarks + gluons. But gluons are charged, so they also emit gluons, ad infinitum...

Also, while in classical simulation you make lattice smaller and smaller you get more and more precise approximation. In QM, when you make lattice smaller, you need to take into account what happens in shorter and shorter time intervals - and the smaller interalis the weirder things can happen, like (t+ - t-) virtual pairs et cetera

 

[MathematicalPhysicist]

We even don't have a model of a real kerr black hole. Real - means that there is an incoming mass and also light of the residual matter, orbiting and falling inside the ring, attempting to escape but captured by the Cauchy horizon

What Cauchy has got to do with gravity?, as far as I remember he didn't have anything to do with physics.

 

[Dmitry67]

What Cauchy has got to do with gravity?, as far as I remember he didn't have anything to do with physics.

Theoretically - nothing.
However, as it blocks the light from escaping, it creates a 'blue sheet' - an area with an infinite energy density and hence starts to curve spacetime and can create spacetime singularity