When can we expect to see numerical simulations of GR for cases that are not highly symmetric? Say 10^3 blobs of matter in an arbitrary initial configuration.
Estimate parameters of the problem. Size of the simulation region? Resolution? Boundary conditions? Duration of simulation?
resolution - 1 meter size of region - 10^6 steps = 10^6 meters duration - 100 Falltime boundary conditions - this I do not know how to do masses - large enough to significantly bend space on a scale of 10^5 steps (10^5 meters) Falltime = the time it takes to fall together -- if we took half the mass in the simulation and placed it 10^5 steps (meters) from the other half the mass in the simulation
Out of curiosity, why are you thinking about this? Seems very random.. What situation is there where there are 10^3 relativistic, interacting masses?
You'd need 40 exabytes (40,000,000 terabytes) of memory or disk storage just to store all values of metric tensor on one timeslice. I don't have recent numbers, but I believe that largest supercomputers and storage systems in the world (e.g. Google) have on the order of 10,000 terabytes. Do you need that fine a resolution, though? You need to think about the answer that you want to get, about desired accuracy, and about the relationship between accuracy and space/time resolution.
I am thinking of times near the big bang. So whatever t gives us 10^3 objects within 10^6 meters. OK it may be the case that things were less clumpy and more uniform in that case we would need more objects I am just trying to understand why people do not do numerical simulation of GR. I see that 10^6^3 is not doable. So let's go to 10^4^3 region 10^4 steps (meters). Can we simulate the physics using numerical methods and Einstein's Equation?
We certainly do. I make (and am currently working with identical data, actually!) movies like the one hamster143 just posted, as a matter of fact! The project is called sxs, and you can find the website here: www.black-holes.org . That's where that video comes from, although it's a few years dated at this point. We also do things like neutron star - black hole mergers, but I don't know of any simulations anyone in our group is doing or has done with 3 or more celestial bodies.
Nabeshin, How do people even do numerical simulations? Naively, when I look at Einstein's equations, it only gives information for the Ricci curvature ... so what determines the Weyl curvature evolution? Furthermore, if you don't know the global topology ahead of time, and instead only know the "topology" of a spacelike slice ... how can you run the equations forward at all? Einstein's equations are local evolution rules, so how can local evolution dictate global topology (whether a spatial point like singularity or ring singularity, or causal horizon, etc appears)? For example the people doing numerical simulations looking at whether naked singularities can form. How can they do it without putting in the topology ahead of time? In a really fun case, how could you "solve" to see if a wormhole appears ... since it seems you'd have to put the topology in ahead of time, which would mean putting in the answer ahead of time? Most differential equations it seems at least intuitively obvious how one would go about simulating it (even if the actual details of actually doing it are often quite involved). It is not obvious to me here at all. It really fascinates me!
Nabeshin might answer in more detail (and more accurately), but, from what I recall, this is in some way a trial-and-error exercise. You start by assuming a 3+1 decomposition of spacetime (i.e. fixing a gauge). There is a formulation of EFE that allows you to "evolve" spatial geometry and matter content of a 3-d hypersurface. Numerical solutions of this formulation are badly prone to formation of coordinate singularities. Once you hit a singularity, you look for a different gauge fixing and a different decomposition that stays continuous in the area. Once you're done, you can end up with a piecewise defined manifold that can, in principle, have nontrivial 4-d topology.
I must confess that I am only a 2nd year undergraduate and my knowledge of a lot of the methodology behind how we solve and evolve the Einstein equations is minimal (I mostly just do visualization of the data to make the movies like you saw above). However, I believe hamster143's explanation is correct, at least in spirit if not in detail. One starts with a set of additional conditions and constraints, and then you solve spatial slices always enforcing (or checking) the constraints. Sorry I can't give a better explanation, perhaps in a couple of years! If you want to investigate on your own, you can check out any of the papers that come out of the research group. Here's one, for example: http://arxiv.org/PS_cache/arxiv/pdf/0809/0809.0002v2.pdf
I am in New York in Rhinebeck south of Albany. I would love to visit Cornel and see what you are doing.
I can do the same thing with Newton's law of gravity. - Is there any numerical simulation of *Solar system* done with GR? Can you point any such software and write down the equation?
Can you point any software that uses GR to model Solar system? To model just Earth's orbit around the Sun, what is QM equation for that? - This is my equation: F = m*a = k* m1m2/r^2, and it can model planets and all their moons pretty accurately, so I do not see how can QM equations can be any better and what could possibly be the difference?
What's your point? You're correct, you do not need full GR to model the solar system. To leading order, you probably don't need GR at all. But for planets like mercury, you can probably use a newtonian approximation to GR, or some other such approximation, in order to get a result within the desired precision. Why are you mentioning QM? Typo? I don't understand what the point of your post is... Do you want someone to do a solar system simulation using full GR? If so, this would be a colossal waste of computing time. Edpell - I'm not sure, but you could probably drop by and chit-chat with the folks I work with. What I personally do is boring undergraduate slave labor, so I'm not a terribly interesting case!
No you cannot. That video shows decaying orbits due to gravitational radiation. Neither is possible in Newton's law of gravity. First off, these are not quantum mechanics calculations. It is GR, but still classical. And second, even in Einstein's time, astronomy measurements of bodies in the solar system showed deviations from Newton's Laws. The data fits GR though. Also, with current measurements, the deviations from Newton's laws can be even more interesting. Maybe your question is: Why do GR simulations? Because while we can solve the two body problem in Newtonian mechanics (but have trouble with the three body and above). We can't solve the two body problem in GR! So simulations are very important. It currently is the only way to make contact between experiment and theory in many cases (especially in the gravitational wave calculation like in that video).
Even so, it sounds like you are getting some interesting exposure to these things. I never learned stuff like that in undergrad. Is there anyway you could coax a gradstudent to come on here and answer a few questions for the curious folks? Maybe they'd enjoy bragging about their work for a bit :)
I know someone (online) who is a numerical relativist working on the 2-body problem at The API in Jena (Germany), but if he's on this forum I don't know what his nickname is. He's a recent PhD so I'd say that would work... maybe I can ask him to come here, or I can relay a question to him if you like?