Theory in brief...now we're getting technical

[Chey]

Here i am... 0403 hrs...i will let you in on a secret, the, so much talked about, fabric of space is time itself. furthermore, time doesn't flow, it is our motion through time (space) which is creating this self-perpetuating experience of the flow of time.

don't believe me, huh?

One thing that has been expanding since the ill famous "big bang", which eventually turned into a "big hum" is space. We have more than enough proof that this notion is correct. However, turn this upside down and what do you get? A space, which is just there, a vast expanse of so called "nothingness". Somewhere the fluctuations in this "nothingness" created a bubble (I will not go too deep into this but you can reference it to the latest developments in string theory) and viola, war between particles and anti-particles ensued, trying to stabilise the region and bring it back to "nothingness". The war was successful. In a different region somewhere else another bubble and another and another... on anyone of those occasions matter, even anti-matter could have prevailed, well it did otherwise we wouldn't be here discussing it. Now we have this bubble, which is expanding, particles, particles everywhere... the rest of the story you all know well.

What we have now is hydrogen, helium and other stuff being driven in all directions. They are moving through space (which really is time). All these particles and atoms have just bought themselves a one-way ticket away from the place of their creation. Others and we like us ended up on the same train being carried along. This is where illusion of time flowing comes from. It is like being in a huge ship looking at the harbour. For a moment there one cannot feel oneself to be moving in a same sense as like being in the car, and in that moment you ask yourself who is really moving, the harbour or me? Logic tells us that since we are on a boat intending to go elsewhere we are the ones moving and not the harbour. Replace the boat with a planet and harbour with a distant object in space and your logic tells you that we are moving. Now replace the sea with the time itself, the fabric of space. Who is moving now? We are, not the time. We are moving through the time and it really doesn't matter in which direction we're going. We will always move forward.

The difference between local time and "base time" is in motion and its direction. I am inclined to say that the "base time" is Hubble’s constant. The rate of expansion. Add to it our motion and direction and you have a "local time frame". Slow down our motion through space (time) and as far as we are concerned time hasn't altered. Why? Because all local time has altered. What used to be a second on the clock based on caesium atoms is still correct since caesium atoms now measure at different rate. Meaning, we would never physically be able to notice that time "flows" slower or faster but we could only say that the rate of "expansion" has altered. Our day would still be 24 hours long -0.03 seconds. It is that basic principle of time separation; when you send a pilot to travel for, say 2 hrs, at fantastical speeds. When he comes back his clock and his perception of time is that two hours have elapsed, just as they would if he was at the bar having a drink with friends. But upon his return when his clock and ours are compared they would be different. Our clock would say that 3 hours have elapsed or whatever time difference it really is.

What is time, how do you describe the fabric of space? Or gravity for that matter? Imagine the mesh in 3D. Little nodules connected forming squares. What gravity does is compressing those nodules closer distorting the space. For the sake of the argument let’s assume that each of those space (time) nodules are 1 second. It becomes obvious that as we move and cross over the nodules we feel as though 1 second has elapsed. The closer these nodules are the quicker seconds tick away. Leave this system and return back to the rate of division between the nodules where gravity does not have as much influence and those seconds tick slower, not that our pilot would notice any difference till he compares it to something else.

True, moving faster means crossing those time nodules quicker, but what is the actual distance between those nodules in an area where there is no gravity? We can assume that, in this case of the argument where each nodule represents 1 second, the distance would be what we call a 1 light second. This is contradictory to our belief that at the speed of light time would stop. Now I will argue that the light, or photons, does not cross over the nodules but squeeze between them. The faster your motion is the further you deviate from the outline of our little squares, which connected nodules, create and move through the centre. After all there should be far less resistance there and everything travels in the line of less resistance and not the shortest route. How do we differentiate between different rates of "time"? Easy, I will again assume that each of those time nodules exert a "time field", just as we have magnetic or electric fields. The slower one moves the more one is forced to take the outline route. It would be similar to the gravity effect, where you require certain speed to escape the gravity well. Photons can do it due to their speed. This still doesn't explain everything in full, now I have to assume that the "time field" if drawn on paper looks like a target. Circles surrounding the centre. Each of these circles representing a fraction of our second that rests in the centre of it, hence faster you move further you deviate less time added to your clock at each nodule...

