Numerical Simulations of General Relativity: 1000 Blobs

[Stingray]

Complexity does not scare me, but I acknowledge you're haste to underestimate a complete stranger. The question is rather if you can write down the meanings of the terms in those equations and their relation to physical properties so we know how to input real numerical values and use it in practical case scenario.

Just getting numerical simulations of Einstein's equation that didn't crash almost instantaneously was a major research area for years. This has only recently been solved. The issues are very different from any other field of simulation I'm aware of. It is not a stretch to assume to that you do not have the background to figure this out. You clearly do not understand the physics at all. You cannot simulate this system without that background. It is not (to repeat again) just solving a bunch of ODEs.

At time t0 relative velocity between mass M1 and M2 is zero, the distance between them is r, what is their velocity and distance at time t0+10 seconds? - Can you show me how the terms in GR equation relate to this and what would be their numerical value here? That's all I need to know.

No. That's an extremely complicated question. There is no direct translation of Newtonian concepts in full GR. If you're really interested, find a review paper and try to understand the field that way. http://relativity.livingreviews.org/" is a good place to start.

Can you write down this "correction equation" you are talking about so I can see the physical and mathematical meaning of that correction? - I do not believe any corrections of any kind are included in any simulation of any solar system, can you point any such software?

The basics of low-order post-Newtonian simulations are reasonably straightforward and could be explained to someone who is not an expert. I really don't have time to do this here. I can't point to any downloadable software for you (unless you just want a bunch of test bodies moving in the field of the Sun with no mutual interaction). As examples that these things are done, take a look at:

http://arxiv4.library.cornell.edu/abs/0802.3371" : This paper shows that the solar system is significantly more stable over very long time scales with post-Newtonian corrections.

"trs-new.jpl.nasa.gov/dspace/bitstream/2014/8903/1/02-1476.pdf"[/URL]: This states that PN effects have been included in ephemeris calculations needed for spacecraft navigation since the 1960's.

[PLAIN]http://arxiv.org/abs/astro-ph/0701612" : PN N-body simulation.

There are (and have been) experiments of many kinds in the solar system that accurately test the PN equations and look for deviations that might signal a problem with GR.

 

[Frame Dragger]

Just getting numerical simulations of Einstein's equation that didn't crash almost instantaneously was a major research area for years. This has only recently been solved.

40 years spent just on that, right? I would also add, @Dunnis: Of course you're not afraid; to be afraid of something requires a basic appreciation of it, and you're literally clueless. You can't be afraid of a "threat" that is beyond your perception. Stop cluttering the thread.

 

[Stingray]

40 years spent just on that, right?

Probably closer to 30, but yeah, it was a really long time.

 

[Ben Niehoff]

           r
M1-------------------M2              r'
M1->-----------<-M2

At time t0 relative velocity between mass M1 and M2 is zero, the distance between them is r, what is their velocity and distance at time t0+10 seconds? - Can you show me how the terms in GR equation relate to this and what would be their numerical value here? That's all I need to know.

It's been pointed out that this question is much more complex than you expect. To elucidate some of the complexity, consider:

1. In GR, you have to solve a minimization problem and do an integral just in order to find the distance between two points! And the answer is not necessarily unique!

2. In GR, gravity is not a force at all. A free-falling body is in a local inertial frame of reference; that is, a free-falling body experiences no acceleration!

3. In GR, the amount of gravitation (i.e. spacetime curvature) between two bodies is not simply a function of distance (such as 1/r^2, etc.). Spacetime itself is a dynamical continuum, and needs to be approximated by some set of finite elements. This is akin to trying to simulate the electric field between N bodies, including radiation effects...except GR is more complicated because the evolution equations for spacetime are highly nonlinear.

4. In GR, gravitation responds not only to mass, but also to energy, pressure, and stress. The mass-energy density itself is not well-defined, but is an observer-dependent quantity. Instead, one needs the entire stress-energy tensor.

This is just the beginning. As others mention, there are also issues with boundary conditions, choices of coordinates, etc.

 

[edpell]

http://relativity.livingreviews.org/Articles/lrr-2000-5/

http://www.astrobiology.ucla.edu/OTHER/SSO/

Atyy thanks the paper on initial conditions is great.

Ed

 

[yuiop]

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?

Hulse and Taylor measured the orbital decay of a binary pulsar which was in agreement with the predictions of GR (and not Newton) and their work earned them a Nobel prize so it must have been reasonably rigorous. See http://www.cv.nrao.edu/course/astr534/PulsarTiming.html

This gravity simulator http://gravit.slowchop.com/ unfortunately does not include PN effects but I could not resist linking to it, because it does produce very beautiful multiple particle gravity simulations that run fast and smooth on an average PC in 3D! Enjoy.

 

[Dunnis]

Again, obviously, when I say that 3+ is unsolvable, I am again speaking of EXACT solutions.

