Physics and quaternions

[thomheg]

Hi

this is my first post here.
I've written a text about quaternions in physics.
It has grown a bit. I would like to hear some comments about this text.
The topic is quite interesting. You can do everything with quaternions, as far as I can see.
The text is a google.doc presentation. It is quite annoying, but google. docs have benefits, i.e. there is a build in chat system, that you can access if you have a google account.


http://docs.google.com/Present?docid...&skipauth=true

Thomas Heger

[Bogie]

Hi

this is my first post here.
I've written a text about quaternions in physics.
It has grown a bit. I would like to hear some comments about this text.
The topic is quite interesting. You can do everything with quaternions, as far as I can see.
The text is a google.doc presentation. It is quite annoying, but google. docs have benefits, i.e. there is a build in chat system, that you can access if you have a google account.


http://docs.google.com/Present?docid...&skipauth=true

Thomas Heger

Hi Thomas;

I looked at the Google presentation. You put a lot of effort into it and I hope someone more qualified than I am will comment.

[yawyaw]

Welcome to ATM. Good to have more insight into Quaternions and their benefit to Physics.

[Fortis]

Welcome to ATM. Good to have more insight into Quaternions and their benefit to Physics.

By the way, you still have some questions outstanding in the thread about the deflection of starlight by the Sun. Do you intend to address them?

[thomheg]

To describe this idea in a nutshell:
let's construct a kind of pseudo-geometry for a three-dimensional space and encode time into rotation.
Pseudo-geometry means: you have relations as basic principle and not geometry. Distance is a length and the equivalent in that space is an interval. The equivalent to velocity is an angle.
But I make it equivalent to our space of three dimensions according to some simple rules: you 'stack' spaces. That sounds quite impossible, but you could make it possible, if you think of hypersheets. Then our space is a moving hypersheet in that space.
The rules are, that to any possible observer you stack the following events atop the past within the future light cone. That is done by quaternion rotation and dilation.
It is shown by Mr Sweetser, that you could regard that space as the spacetime of GR.
Maxwell himself formulated his equations with quaternions.
What I do now is trying to show, that you could do all kind of quantum physics with such a model. You could not easily calculate something, but that model is assumed to be fundamental, because of the observer-invariance of spacetime.
Now this model is based on GR and could be connected to QM and is expected to be fundamental.
There are very strange predictions possible with this model. That space has three axes, that -in respect to an arbitrary observer- are called plus, north and later.
A shift of the axis of observation makes an inverse out of a relation. Out of mass you get charge, and out of that you get magnetism. If you'd take the principle of relativity into account, then you could imagine, to jump to other places our shift around those axes, by putting your observer to some speed. In combination, you could envision matter from the 'side' and could turn matter into radiation and back.
That model looks quite like our world of observation, if we transform it back into our world by a kind of projection. Out of the multiplicative relations of spacetime we make additive spaces of quantum-mechanics. Those are the same, but describe the world with different methods, one uses multiplications of complex numbers, the other sums of vectors. So one theory describe the vertical 'stripes' and the other the horizontal.

[worldcruiser]

I think you neither understand GR nor QM and you don't have the physic basic knowledge to make a useful critic of mainstream physics.
Just my 2 cents.

[thomheg]

I think you neither understand GR nor QM and you don't have the physic basic knowledge to make a useful critic of mainstream physics.
Just my 2 cents.

Where do you find critic in my statement, and why is my knowledge important?

[worldcruiser]

you have fallen in love with Quaternions and try to restructure all known physics.
How can you do this without knowledge of the currently best theories?

[thomheg]

While, you're doing that, however, you should find out what the principle of equivalence is, as I think that you may have it a bit confused...