[ogg]

I was wondering when the cranks would find this forum

[Don. Edwards]

Esoteric, very esoteric if you ask me!

The next question is.
Does Chey really exsist at all in our own time and space?

[t00fri]

...
don't believe me, huh?
...
.

NO;-)

Just tell us a little about the role of (general) relativity in your 'ferry tail'...

Bye Fridger

[Chey]

well Don.Edwards, chey resides in the universe with a totally different reality system, if you must know. as for it being esoteric, i don't think so, it is just a different view, like changing the perspective by standing on the tabletop. try it.

as for t00fri, ferry? i only use it when i travel off shore and since i haven't been anywhere lately i have no tales to tell either, as for TR, it does not interfere with anything i have said so far, further more i do not need to invent imaginary numbers to fit into any of my equations

ogg, i am not a crank, I quite simply do not take anything for granted regardless of how "big" a name stands behind a published and accepted ideas which larger portion of scientific world portrays as a bible.

anyhow, the reason why i posted this is coz as of yet i have heard far more ludicrous ideas which unfortunately have been published..

prime example:
problem regarding collapse of the wave function, someone, i will not even lower myself to that level to say the name of the person in question stated:" The moon exists because we are looking at it." apparently, as this individual will have you believe, only self conscious being can collaps the wave function and bring the moon and whatever else into existance.

everybody knows the thought experiment "cat in the box", honestly, what above unnamed individual has come up with is ludicrous at least what i have come up with after detailed investigations, spaning numerous fields, in not the truth, it is just a possibility, and not everything should hold true either. some stuff is just a bigining of an investigatin. be it right or wrong i am at least not affraid to suggest any one of billions possibilities which may eventually lead me from the "wrong" path to the path of better understanding.. anyways, can you proove that any of what i have said is not true? you cannot, but you would rather carry on adding ever more dimentions and complicating things ever further to feel a sense that the universe is not simple? why? after all, is it not true that the most simplest thing eludes educated mind the longest?

[t00fri]

as for t00fri, ferry? i only use it when i travel off shore and since i haven't been anywhere lately i have no tales to tell either

English is not my mother tongue, but theoretical physics is;-)

You have not answered my request about explaining the consistency of your views with Einstein's (general) relativity...

ogg, i am not a crank,

obviously;-)

Bye Fridger

[Christophe]

Well Fridger, after all it was you who complained about the lack of serious discussions on the forums.

So don't come complaining now ;-))

'Ferry tail' is kind of appropriate, I actually thought it was a joke of yours.

[JackHiggins]

For some reason, i'm not sure why, I find it very hard to follow the top post...

Chey;

Fridger is a physicst, so he'd really be the right person to debate this with. It'll be interesting to see how this one goes...

as for t00fri, ferry?

larger portion of scientific world portrays as a bible.

From what i've seen, scientists question EVERYTHING too. The whole point is to find out what's really going on! A theory only becomes accepted as "bible" like you say, when it has been proved true over many, many years. Simple example: The theory that the earth revolves around the sun. It's been proved true because it's the best way to explain the motion of the planets around the sun in elliptical orbits, we've been able to actually observe it with spacecraft, and it explains parallax in nearby stars!

prime example:

Well, just because a lot of crazy ideas have got published online, and some are less crazy than others, doesn't make them any more right, now does it?!

everybody knows the thought experiment "cat in the box", honestly, what above unnamed individual has come up with is ludicrous at least what i have come up with after detailed investigations, spaning numerous fields, in not the truth, it is just a possibility, and not everything should hold true either.

