Numerical Simulations of General Relativity: 1000 Blobs

[edpell]

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.

 

[hamster143]

Estimate parameters of the problem.

Size of the simulation region?

Resolution?

Boundary conditions?

Duration of simulation?

 

[edpell]

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?

 

[hamster143]

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.

 

[edpell]

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?

 

[hamster143]

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

But they do.

 

[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.

 

[JustinLevy]

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!

 

[hamster143]

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

 

[edpell]

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

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

 

[Nabeshin]

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

New York.

 

[edpell]

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

 

[Dunnis]

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?

 

[Dunnis]

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!

 

[JustinLevy]

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).

 

[JustinLevy]

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 :)

 

[Frame Dragger]

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?

 

[Dunnis]

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?

 

[Dunnis]

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

What's your point?

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 ?

 

[hamster143]

It is not necessary to use numerical GR to model solar system, because solar system can be modeled 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 modeled analytically, numerical simulations are the way to go.

 

[JustinLevy]

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.

 

[Frame Dragger]

@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?

 

[JustinLevy]

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?

If he'd be willing to come talk basics about his research, that would be a lot of fun. I've always been curious how they do numerical GR.

If you can only relay questions, I guess what I wrote in https://www.physicsforums.com/showpost.php?p=2646415&postcount=9" is along the lines of what I'm curious about. I have a feeling my ignorance of the field would require some translating before those are useful questions though.

Probably the most approachable question is this:
Naively, when I look at Einstein's equations, it only gives information for the Ricci curvature ... so how do you determine the Weyl curvature evolution in numerical GR?

My understanding is that in analytic solutions they use symmetry arguments and boundary conditions at infinity to constrain the form of the metric, which effectively puts in the Weyl terms. Maybe that is not correct, but even if it is along the right track, in dynamic situations you don't have those luxuries. Naively it looks like the Weyl curvature can just evolve however it wants (I assume that is wrong for some reason though).

 

[edpell]

My understanding is that in analytic solutions they use symmetry arguments and boundary conditions at infinity to constrain the form of the metric, which effectively puts in the Weyl terms. Maybe that is not correct, but even if it is along the right track, in dynamic situations you don't have those luxuries. Naively it looks like the Weyl curvature can just evolve however it wants (I assume that is wrong for some reason though).

I also have a very limited understanding of the issue for GR simulation. But it does seem like two of the issues are initial conditions and coordinate systems.

I would hope that there is some way to give an initial placement and velocity to N-bodies and then through some sort of relaxation technique take an initial guessed (known wrong) space-time condition and relax it to the correct space-time condition.

On the coordinates the paper offered in an earlier post covers some of this. They use two systems of coordinates one fixed and one deformable. The issue is in GR you do not have a fixed cartesian (sic) grid the space-time deforms! So what do you simulate? Give me a lever and a place to stand and I will move the Earth, in this case give me a place to stand and I will simulate GR.

 

[Stingray]

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?

Einstein's equation: $R_{ab} - \frac{1}{2} g_{ab} R = 8\pi T_{ab}$. This is a nonlinear PDE. Unlike in Newtonian gravity, the field has its own dynamical degrees of freedom in GR. You can not code up anything to solve this in 5 hours. You need a great deal of theoretical knowledge even to turn it into something you might try to write into a program. It is still extremely difficult and time consuming to get high quality simulations. The problem is VASTLY more complicated than solving a problem in Newtonian gravity.

That said, something like the solar system does not need full GR. It is adequate to use what is called the post-Newtonian approximation of it. This assume weak fields and slow speeds to analytically simplify the equations. The result can be simulated without much effort. The very lowest order corrections have a similar effect to making the gravitational field of the Sun look like it is coming from a somewhat more oblate object. This kind of thing is included in modern simulations of the solar system.

 

[atyy]

I also have a very limited understanding of the issue for GR simulation. But it does seem like two of the issues are initial conditions and coordinate systems.

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

http://www.astrobiology.ucla.edu/OTHER/SSO/ [Broken]

 

[Stingray]

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?

You split the spacetime up into a stack of spacelike hypersurfaces representing surfaces of constant time. It is usually assumed that each of these hypersurfaces has the same topology, so the manifold looks like $M = \Sigma \times \mathbb{R}$. You now specify initial data on one of the hypersurfaces and evolve it forward.

This requires writing Einstein's equation in terms of geometric objects intrinsic to the hypersurfaces. They all have an intrinsic (3D) metric as well as an extrinsic curvature. These things can be related to the 4D curvature (and therefore Einstein's equation) using the Gauss-Codazzi equations. You also have to worry about how the hypersurfaces stack on top of each other. This is parameterized by a lapse function ("relative time separation between leaves") and a shift vector ("shearing between leaves"). The lapse and shift are not constrained by Einstein's equation, and must be specified. Bad choices quickly lead to singular coordinate systems (and crashed computers).

In terms of these variables, Einstein's equation reduces to a pair of constraint equations and a pair of evolution equations for 3-metric and extrinsic curvature. The constraints are analogous to Laplace equations, and do not involve time derivatives. If they are satisfied on one hypersurface, one can show that use of the evolution equations alone guarantees that they are satisfied on all other hypersurfaces.

The problem of specifying initial data is very difficult. So is the boundary problem. There are also a lot of subtleties with precisely which form of the Einstein equation to use, which variables are most efficient (the ones I outlined above aren't the best for numerical stability), lapse and shift choices, etc.