I prefer pictures, even linguistic. So let me explain my idea about the principle of relativity (oops, I've wrote equivalence principle) with a picture.
Imagine a piece of spacetime. It's four-dimensional, but that is hard to imagine. But it's possible to 'cheat' a bit. We could leave out one dimension and make it three-dimensional and and encode time into rotation.
So now think about a cube of spacetime of some extension.
The principle of relativity now says, that relative to any such path you draw through spacetime, the laws of physics are the same. More metaphoric: space does look everywhere like space, however and where ever you move. Now I restrict myself to free-falling bodies. That is not such a tough restriction, since that would include all celestial bodies.
So for all kinds of free-falling bodies the universe looks like the universe looks to us. Some differences in the arrangement, but more or less like ours.
That is related to lines that you could draw through this cube of spacetime: all worldlines drawn through this cube must gain a possible universe, with the same laws of physics than those we have in our region of space.
The EP says, that there is nothing special about our way through space and time. So any others would be possible, too. But that very principle would exclude a big-bang. How? Well, simply by showing, that worldlines are possible, that do not intersect with our own history. This argument with expanding space is (I'm sorry!) crap, but I didn't talk about space. Space is what we make of our own way through spacetime. So time and space are to some extend interchangeable. Interesting is, that there is no obvious reason for any limit of the 'some'. What is by the way another misguiding concept, to say, that speed of light is a limit in general. It's a limit for light. That's why I have chosen those 'hops'. At any such hop, it starts counting again from zero. So it is in fact possible to construct valid worldlines, that could by no means share the same history with ours.
(An other btw.: the same argument would exclude black holes. I see it like this: there is a terrible mistake made for a long time and by many: see above. I talked about free-falling bodies. That is with reason. You HAVE to center the reference frame at the observer. If you did, the black hole in the center of our galaxy would be gone and seen from there it would be here.)

Thomas Heger

[thomheg]

you have fallen in love with Quaternions and try to restructure all known physics.
How can you do this without knowledge of the currently best theories?

Quaternions are -of cause- not my choice. I have chosen, what would fit to known facts.
To discuss my personal abilities is useless. I have strong sides and there are way more things, that I don't know anything about. But that's so for everybody, hence gives no means to disqualify anybody to make a statement. You could disqualify the statement, what would make sense, but not the person.
In fact I think, I know a lot, but I've never claimed to be a physicist. I started this project as a challenging task and a kind of hobby.

[tusenfem]

It's four-dimensional, but that is hard to imagine. But it's possible to 'cheat' a bit. We could leave out one dimension and make it three-dimensional and and encode time into rotation.

why would you make time a "rotation"? why not a translation? why not a combination of the two?
4D may be hard to imagine in your mind, but math has made a very interesting way of dealing with that problem, it invented indices A = (a₁, a₂, a₃, a₄), that makes it rather easy to deal with 4D (or 5, 6,7 8....)

So now think about a cube of spacetime of some extension.
The principle of relativity now says, that relative to any such path you draw through spacetime, the laws of physics are the same. More metaphoric: space does look everywhere like space, however and where ever you move. Now I restrict myself to free-falling bodies. That is not such a tough restriction, since that would include all celestial bodies.

No, that is NOT what relativity says. Relativity says (equivalence principle) that the laws of physics are the same for any inertial observer. These observers can either be at rest with respect to each other or move at a constant velocity. And an inertial frame is a frame in which bodies move at a constant velocity or remain at rest when no external forces are working on them.

Please read an introductory book on special relativity, and an introductory book on general physics. Please, understand physics like any ordinary person learns it at school, and when you comprehend that, you can move to try and redefine physics in quaternions.

[thomheg]

why would you make time a "rotation"? why not a translation? why not a combination of the two?
4D may be hard to imagine in your mind, but math has made a very interesting way of dealing with that problem, it invented indices A = (a₁, a₂, a₃, a₄), that makes it rather easy to deal with 4D (or 5, 6,7 8....)

I guess you mean vectors. But that is a critical thing to use in general relativity. I use intervals, that represent not form, but some more abstract relation what is defined by ds²=(cdt)²-dx²-dy²-dz². If you'd take the square root of that and name the components (w,x,y,z) that's it. The indices do not mean coordinates. They have a relation to coordinates I want to show.

No, that is NOT what relativity says. Relativity says (equivalence principle) that the laws of physics are the same for any inertial observer. These observers can either be at rest with respect to each other or move at a constant velocity. And an inertial frame is a frame in which bodies move at a constant velocity or remain at rest when no external forces are working on them.

Please read an introductory book on special relativity, and an introductory book on general physics. Please, understand physics like any ordinary person learns it at school, and when you comprehend that, you can move to try and redefine physics in quaternions.

I try to avoid special theory of relativity and use only general theory of relativity as basis. But I don't like tensors and prefer a quaternion alternative.
SRT is somehow not what I want, because it compares two observations instead of kicking out the observer.

[tusenfem]

I guess you mean vectors. But that is a critical thing to use in general relativity. I use intervals, that represent not form, but some more abstract relation what is defined by ds²=(cdt)²-dx²-dy²-dz². If you'd take the square root of that and name the components (w,x,y,z) that's it. The indices do not mean coordinates. They have a relation to coordinates I want to show.