Schr?dingers's cat was something he came up with to show how bizzare quantum theory can be- he wasn't speaking literally- although many people took it WAAY too seriously!! And Schr?dinger came up with a LOT of other great ideas too- so don't diss him!

can you proove that any of what i have said is not true?

Another thought experiment- you may have seen this elsewhere online.

I have a pot plant on my windowsill, next to my PC. I go downstairs to watch TV for a while, and suddenly i hear a crash from my bedroom. I run in to find the pot plant has fallen & broken on the floor. I theorise that aliens have come in, broken the pot plant, and left again.
I then ring my friend to tell him what's happened. He is a bit freaked out by my sudden turn, he comes up with an alternate theory that my cat knocked it over. I tell him that no, my cat was next to me while watching tv!

Does this prove that my theory about the aliens is correct? Of course not! It is still a possibility, but I need to offer some VERY strong proof to show that it is true. See what i'm getting at here?!

but you would rather carry on adding ever more dimentions and complicating things ever further to feel a sense that the universe is not simple? why? after all, is it not true that the most simplest thing eludes educated mind the longest?

Yeah... But who said the universe was simple? Just because it could be, doesn't mean it is... If it was so simple, someone really would have realised it by now...! Occams razor & all that...

I'm not saying your theory is rubbish or anything- but i'll trust Fridger's opinions on this!!

[t00fri]

Well Fridger, after all it was you who complained about the lack of serious discussions on the forums.

So don't come complaining now )

'Ferry tail' is kind of appropriate, I actually thought it was a joke of yours.

You got me...
Usually I am least serious when I appear serious;-)

Bye Fridger

[t00fri]

Chey,

the problem I was alluding to concerns the fact that whenever one attiributes a special role to space or time, one tends to destroy relativistic invariance of the theory. You don't want to do that, do you?

Bye Fridger

[ogg]

ogg, i am not a crank, I quite simply do not take anything for granted regardless of how "big" a name stands behind a published and accepted ideas which larger portion of scientific world portrays as a bible.

{snip}

... be it right or wrong i am at least not affraid to suggest any one of billions possibilities which may eventually lead me from the "wrong" path to the path of better understanding.. anyways, can you proove that any of what i have said is not true? you cannot, but you would rather carry on adding ever more dimentions and complicating things ever further to feel a sense that the universe is not simple? why? after all, is it not true that the most simplest thing eludes educated mind the longest?

The point of 'big' names is that they're the ones with the track record of successful work. So of course we're going to take them more seriously then we take you.

At first I regretted coming out calling you a crank, but when confronted you simply lapsed into the kind of radical solipsism that's fairly distinctive of them so I stand by the assessment.

- If you really thought that *any* idea vague enough to evade falsification (and there are billions of them, as you say) was just as valid as any other, then why would you ever bother suggesting one over any others? If that's the defence you intend to mount for your idea, then you're committed to there being only one in a billion chance of *this* idea being right. If you think considering it might move you toward the right path then fine, think as you will, but don't waste our time with your intellectual masturbation.

You're shifting the goalposts of epistemic warrant to suit yourself as you go along, and *that's* what makes you a crank.

[ogg]

This is somewhat of a plug for one of my professors, but I'd suggest the following book as an introductory text for anyone interested in philosophy of science:

Theory and Reality: An Introduction to the Philosophy of Science
by Peter Godfrey-Smith

http://www.amazon.com/exec/obidos/tg/detail/-/0226300633/qid=1069724094/sr=1-1/ref=sr_1_1/103-4326289-3797436?v=glance&s=books

[Chey]

let me go back to the begining since i did not get any constructive criticism as was my intention

What is time? How do you describe time? Does it have a mass, shape, charge any physical property that would suffice it to be real? How fast does time flow? What do you measure time against, time itself? Is it scientificly acceptable for me to say that time flows at the rate of 1 second per second?