Exact, eh? At first I thought it was ridiculous to even talk about dynamics and numerical modeling in terms of EXACT solutions, and then I realized your nonsensical objection is self-refuting.



1.) 3+ body problem is unsolvable EXACTLY

2.) Planetary distances, velocities, masses and densities are unmeasurable EXACTLY
--------------------------------------------------------------------------------------

* During 17th to 19th century, some people, different people, somewhere, somehow measured the rate of precession of the perihelion of Mercury's orbit, which led to the final conclusion this rate to be around 5599 arc seconds per century.

* Urbain Le Verrier reported that the slow precession of Mercury’s orbit around the Sun could not be completely explained by Newtonian mechanics and perturbations by the KNOWN planets and masses in the Solar system and according to KNOWN calculation.

These calculations were obviously never supposed to be exact, but what is worse you were not there to tell them it was IMPOSSIBLE, plus they used some analytical and not dynamical methods as there were no computers and numerical modeling back then. Their "calculations" were far from being exact, even appropriate, and their measurements of distances, planetary mass and densities were not EXACTLY what we know today, as they were oblivious to many moons and many other masses we later discovered in the Solar system and measured with much better accuracy. Nevertheless, they got some pretty numbers...


5025.6 - due to precession of the equinoxes - EXACT?
+531.4 - due to gravitational tug of other planets - EXACT?
------------------------------------------------------------------
5557.0(calculated) - 5599.7(observed - EXACT?)

= 42.7 arc SECONDS per CENTURY... give or take 900", eh?


First they use Newtons law of gravity to obtain some result they presume to be vey accurate, just to later use it against itself and prove it was never correct to start with?! - How in the world could anyone think they were able to exactly solve the precession of the equinoxes and gravitational tug of other planets?! Analytically?? Without computers and with inaccurate and incomplete measurements that have much less accuracy than the final result?


Let me tell you, if you take MODERN measurements and dynamically integrate all the masses via volumes and densities, then you still can play around *within measurement error* and obtain wide range of conditions that will satisfy Mercury’s precession. For example, you can tweak and shift mass distribution of overall system, within measurement error, and get the precession due to equinoxes to be, say 1207 arc seconds, and due to gravitational tug of other planets 4393 arc seconds per century, which would agree with observation.

Yes, dynamical approximations can be far more "exact" than any of those measurements, otherwise we could not obtain stable orbits for millions of simulated years that underwent billions of perturbations and interaction with all the other planets and still match "exact" position at "exact" time of each planet or moon, where "exact" means that the difference between solution and observation is no more than the "error" in measurements themselves, and that is "exact" as you can get.

 

[Dunnis]

It's been pointed out that this question is much more complex than you expect. To elucidate some of the complexity, consider:

Don't worry about me, I like complexity. What you telling us is your own perception, and if you find it so complex that you can not solve the simplest problem, then maybe you should not be attempting to give advice about it at all.

You go ahead and pick any problem, example or equation from classical physics or dynamical modeling and computer simulation, and I will precisely tell you all the numerical values of all the constants and explain how all the terms relate to physical properties, and this is all I'm asking to know about GR equations - how to apply them on the simplest case scenario, NO MATTER HOW COMPLEX IT APPEARS TO ANYONE, it is either solvable or not.

1. In GR, you have to solve a minimization problem and do an integral just in order to find the distance between two points! And the answer is not necessarily unique!

You are not making sense, you just said the problem can not be solved.

Please provide reference since you seem to be misinterpreting something.

2. In GR, gravity is not a force at all. A free-falling body is in a local inertial frame of reference; that is, a free-falling body experiences no acceleration!

I do not care what you name your terms, I just want to see how they relate to practical case scenario, then I'll figure out the rest and what is what.

3. In GR, the amount of gravitation (i.e. spacetime curvature) between two bodies is not simply a function of distance (such as 1/r^2, etc.). Spacetime itself is a dynamical continuum, and needs to be approximated by some set of finite elements. This is akin to trying to simulate the electric field between N bodies, including radiation effects...except GR is more complicated because the evolution equations for spacetime are highly nonlinear.

Ok. Now, you either can solve that problem or not. If you can, than you should be able to print those equations down and show how to use them in this simple case scenario.

4. In GR, gravitation responds not only to mass, but also to energy, pressure, and stress. The mass-energy density itself is not well-defined, but is an observer-dependent quantity. Instead, one needs the entire stress-energy tensor. This is just the beginning. As others mention, there are also issues with boundary conditions, choices of coordinates, etc.

Sounds interesting. So, do you know how to apply those equations to this example, or not?