One cannot take the square root of ds² and then get four components, because
ds² is a scalar.
You need not call it coordinates, it is just components of a vector. Stop using confusing words and write down equations, which show much more than your "pictures in words."

What are we to understand from the following you write above; "I use intervals, that represent not form, but some more abstract relation what is defined by ds²=(cdt)²-dx²-dy²-dz²?"
Intervals that represent not form? -> squze me, does not compute
Maybe you should write it in German and I will translate it into English, depending I understand what you write in German.

There is a wiki page on quaternions, which give all the equations that you probably need. So why not use that in trying to explain to us what you want to explain?

I try to avoid special theory of relativity and use only general theory of relativity as basis. But I don't like tensors and prefer a quaternion alternative.
SRT is somehow not what I want, because it compares two observations instead of kicking out the observer.

But the special theory of relativity is contained in the general theory of relativity. I just said you should start at the special one, because it is easier to understand, and can be used as a base for understanding the general theory.
Why do you not like tensors? In what way are quaternions different? The example you gave in the electric earthquake thread that the transformation in q-space looks like:

v' = q * v * q^-1

looks exactly like transforming a vector v into a new coordinate system with directions given by the columns of the matrix q. But as you keep from telling us what exactly v is we can only guess at what you really understand and what you want to do. According to the wiki page a quaternion is defined as: H = (a + bi + cj + dk) with a,b,c,d real numbers and i² = j² = k² = ijk = -1. Which is basically a vector of which one component is always real (a) and three components that are along imaginary axes i, j and k, and there are some special operants like ij=k which in normal math would be that the cross product of x and y gives z.

All this kind of information we have to find out for ourselves, but it is YOUR JOB to explain it to us clearly, not with all that "pictures in words".

[yawyaw]

you have fallen in love with Quaternions and try to restructure all known physics.
How can you do this without knowledge of the currently best theories?

The Four-vectors in Relativity seem to be quaternions by another name.
See the quote in the link below:

" Note that this differs from the scalar product of vectors because of the minus sign. That minus sign is necessary for the property of invariance of the length of the 4-vectors."

http://hyperphysics.phy-astr.gsu.edu...ativ/vec4.html

These 4-vectors ARE quaternions as Hamilton defined them. Could it be that Relativity is all about quaternions but todays physicists have rediscovered them as 4-vectors? It could be that Quaternions are the real Relativity Theory and Einstein's and Minkowsi's mathematics is an attempt to understand Relativity with the wrong mathematics.

That is my opinion based on 4-vectors. The momentum energy 4-vector is also a quaternion and it is a quaternion energy E=-mu/R + mcv for gravity Theory.

[papageno]

That is my opinion based on 4-vectors. The momentum energy 4-vector is also a quaternion and it is a quaternion energy E=-mu/R + mcv for gravity Theory.

I guess that answers one of my questions in your thread...

I have never seen quaternions-worshippers actually prove that quaternions do a better job that four-vectors, or can provide more information.

[thomheg]

I guess that answers one of my questions in your thread...

I have never seen quaternions-worshippers actually prove that quaternions do a better job that four-vectors, or can provide more information.

It's not worshiping. Those numbers fit to the problem of interest. In fact, you do not need to know how to manage quaternions in practice.
Just forget about that, just only think about anti-symmetry and four dimensions. That's how relativity actually IS. No doubt is possible. If you'd take into account, that somehow the stress-energy tensor has to be filled, by things that influence gravity, than everything does. So everything has to be found in spacetime somehow.
So everything has to be modeled by quaternions. And, believe it or not, that is possible.
It would be nice to know, how you could really calculate in such a system, but till now, I couldn't find the right tools.

TH

[thomheg]

The Four-vectors in Relativity seem to be quaternions by another name.
See the quote in the link below:

" Note that this differs from the scalar product of vectors because of the minus sign. That minus sign is necessary for the property of invariance of the length of the 4-vectors."

http://hyperphysics.phy-astr.gsu.edu...ativ/vec4.html

These 4-vectors ARE quaternions as Hamilton defined them. Could it be that Relativity is all about quaternions but todays physicists have rediscovered them as 4-vectors? It could be that Quaternions are the real Relativity Theory and Einstein's and Minkowsi's mathematics is an attempt to understand Relativity with the wrong mathematics.