These questions do not have a real answer, hence, i drew analogy to explain to my primitive brain what time is. Analogy is as follows...

In Minowski's words:"Space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will persevere an independent reality." He did a whole study over all the transformations based on experimental physics. when i read that paper a little light bulb switched itself on...

I shall assume that you know what i am talking about, but for those who don't, Minowski used pure mathematics to transform equations of Newton's mechanics, which in themselves exhibit a two-folded invariance. Namely if "we subject the underlying system of spatial coordinates to any arbitrary change of position and if we change it's state of motion by importing to it any uniform translatory motion." Minowski, however decided to respect the dogma over time and space retaining their independent significance. I really wish he didn't. Anyhow, lets take a system of coordinates with values x,y,z,t, this would be a point of space in a point of time, we can easily imagine such a point with values x,y,z at any other point of time. For the equations to work these new co-ordinates have a variation value which we will express as dx, dy, dz, as such they corespond to a time element dt. This is nothing more but a time-line, a curve which can be refered to unequivocally to the parameter t from -infinity to +infinity. At the very onset at t=0 we can subject the x, y, z, to any rotation we choose, without changing any laws of mechanics we can replace x,y,z, with any constant values A,B,C such that we get Ax, By, Cz giving the time-line any direction we choose towards the upper half of the world where t>0.

So far, what this means is that we create the time-line, not by altering time but by altering the values of the system coordinates. If we designate time t', then out of necessity, in connexion therewith, we must also define space by the manifold of the three parameters x', y, z in which case physical laws are expressed in exactly the same by means of x',y,z,t' as they are by means of x,y,z,t. Now we have no longer space but an infinite number of spaces, each with it's unique set of coordinate values.
If we take any world-point O as the zero-point of space-time and add to it 2 cones representing "before" and "after" respectivly, using hyperboloidal figures we imediately create branches of all possibe hyperbolas with O as center. One of these branches would represent a motion. Use the analogy of vectors in space (call a directed length in the manifold x, y, z, t a vector) and you are forced to distinguish time-like vectors with direction from O to +F=1, and the space-like vectors with directions from O to -F=1, thus the time axis can run parallel to any vector described in such manner. Use same equations at at the value of c, we get complete flattening of the cones into the plane manifold at t=0. Make these steps incrementional, such as that they exhibit the manifold of all in-betweens and each can be described by its own equation with it's own set of values, analogous to a plane of field with it's own distinct characteristics.

Since we have created a vectors from from components x,y,z,t we can also brake such a vector back to its component, all vectors are said to be normal where they satisfy the conditions of a radius vector between point O and and one of the surfaces F=1 and its tangent. for measurement of vectors in different directions we have to create a space-like vector and assign fixed magnitudes to it, lets call this vector AB. On such a world-line we regard the x, y, z, t to be components of the vector AB and to act as functions of the "proper time T". Now we can give differential coefficients to any set of coordinates in regard to T. Call the derivative vector of AB a velocity vector with respect to T, meaning any two sets of coordinates which can be viewed as a vector. Since any velocity vector is a time-like vector in the direction of the world-line then the acceleration vector at any point on the time-like vector is a space-like vector.

What we have now is a collection of mathematical figures, represented in any given way, that give us the set of coordinates x, y, z in direct corelation to time coordinate t. Let the time coordinate be fully fixed and ultimately stationary, by virtue of motion, such that we can obtain a physical description in the three dimentional space with values of x, y, z and due to our own motion we continually obtain further coordinates described by the difference as dx, dy, dz. Such a vector would obtain its time coordinate automatically given that no laws would be broken only for as long as x, y, z, fall on the same time-line as dx, dy, dz,. Hence, the only parameter which in itself is preset is time itself governed by the fact that as we move and change our spatial coordinates we have obtained a unique description of that spatial coordinate by the value of underlying t. In a sense, each vector with the origin in "big bang" has its own unique value of t given by the spatial coordinate which is unique in the system of coordinates describing the spatial and physical aspects of space namely x, y, and z. Hence, as we move through the value of coordinates in any given direction and alter the values of x, y, z we obtain the value of t. set the series of such time-like points t as a unique vector and assume that the vector is really only a discription of a line which is stationary along which we move and you get the notion of time flowing since each phenomena becomes tied up to the precise point on such a line. As is customary to describe each phenomena as taking place in a point of time t it become analogous to saying that the phenomena is taking the place at the point of space with coordinates x, y, z.