 

[atyy]

Let me tell you, if you take MODERN measurements and dynamically integrate all the masses via volumes and densities, then you still can play around *within measurement error* and obtain wide range of conditions that will satisfy Mercury’s precession. For example, you can tweak and shift mass distribution of overall system, within measurement error, and get the precession due to equinoxes to be, say 1207 arc seconds, and due to gravitational tug of other planets 4393 arc seconds per century, which would agree with observation.

See section 3.5 of http://relativity.livingreviews.org/Articles/lrr-2006-3/

 

[atyy]

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?

Maybe you can find some references from http://relativity.livingreviews.org/Articles/lrr-2000-5/ Section 3:

"With no matter to support the gravitational field, we find that we must usually use a spacetime with a non-trivial topology, although a black hole can be supported by a compact gravitational wave [2]. It is certainly possible to construct black-hole solutions supported by matter [90, 91, 5], but it is often desirable to avoid the complications of matter sources.

This raises a point about solutions of Einstein’s equations which we have not yet mentioned. When constructing solutions of Einstein’s initial-value equations, we are free to specify the topology of the initial-data hypersurface. Einstein’s equations of general relativity place no constraints on the topology of the spacetime they describe or of spacelike hypersurfaces that foliate it. For astrophysical black holes (i.e., black holes in an asymptotically flat spacetime), the freedom in the choice of the topology has relatively minor consequences. The primary effects of different topology choices are hidden within the black hole’s event horizon."

 

[Dunnis]

See section 3.5 of http://relativity.livingreviews.org/...es/lrr-2006-3/

-"Now, the measured perihelion shift of Mercury is known accurately: After the perturbing effects of the other planets have been accounted for, the excess shift is known to about 0.1 percent from radar observations of Mercury between 1966 and 1990."

These "perturbing effects" are calculated with classical methods, so in essence that sentence claims the great deal of accuracy comes from some "dubious" calculations, where the error nonetheless is only a couple of percents, it's contradicting.

We have three different values here, and all three of them come with their own method of acquisition and error built within it, where the second two values require even more measurements that all bring their own errors with them.1. "NUMBER A", perihelion measurements - observation error ?%


2. "NUMBER "CX", precession of the equinoxes - calculation error ?%
+ liner and spin velocity, density, mass distribution - observation error ?%

3. "NUMBER "CY", gravity tug of all the other planets -calculation error ?%
+ velocities and mass distribution of all the other objects - observation error ?%
-----------------------------------------------------------------------------So the error of 43 per 5600 arc SECONDS per CENTURY proves very, very, very good accuracy of Newton's gravity actually. See if GR can account for the perturbation and gravity tug of all the other planets and get the accuracy withing 43 second per century.So, first we say 3-body problem is IMPOSSIBLE to solve and when it nevertheless gets solved with 99% accuracy with Newton's gravity, then we say it is wrong because it was not 100% accurate, right? - Try to find the error for each of those numbers in calculations and measurements and you will see that 43 SECONDS per CENTURY is far, far less than what would be the expected error, or indeed far less than the error that comes along with any of those measurements.

 

[Frame Dragger]

Exact, eh? At first I thought it was ridiculous to even talk about dynamics and numerical modeling in terms of EXACT solutions, and then I realized your nonsensical objection is self-refuting.

You don't even know what is meant by "exact" do you? Lord you're completely out of your depth. http://en.wikipedia.org/wiki/Exact_differential_equation

 

[yuiop]

This is the closest thing to a simulation of an orbit in GR available on the internet http://www.fourmilab.ch/gravitation/orbits/ for the very limited case of a low mass test particle orbiting a massive body that has a mass that is many orders of magnitude larger than the mass of the test particle. The source of the applet is downloadable for anyone who wants to see the equations used. It does not seem to include orbital decay due to gravitational radiation but that is probably reasonable for a test particle of insignificant mass. I think the program uses exact GR equations which are possible for this very limited case, rather than numerical aproximations, but it might still be of interest.

 

[Frame Dragger]

On a note unrelated to Dunnis...

@Justin Levy:

From my friend working at API in Germany as a PhD in NR. He is working on the 2-body problem.

Hi FD,

This question is not simple to answer. First, though, I would point out that the Weyl tensor is not a primary variable of relativity. All information about a spacetime is contained in the metric, from which the Ricci tensor and Weyl tensor are computed. Just like any mathematical theory, Einstein's equations alone are not enough to determine the solution. We must make choices, such as the inclusion of black holes, or matter, along with spins, angular momentum and masses, not to mention gauge. Indeed, choices must be made about gravitational wave content. Those choices have an impact on the metric. Given proper choices, one can have enough data to use Einstein's equations to uniquely determine the spacetime metric.

Here's the way to look at it. General Relativity is a theory that admits all metrics which conform to Einstein's equations. That is, Einstein's equations give us a set of rules which determine whether a metric is allowable. The physicist must then see to it to choose a spacetime representing the desired physical solution which is allowable.