To call that a four-vector was not a good idea, because it covered the relation between vectors (what are based on geometry, and that on sums) and relations. Relations should be modeled as intervals, what are not vectors, since the relations don't have a orientation (in respect to an observer, what a vector has). Spacetime is able to model curvature. So a worldline that is straight to one observer, could be curve for another. That would exclude the use of vectors, since you'd have otherwise to introduce a frame of reference, but that would contradict the principle of relativity.
Intervals are the basis of spacetime and don't mean length. So spacetime is not geometric in the sense of a vector. It's based on multiplications, that connect neighboring 'points'. Those points are the quaternions, but they are not defined by their position, but by their interval to other such 'points'. A fundamental process could way easier deal with intervals than with vectors, because a vector needs an observer, but those processes don't.
If spacetime would be treated *really* relativistic, than all those strange mixed-mode models would vanish. Amongst those are things like black-holes...


TH

[papageno]

It's not worshiping. Those numbers fit to the problem of interest. In fact, you do not need to know how to manage quaternions in practice.

[SNIP!]

What you wrote does not even remotely address what I wrote in response to yawyaw.

[thomheg]

What you wrote does not even remotely address what I wrote in response to yawyaw.

Maybe he will answer them himself. I just wanted to comment the term '...worshiping...', what is attempting a kind of irony.

[papageno]

I just wanted to comment the term '...worshiping...', what is attempting a kind of irony.

No irony intended.
The quaternion fanboys usually show an exaggerated enthusiasm about the use of quaternions in physics and a severe lack of understanding of the maths more commonly used.
They seem to support quaternions not because they do a better job, but because it is against the mainstream.

[agingjb]

One reason why people are attracted to quaternions is because they are unique mathematical objects, and on the face of it they should be a useful extension of the complex numbers.

But it turns out that vectors, matrices, and tensors form more useful and general ways of assembling quantities (including the fact that quaternions can be defined as sets of particular matrices).

As for octonions...

[papageno]

I am posting this here, because that's where thomheg said he would reply.
The original post of mine is here; thomheg's reply is here.

I treat all observers as at rest and in free fall. That does exclude us humans here on earth, since we are not in free fall (usually). To calculate our state of motion Newtons laws are the tool of choice.
For a free falling body the world 'moves' and let him stay at rest. The 'movement' of the world is in fact performed by integrals.
It's a bit double headed: the relations are modeled by intervals and multiplying quaternions. To regain the world you sum over spacelike hypersheets. There you sum over the amplitudes you get by quaternion multiplications. So these sheets 'move' in timelike manner and you regain the world.

This has absolutely nothing to do with my point:

The question was about "calculate how far a falling body moves over time knowing all the forces without integrating". You indicated Doug Sweetser as a source to answer the question.
My point is: in the thread I linked, Sweetser has proved himself unable to distinguish time-dependent and time-independent equations in mechanics.
So, my rather pertinent question is: what makes you think that Sweetser is a good source?

I don't have knowledge about what exactly he does with his equations in mechanics. That does not bother me much, since it goes with the infinity of other things, that I do not know. But I'll try to check it out.

The question "calculate how far a falling body moves over time knowing all the forces without integrating" is exactly about mechanics. If you don't know what Sweetser does about mechanics, why did you point to him for an answer?

Doesn't it bother you that he failed to support his claims in the thread I linked?

No

So, pointing to Sweetser was just a diversion, not an actual answer...

I examined some of his claims, and I found his arguments lacking.
He thinks that he can get physically meaningful results from juggling with symbols.

That is a meaningful critic. But try to see it like this: he's doing mathematics for physics and for physicists.

He cannot, because he has little grasp of physics.

He's doing a good job, as far as I can see.

But you admitted you don't really know what he is doing about mechanics, so how would you know he is doing a good job?

I wouldn't call mathematics 'juggling with symbols', what it is in a way.

I never said that maths is "juggling with symbols". I said that what Sweetser does is "juggling with symbols". And I have serious doubts that what he is doing is meaningful maths.

But that's not really fair, because I guess it is a tough work.

Apparently it is tougher to understand the commonly used methods in physics, than playing with quaternions...

It is obvious that you pointed to Sweetser not because you examined his work, but because he uses quaternions:

They seem to support quaternions not because they do a better job, but because it is against the mainstream.

[tusenfem]

In fact, you do not need to know how to manage quaternions in practice. Just forget about that, just only think about anti-symmetry and four dimensions.

Please, Thomas, tell me that you are joking here.

That's how relativity actually IS. No doubt is possible. If you'd take into account, that somehow the stress-energy tensor has to be filled, by things that influence gravity, than everything does. So everything has to be found in spacetime somehow.