So what is time? time is a unique point in a spatial system of coordinates given by the values of x, y, z.
How fast does the time flow? It "flows" at the rate of change of coordinates on any space-like vector due to motion.

now, if i have gotten something wrong than explain it to me. do not forget this is not a description on a small, local scale, this is representative of the universe at large.[/i]

[t00fri]

let me go back to the begining since i did not get any constructive criticism as was my intention

What is time? How do you describe time?
...

Chey,

I really do not know on what level I should communicate with you. It appears quite obvious from the 'crummy' way you try to explain elementary facts about Minkowsky space-time that you never went (successfully) through a systematic training in theoretical physics. While contemplating about the nature of time from such a background can never 'hurt', it has little chance of leading anywhere. The basket simply is hanging too high...

Bye Fridger

PS:
Some 'trivial' corrections of your above 'explanations':

-- Minowsky -> Minkowsky

-- Lorentz covariant space-time 4-vectors are not

x_mu=(x,y,z,t) but rather x_mu=(ict,x,y,z)

such that their length^2

x_mu x^mu = x^2 + y^2 + z^2 - c^2 t^2

is left invariant by an arbitrary Lorentz transformation. Of course, you may always put the speed of light c=1 but the i, with i^2 = -1 is crucial to get the right metric in 4-dimensional Minkowski space.

Your 'theory' among lots of other deficiencies, dispenses with all the basic ideas of Einstein's relativity...It appears to me as if you never really understood what it was about.

[Chey]

Since you cannot stop mentioning general relativity let us reflect on what has already been said. Have a look at these extracts from the man himself to whom you describe the whole of the reality and how nothing can accout for anything that deviates from the line of thought which Einsten himself alterd so much that I have to think long and hard what else would he changed had he lived longer. Furthermore how much of his theories would sit on the shoulders of an imaginary giant...

"Einstein had changed everything by showing with mathematics that time and space are changing with velocity. Einstein also wrote, in fact, he decided, that ".... the velocity of C plays the part of a limiting velocity, which can neither be reached nor exceeded by any real body." (Relativity, The Special and the General Theory, the Fifteenth Edition, 1961, by A. Einstein, Wings Books, N.Y. ISBN 0-517-029618).... It is upon this single statement that both theories of relativity depend, for if any "real body" in the universe, including masses or particles of any kind, can travel faster than the speed of light, then the mathematics of relativity theory would be rendered imaginary and unreal, which would mean the collapse of Relativity....

After publishing more than a dozen editions of Relativity, Einstein never specified that only signals or information cannot exceed the speed of light. In fact, he clearly stressed in his book that the speed of light "can neither be reached nor exceeded by any real body....". Actually, both the Special and the General Relativities depend on equations designed for the "Measuring Rod" and "Atom Clock", which are real bodies and not signals or information. Einstein never distinguished the difference between directional "Velocity" caused by linear force and "Speed" of wave propagation, which transmits forces in all directions. Wave propagations can reach a top limiting speed as well as energy, when wavelength approaches zero and frequency reaches infinity, but linear forces do not have top limits at all.... In his famous Principle of Equivalence Einstein declared that there is no difference whatsoever between the linear acceleration and the gravitational wave. In many parts of his book he refers to "the velocity of C" and not speed of C..... In Lorentz Transformation the linear velocity of real body in motion is mingled with the speed of light C (in vacuum) in the same mathematical equation. The same ambiguity is also extended to the use of E = MC?, where acceleration of particles by electric or magnetic wave energies (particle accelerators) are often not distinguished from direct propulsion by mechanical impact or contact caused by chemical energies (bullets and rockets)... (Apparently, all forces/energies created in earth laboratories, i.e. mechanical, chemical, electrical and magnetic forces/energies, are small compared with forces/energies created at the center of galaxies. Light wave and all electromagnetic waves are created and emitted from galaxies... )