In numerical relativity, these choices are larely made in the initial data. While I don't choose the gravitational wave content directly, choices I make, such as conformal flatness, has a direct effect on the gravitational wave content. We then use Einstein's equations to evolve the metric. We're not evolving the Weyl Tensor. We simply calculate the Weyl tensor using the information we possess.

cheers,

grigjd3

I hope this is in some way helpful!

 

[Dunnis]

Just getting numerical simulations of Einstein's equation that didn't crash almost instantaneously was a major research area for years. This has only recently been solved.

Perhaps they should first learn how to write code, handle exceptions and debug before attempting to integrate anything, but I would not blame anyone for such funny failure if the function to be implemented is not temporally continuous, causal and sequentially deterministic.

The issues are very different from any other field of simulation I'm aware of. It is not a stretch to assume to that you do not have the background to figure this out. You clearly do not understand the physics at all. You cannot simulate this system without that background. It is not (to repeat again) just solving a bunch of ODEs.

I find your lack of faith disturbing. As I said, I happen to be a master of the dynamical systems, numerical modeling, animation and integration algorithms. We do not really need to talk about me anymore, so relax and concentrate on the discussion.

I did not mention any ODEs, solve the problem any way you can, if you can.

No. That's an extremely complicated question. There is no direct translation of Newtonian concepts in full GR.

M1= 950kg; M2= 730kg

          r= 25m
M1-------------------M2     


At time t0 relative velocity between mass M1 and M2 is zero.
Q: What is their velocity and distance at time t0+10 seconds?

Are you saying it would take you more than 10-15min to solve this by hand? I was not asking you to translate anything into anything, but simply to assign those variables to the terms in that GR equation. Ok, newer mind that, but can you then print down that other equation, easy one, the one you call "PN"?

The basics of low-order post-Newtonian simulations are reasonably straightforward and could be explained to someone who is not an expert. I really don't have time to do this here. I can't point to any downloadable software for you (unless you just want a bunch of test bodies moving in the field of the Sun with no mutual interaction). As examples that these things are done, take a look at:

http://arxiv4.library.cornell.edu/abs/0802.3371: This paper shows that the solar system is significantly more stable over very long time scales with post-Newtonian corrections.

http://arxiv.org/abs/astro-ph/0701612: PN N-body simulation.

Don't be too proud of this technological terror you've constructed. The ability to destroy a planet is insignificant next to the power of the Force... gravity force and vector calculus that is. These two papers, that's not really about any relativistic corrections as long as they not first learn how to make correct integration algorithm - they have unstable orbits, and I automatically know what rubbish integration methods they use, amateurs.

trs-new.jpl.nasa.gov/dspace/bitstream/2014/8903/1/02-1476.pdf: This states that PN effects have been included in ephemeris calculations needed for spacecraft navigation since the 1960's.

This is something I would like to see, but the link does not work.

 

[Dunnis]

This is the closest thing to a simulation of an orbit in GR available on the internet http://www.fourmilab.ch/gravitation/orbits/ for the very limited case of a low mass test particle orbiting a massive body that has a mass that is many orders of magnitude larger than the mass of the test particle. The source of the applet is downloadable for anyone who wants to see the equations used. It does not seem to include orbital decay due to gravitational radiation but that is probably reasonable for a test particle of insignificant mass. I think the program uses exact GR equations which are possible for this very limited case, rather than numerical aproximations, but it might still be of interest.

Hi, I appreciate all you links. Perhaps after having some basic knowledge, then such source code could save months, but if you don't even know what to look for, then I would not do that unless I wanted to torture myself. -- The best way to learn how to use some equation is to see someone apply it on some simple case scenario. Teach a man how to fish...

 

[atyy]

I find your lack of faith disturbing. As I said, I happen to be a master of the dynamical systems, numerical modeling, animation and integration algorithms. We do not really need to talk about me anymore, so relax and concentrate on the discussion.

So what do you think of the book "Numerical recipes" http://www.nr.com/ ?

 

[Frame Dragger]

You know, I'm glad my friend didn't come over and actually participate in this thread. He'd have been utterly disgusted by Dunnis, and probably wondered what the hell I saw in this place.

@Dunnis: http://en.wikipedia.org/wiki/Dunning–Kruger_effect
http://en.wikipedia.org/wiki/Crank_(person)#The_psychology_of_cranks

You reading those is like you looking into a mirror. Relish the experience. By the way, "We do not really need to talk about me anymore" as you post more drivel, is a truly pathetic cry for attention.

 

[Stingray]

Dunnis, you don't know what you're talking about. We couldn't teach you even the basics of an entire field of research on a message board. Your claimed "mastery" of numerical modeling is irrelevant. It has very little to do with this problem, and you refuse to accept that. Your criticisms of others' work is also silly. I am done responding to your rudeness.

 

[ZapperZ]

This thread is going nowhere fast. It is done.

Zz.