Although I'm not working in numerical relativity, I have a fair bit of knowledge about it if you have more specific questions.

 

[Dunnis]

Einstein's equation: $$R_{ab} - \\frac{1}{2} g_{ab} R = 8\\pi T_{ab}$$ . This is a nonlinear PDE. Unlike in Newtonian gravity, the field has its own dynamical degrees of freedom in GR. You can not code up anything to solve this in 5 hours. You need a great deal of theoretical knowledge even to turn it into something you might try to write into a program. It is still extremely difficult and time consuming to get high quality simulations. The problem is VASTLY more complicated than solving a problem in Newtonian gravity.

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.

           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.

That said, something like the solar system does not need full GR. It is adequate to use what is called the post-Newtonian approximation of it. This assume weak fields and slow speeds to analytically simplify the equations. The result can be simulated without much effort. The very lowest order corrections have a similar effect to making the gravitational field of the Sun look like it is coming from a somewhat more oblate object. This kind of thing is included in modern simulations of the solar system.

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?

 

[Dunnis]

@Dragger: My angry friend, with slight amusement I acknowledge your emotional distress, but I do not recall to have been talking to you before, so can you just tell me what is it we are arguing about and what did I say to make you cry? -- You are confusing analytical and dynamical "solution", former is 'exact' and later is 'approximation', by definition. You surely do not mean to deny that classical modeling of the sun, planets and all their moons - with high degree of precision and absolute stability through millions of years of simulated time - would be impossible, do you?

It is not necessary to use numerical GR to model solar system, because solar system can be modeled analytically to a high degree of accuracy.

Solar system can not be modeled analytically, but it can be approximated dynamically... with high degree of precision and absolute stability through millions of years of simulated time. -- This is not the question about necessity, but about ability. Let's make it really, really simple, let's forget all the moons and all the other planets, so let's just model Sun and Mercury, can anyone print down GR equation which can model this?

GR corrections to Newton's law are in good agreement with experiment.

What experiment? What correction?

 

[Frame Dragger]

@Dragger: My angry friend, with slight amusement I acknowledge your emotional distress, but I do not recall to have been talking to you before, so can you just tell me what is it we are arguing about and what did I say to make you cry?

Ah, Dunnis, that was the first time I've ever "spoken" to you. What you've interpreted as anger (and crying?) is just a global frusteration with the bulk of your posts. If you want to play the psychologist, let me help... I suppose there was some referred anger which I generally have aimed towards all people who talk, and talk, but choose not to listen to, or understand the responses given.

I think that's enough playing footsie, don't you? The point of the 2-body problem, beyond the actual model, is that when we talk about it we mean the EXACT solution. Again, obviously, when I say that 3+ is unsolvable, I am again speaking of EXACT solutions. You seem to be a code-jockey with delusions "above your station" so to speak. You were told quite a while ago that you were pursuing a dead-end, a "waste of computer time", and instead of accepting that as the state of affairs, you've reformulated your original pointless query into a new one.

The answer is the same, and it's the one Nabeshin, Justin Levy, and Hamster gave you. Like it or not. You're acting like a moderately well educated (I say moderate given "you're" instead of "your"... always a giveaway) brat. If what you're asking for is so easy, why not hit the old internet and find that equation? Better yet, find something simpler and see if you really CAN do any of what you claim.

Btw, this whole post is absolutely SOAKED in my angry tears. :rofl:





--------------------------------------------------------------------------------
@JustinLevy: I've invited him to join us here, or if he'd prefer not to, to relay your question. I suspect he might come over here, if he isn't already however.

 

[Dunnis]

It would be a waste of computer time because the gravity is so weak in the solar system...

...it is a waste of computer time because you don't seem to understand how massive these calculations are.

Gravity is sooo weak.. I dooon't understand.. calculations are maaassive..
..it's waaaaste of time... it's sooo unnecessary. Awww, you had me at "hello".

Since you clearly do not know this field...

:rofl:

[edit: someone actually edited my post and removed the part where I said how amused I was with this comment, so I'll just follow Dragger's example, put 'rotating head' and call this person delusional, as that seem to the a proper and allowed way to communicate around here. Hilarious indeed.]

Let's make this very clear right now.

Let's do that, but let's also not forget that previously you said "video shows decaying orbits due to gravitational radiation", so again - can you support that statement with some reference?

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?

No, but I would like to know how was that conclusion made, when and by whom. What I can tell you though, is that I can model precession of Mercury in many different ways with Newton's gravity, since once all the planets and moons are in the simulation, all interacting simultaneously with each other, and once approximation is upgraded from point masses to volumes and densities, then everything works just fine.

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

No, I'm here to learn how GR does numerical modeling, what is the meaning of the terms in those equations and how to apply it on the simple case scenario with only two planets.

 

[Frame Dragger]

@Dunnis: "The simple case scenario with only two planets" :Rofl: ... Well, it's been nice knowing you on this site Dunnis, but I think the rest of us will be wishing you Adieu fairly soon. Insulting other users (one of which, not me) was trying to help you, is a quick way to feel the wintery freshness of BAN.

Oh, and... "No, but I would like to know how was that conclusion made, when and by whom."... ok... try Google, or a textbook, or school! You want to be spoon-fed GR... not going to happen, for you, or anyone.