Einstein wrote a new Note as the 5th appendix to the 15th Edition of his book titled "Relativity, the Special and the General Theory" (see above reference), which was dated many decades after he first published the theory. This 5th Appendix, titled " Relativity and the Problem of Space ", runs over 20 pages, but it was summed up in page 7 in the "Note to the Fifteen Edition", in which Einstein wrote :-
"In this edition I have added, as a fifth appendix, a presentation of my views on the problem of space in general and on the gradual modification of our ideas on space resulting from the influence of the relativistic view-point. I wish to show that space-time is not necessarily something to which one can ascribe a separate existence, independently of the actual objects of physical reality. Physical objects are not in space, but these objects are spatially extended. In this way the concept 'empty space' loses meaning"... Signed A. Einstein dated June 9th, 1952
In the above quote Einstein is talking about "space" then suddenly "space-time" comes in, then he goes back to explain what Space really is. Einstein emphasizes not only Space-time cannot exist separately and independent of a physical object, but Space too cannot be separated from a physical object.... because he then goes on to explain that space is the extension or physical part of objects and therefore, there is no such thing as "empty space"..

In 1905 Einstein presented in his Special relativity Space and Time as separate topic:- In the special relativity Einstein uses the term "Measuring Rods" to represent Space in the first equation of the Lorentz transformation, showing that the Space shrinks with increased velocity, thus Space warps and is therefore dynamic. He then presents Time as Clocks in Motion separately using the first and fourth equations of Lorentz transformation to show that time dilates also with increasing velocity, thus Time too is dynamic. These two famous ideas form the fundamental postulates of the theory of relativity ever since. However, in the theory there is no specification of the particular materials or substances the measuring rods are made of.. Therefore, Space is treated as a general reference, or an abstract noun in his mathematical formalism. But in practice different materials of rods in different physical states, i.e. solids, liquids, or gases have different physical properties and would shrink differently. Consequently, such pure mathematical or metaphysical Space and Time are not necessarily representative of practical reality....

In 1905 Einstein also wrote in his first paper of relativity these words "The introduction of a luminiferous ether will prove to be superfluous.... " Apparently in 1905 Einstein did not think it was necessary to endow his Space, or Measuring Rod x, with a particular physical reference or identity, i.e. he did not specify whether the rod is made of wood or steel. Obviously at that time, such non-physical Space of the relativity theory would not pass well with the postulate of physical ?ther, so Einstein deemed it to be superfluous... When Einstein first presented special relativity he was not thinking about Spacetime or empty space yet. He merely adopted Lorentz Transformation which refers Space and Time as only separate mathematical entities.

Then enters Minkovski-Einstein's Spacetime as an inseparable Continuum of Time and Space. What's more, its Time component is represented by the imaginary term v(-1).ct. Thus the normal mathematical co-ordinates for the 3 space dimensions and time, as expressed by x1, x2, x3, t now in general relativity have become x1, x2, x3,v(-1).ct instead, where time co-ordinate t has been substituted by the 4th space dimension, which is an imaginary number v(-1).ct. So Spacetime now in fact is the union of 3 real dimensions of Space and one imaginary 4th space dimension in lieu of Time. Spacetime Continuum is therefore strictly speaking partly real and partly imaginary.... (In the context here the term "Imaginary" or "Imaginary Number" should not be mixed with a mathematical "Complex Number". Because in the real world no one can ever turn time into the 4th space dimension)... This led to Einstein's work on General Relativity...

