# Numerical GR

## Main Question or Discussion Point

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.

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Estimate parameters of the problem.

Size of the simulation region?

Resolution?

Boundary conditions?

Duration of simulation?

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

Nabeshin
Out of curiosity, why are you thinking about this? Seems very random.. What situation is there where there are 10^3 relativistic, interacting masses?

resolution - 1 meter
size of region - 10^6 steps = 10^6 meters
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.

Out of curiosity, why are you thinking about this? Seems very random.. What situation is there where there are 10^3 relativistic, interacting masses?
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?

I am just trying to understand why people do not do numerical simulation of GR.
But they do.

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Nabeshin
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.

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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.

Nabeshin
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

Nabeshin
Thank you. Cool paper. Are you in New York or California?
New York.

I am in New York in Rhinebeck south of Albany. I would love to visit Cornel and see what you are doing.

But they do.
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?

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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.
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?

Nabeshin
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!

I can do the same thing with Newton's law of gravity.
No you cannot.
That video shows decaying orbits due to gravitational radiation. Neither is possible in Newton's law of gravity.

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?
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).

What I personally do is boring undergraduate slave labor, so I'm not a terribly interesting case!
Even so, it sounds like you are getting some interesting exposure to these things. I never learned stuff like that in undergrad.

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!
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?

No you cannot.
That video shows decaying orbits due to gravitational radiation. Neither is possible in Newton's law of gravity.
Gravitational radiation? Can you support that statement with some reference where I can see exact equations used in such simulations, or if you can just write the equation down here, please?

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.
What are you talking about? You have yet to show me any GR software that can simulate the complete Solar system, while there is thousands of them that can do it Newton's laws of motion and gravity, and with great precision even through millions of years of simulated time. The "error" then is obviously not in the equation, but in our measurements, estimates, approximation of point masses and computer precision.

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).
It is not a problem to be solved exactly in any case, it's a chaotic system, hence numerical integration. N-body problem, like any other time-integral, is "problem" only because we need to make discrete time steps in our simulation, so the real problem is approximation of the time intervals and computer power/precision.

We can't solve the two body problem in GR!
Can, or can not? You should be able to solve n-body problems with GR just like it can be solved (approximated) classically, equations can be different but the time integration algorithm stays pretty much the same, and all the problems from above apply just the same.

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).
I absolutely agree, that's how I know Newton's law of gravity works amazingly well. On the other hand, I wonder how you can confirm that simulation of two black holes is valid?

Simulation = n-body numerical integration.

So is there some GR software that can do Solar system or is GR good just for black holes?

I have no idea why I typed QM where I clearly meant to say GR.

As with any other question, the point is to get answer.
- Can GR model the Solar system and can you point any such software?

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.
It's not the question of what I "need", but what GR can or can not do.

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.
Huh?!?! Why would that be a waste of computer time?

Write down the equation and I will do it in less than 5 hours.

Are you not a programmer? Surely once you have a function to evolute motion of two bodies, like two black holes, then of course you should be able to plug in any number bodies and solve the n-body problem two by two, that's what computers do, why would that be any more waste than doing spinning black holes?

Are you really saying that no one ever even bothered to check those GR equations by simulating complete Solar system? Why then do you think those equations are better than Newton's equations, how can you verify them otherwise, by observing black holes collisions ?

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It is not necessary to use numerical GR to model solar system, because solar system can be modelled analytically to a high degree of accuracy. GR corrections to Newton's law are in good agreement with experiment.

Black hole collisions, on the other hand, can't be modelled analytically, numerical simulations are the way to go.

Huh?!?! Why would that be a waste of computer time?

Write down the equation and I will do it in less than 5 hours.

Are you not a programmer?
It would be a waste of computer time because the gravity is so weak in the solar system, that you can calculate the metric due to a massive body (we CAN analytically solve some one body problems in GR), and then just treat the planets as test particle moving in this background. I do not know of any orbit measurements in the solar system that this level of approximation doesn't fully cover. Maybe someone in astronomy can comment.

But more than that, it is a waste of computer time because you don't seem to understand how massive these calculations are. Especially with your 5 hour comment. Since you clearly do not know this field, can you please calm the tone down some, for if you are asking questions of "experts"/students you might as well trust their advice in the field you do not know, or why bother asking?

So, to address the implicit question: Why are the computations so involved?
In Newtonian mechanics, the gravitational force is instantaneous at a distance, and spacetime is not an active player. The state is given merely by the position and velocity of the bodies. These are the only things you need to keep track of.

Now in GR the spacetime itself is dynamic. So depending on how finely you want to "grid" spacetime, you have a HUGE number of state variables to keep track of. To see modifications from the simple "1 body approximation" I explained above, to use full GR for the solar system would require a tremendous amount of computation time just to get that extra little corrections (which could probably be done much easier with a different GR approximation, like Linearized GR, which is reminescent of solving electrodynamics equations ... since it is linear, many EM type approximations can be applied).

Are you really saying that no one ever even bothered to check those GR equations by simulating complete Solar system? Why then do you think those equations are better than Newton's equations, how can you verify them otherwise, by observing black holes collisions ?
We're saying the predictions were made with GR to within the experimental limits. There is no reason they must solve everything the way you are suggesting, as that is often overkill.

And why do we think GR is better than Newtonian gravity? It has already been explained to you multiple times now that Newtonian gravity already couldn't explain the planetary motion to the degree of experimental accuracy in Einstein's time when he proposed GR.

You have yet to show me any GR software that can simulate the complete Solar system, while there is thousands of them that can do it Newton's laws of motion and gravity, and with great precision even through millions of years of simulated time. The "error" then is obviously not in the equation, but in our measurements, estimates, approximation of point masses and computer precision.
Ugh.
Let's make this very clear right now.
Are you denying that Newton's gravity cannot explain the precession of mercury (already mentioned to you previously)? Are you actually claiming these must be error in measurements since it disagrees with Newton?

If you are here to promote the Newtonian view over Relativity, I am not interested in having this discussion any further.

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@Dunnis: You think that ANY n-Body problem is solvable in GR hm? The 2-body is unsolved, and 3+ is considered IMPOSSIBLE. Your ignorance of the field, and the nature of how PDE's work is staggering given your arrogance and boorish manner. You have little to say, no concept of what you're talking about, yet you say it loudly and rudely in the faces of those who tried to help you.

If you want to be deluded, plese be so in private eh?