Really? And in which way does this "you-don't-have-to-know-how-they-work" quaternion description better than the normal regular 4-vector description of relativity?

You still have to respond to my math mail above, methinks quaternions are are just 4-vectors in disguise with some different math, instead of crossp(x,y) = z they have xy = z.

How are quaternions better in describing spacetime? And don't give me word salat, I am quite full already, give us some math examples.

So everything has to be modeled by quaternions. And, believe it or not, that is possible.
It would be nice to know, how you could really calculate in such a system, but till now, I couldn't find the right tools.

Sure, it would be possible, as I already mentioned, it seems that your idea of quaternions is just a normal 4-vector multiplied by "i", for the rest you have shown absolutely no insight only that "they must be the right thing".

If you would take the effort of just looking at the wiki page, then you would get help on how one calculates stuff with quaternions. It's basically just a bunch of new multiplication rules and probably also some differences in derivatives and integrals. Do your effing homework, and maybe we will take you seriously.

[tusenfem]

I was just curious about quaternions, and decided to ask our good friend ADS about published papers. I found the following list. I guess a great help for all quaternion aficionados and other interested persons.

[papageno]

I was just curious about quaternions, and decided to ask our good friend ADS about published papers. I found the following list. I guess a great help for all quaternion aficionados and other interested persons.

Hundreds of papers in the last seventy years, published in mainstream journals, with titles like "Quaternionic quantum field theory", "Quaternion quantum mechanics: Second quantization and gauge fields", "Foundations of Quaternion Quantum Mechanics", "Quaternionic Formulation of Dirac Theory in Special and General Relativity", "Quaternionic Weinberg-Salam Theory", "Quaternions in Relativity"...

I guess that yawyaw and thomheg are a bit late.

[thomheg]

Please, Thomas, tell me that you are joking here.

As said earlier, my aim is more pedagogic. I would like to make it possible to think, what seems impossible. So I try to convince someone, at least a single person, to try to think that way. Hacking something into mathlab is regarded as disturbing the inspiration.
On the other hand I had to find my own way through the jungle and had to be really really confident, because there is no help at all and nobody understands, what you are talking about.
There are 475000 papers on arxiv.org. NOBODY is able to find out, what is right and what is wrong. So I decided to develop my own theory, without listening and putting it up, step by step. This not an easy task and you can't ask anybody, because nobody understands, what the hack you are babbeling. So I worked slow and steady, testing every single assumption and putting it together stepwise. What I've got you see in my paper.
No idea, what that is good for, but maybe someone likes it.

Really? And in which way does this "you-don't-have-to-know-how-they-work" quaternion description better than the normal regular 4-vector description of relativity?

Treat it like Sherlok Holmes, strike out the impossible and the remainder has to be the murderer. I can't proof, that this model is right, but it makes sense and it is very simple. What else could you want.
I particularly didn't like those absurd interpretations like Schroedingers cat, that QM guys like to discuss. So I tried to find a model for particles.
It is interesting, that that's possible.
Next week or so, they switch on the LHC. And I know exactly what they find out. I guess they will rub their eyes after a while, but it's as much as with LIGO and gravity probe B.

You still have to respond to my math mail above, methinks quaternions are are just 4-vectors in disguise with some different math

Well, if you like to call them vectors, then do. But be warned not to mix that up with real vectors. It is very very important to see, that an interval is something different than a distance. A vector points from point A to point B. So A and B have to be points. But in relativity you have relations. All points are ONLY defined by their relation to other points. That makes spacetime 'curvable'. That's one of the mayor flaws in all kind of physics, that relativity is not taken seriously. All kinds of crap is result of that.

How are quaternions better in describing spacetime? And don't give me word salat, I am quite full already, give us some math examples.

Sounds like you are asking me to perform of a few tricks. But I'm not your dog. I give you a link with all that you might need:
http://home.pcisys.net/%7Ebestwork.1/index.html
And, as mentioned before, the choice of quaternions is not, because they are easier. It's the tool of choice, because they fit to the problem. It's in a way counter-intuitive to base everything on such a model, but I guess there is no other possibility.

[tusenfem]

As said earlier, my aim is more pedagogic. I would like to make it possible to think, what seems impossible.

Basically, what you want to say is that you have no idea what you are talking about.

Your in-understandable descriptions of space and quaternions show you have no pedagogic qualities.

The number of papers on Arxiv has nothing to do with this topic. There are many many different kind of disciplines, all with many many researchers. Not everybody has to understand everything.

because nobody understands, what the hack you are babbeling.

that's how it goes when we read your messages, maybe this should be a clue, Sherlock.

Next week or so, they switch on the LHC. And I know exactly what they find out. I guess they will rub their eyes after a while, but it's as much as with LIGO and gravity probe B.

Can't wait to hear your explanation!

Sounds like you are asking me to perform of a few tricks. But I'm not your dog. I give you a link with all that you might need:
http://home.pcisys.net/%7Ebestwork.1/index.html

Thomas, you don't seem to understand the rules of this board. YOU come up with this thread "Quaternions and Physics" and therefore it is YOU who needs to explain WHY we should use Qs and HOW they work and WHAT we can do with them.

How much do you understand of the link you put in this quote? And with the list of published papers on Qs, how come that you cannot even give us a good introduction to what Qs are? And, at the same time with all of these papers and the link above, how does this comment of yours fit in here:

So I decided to develop my own theory, without listening and putting it up, step by step.

You are hardly developing your own theory, you are just claiming that quaternions are the way to go and then give us a lot of wild ramblings about intervals and hyperplanes and other stuff, and when we just push a little bit (only so little that warm butter will not even be dented) it becomes clear you have no idea about the basics or physics in general.

I think this whole thread is rather a waste of time. :-(

[thomheg]

Thomas, you don't seem to understand the rules of this board. YOU come up with this thread "Quaternions and Physics" and therefore it is YOU who needs to explain WHY we should use Qs and HOW they work and WHAT we can do with them.

Of cause I can. I wanted to make spacetime three-dimensional and introduce an arbitrary observer. That has to be possible, because of the principle of relativity. So I have to find a structure, where any left over dimension could be treated as time.
Why this? First, I wanted to find the connection between QM and GR. It has to be somewhere, hence both theories have to fit into such a model.
I started with GR and looked for a model, that would fit to GR, but would provide some structure. The only possible way to do that, is with quaternions and by encoding time into rotation. That is possible only for quaternions.
Why do not leave spacetime four-dimensional? Well, it is four dimensional, but euclidean space isn't. Since I wanted to compare both, I had to find a way, that GR is modeling the same space, but in a different perspective.
Since it is possible this way, I think it is a good model.

How much do you understand of the link you put in this quote? And with the list of published papers on Qs, how come that you cannot even give us a good introduction to what Qs are?

I don't regard it as my duty to teach you. It's up to you, to learn what you want to know. And I don't feel obliged to proof my knowledge, because we aren't in school and you are not my teacher.
If you want to know, what I have to tell, then read my paper:
http://docs.google.com/Present?docid...&skipauth=true

There is a more subtle reason, why don't use a lot of math (you think I can't, but that's not true). I wanted to set up the model first and then calculate it. The reason is, that you'd otherwise don't know what to calculate. First you need a reason and then you apply a tool and not the other way round. But my model is not yet finished and I'm changing it around all the time.
I like to give links to stuff of interest, since I regard it as bad habit to cut and paste. I wanted a clean separation between my stuff and other peoples ideas. So linking is my way of quoting. I guess, that is ok, because it leaves the original under control of its creator and you can't show, what is not yours.

[thomheg]

.. and therefore it is YOU who needs to explain WHY we should use Qs and HOW they work and WHAT we can do with them.

I developed a model, in witch you could do quantum physics with a model known to fit to GR. I show that in similar usage of certain equations or ideas, that lead to same equations within that model. The terms are different so are the interpretations, but the meaning of that, the principle behind is the same.
My model is a bit strange and not GR. The principle is, to take a 'point' that is a quaternion and multiply with all its neighbors and step ahead by a dilation.Than you get a sheet, within that you have to sum over the amplitudes. It has to be done continuously and depending on the problem of interest in more than one direction at the same time (no idea how to find a mathematical solution for that).
The idea is simple itself, but describes a heuristic concept, that can be modeled only as a simulation or I would need strange circular infinite Integrals of complex linear functions. No idea. Do You have a suggestion?
I guess, the quantum guys already have such methods, but I'm not yet able to use them.

[Celestial Mechanic]

[Snip!] The text is a google.doc presentation. It is quite annoying, but google. docs have benefits, i.e. there is a build in chat system, that you can access if you have a google account. [Snip!]

I cannot access this with either of the browsers at my disposal. Could you please use something standard like a PDF? And I mean a real PDF, not links to a web site masquerading as PDF like the Aether Physics Model people have done.

Thanks in advance.