General Relativity was developed to accommodate gravitation, which postulates that gravitation is not a force but a consequence of geometry or spacetime. So, in 1920 Einstein in his Sidelights on Relativity wrote "...To deny the ?ther is ultimately to assume that empty space has no physical qualities whatever... Recapitulating, we may say that according to the theory of relativity space is endowed with physical quality; in this sense, therefore there exists an ether. According to the general theory of relativity space without ether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense... "{Sidelights on Relativity by A. Einstein, translated by G. B. Jeffrey and W. Perret, republished by Dover, New York (1983)}.. Here it is understandable that if Space is not given a physical property, then there can be no gravity, specially gravitation waves, which depend on the physical properties, as in General Relativity... This, Einstein reiterated in June 1952 when he wrote in his 5th Appendix "..Physical objects are not in space, but these objects are spatially extended... ".

We see from 1905 to 1952 there were major changes to the fundamentals of Einstein's theories. These changes include Einstein's calculation first on expanding universe, then changed to Static universe and ended up in the infamous "Cosmological Constant". So, almost half a century had passed since the publication of Special Relativity Einstein was ready to tell the world what he really think about Space, I wonder how many people have read this fifteenth edition of his book. But since Einstein now declared that there is no such thing as "empty space", and since physical object are spatially extended, it means that Space is in fact the physical displacement of physical objects in question. Einstein had come to agree with the likes of Descartes and Huygens. But then what about Lorentz equation's Measuring Rod, what material is it made of?

We can now go back to Time. Actually, it's bad enough to misconceive clocks for time, Einstein and Minkowski made it worse by substituting mathematical Time t with the imaginary term v(-1).ct and then experiment it with a real physical atomic clock.. That is back and forward three layers of imaginary representation for a none existent Time!..."

Now, i would rather go back to Minkowski space-time than have myself live in an imaginary world on a simple premise that equations finally work. Do you remember who said that if you can't explain a result in simple, non-technical terms, then you really do not understand it?

Regardless of how many PhD's I have or do not have, what ever the case may be, I have never taken anything for granted, not even the Relativity Theory, what's more I find the idea of using imaginary values absurd when it comes to describing the true reality of a physical system, let me tell you about the dream i had, honestly it is so true...

[t00fri]

...
We can now go back to Time. Actually, it's bad enough to misconceive clocks for time, Einstein and Minkowski made it worse by substituting mathematical Time t with the imaginary term v(-1).ct and then experiment it with a real physical atomic clock.. That is back and forward three layers of imaginary representation for a none existent Time!..."

Now, i would rather go back to Minkowski space-time than have myself live in an imaginary world on a simple premise that equations finally work. Do you remember who said that if you can't explain a result in simple, non-technical terms, then you really do not understand it?

Regardless of how many PhD's I have or do not have, what ever the case may be, I have never taken anything for granted, not even the Relativity Theory, what's more I find the idea of using imaginary values absurd when it comes to describing the true reality of a physical system, let me tell you about the dream i had, honestly it is so true...

Chey,

your above statements clearly demonstrate that you have not at all understood what it is all about. I had suspected it before, now you have proven it yourself;-)

The use of an imaginary component ict in Minkowski-space 4-vectors is nothing but a convenient mathematical tool to make the underlying Lorentz symmetry group structure manifest! It certainly does not make any statements about the reality of our world.

Physicists always like to expose underlying symmetries in a most manifest way. This often becomes most elegant by complexifying the mathematical language. The invariance of the physics under changes of reference frames according to special relativity, is expressed mathematically in terms of group theory: the complex Lorentz group acts in Minkowsky space on complex base vectors. So whenever we construct theories about our world/universe we can account most easily for relativistic invariance by requiring any observable to be invariant under transformations with elements of the complex Lorentz group!

We know many other familiar cases where 'complexifying' the mathematical language provides a more uniform physical description!

Take for example an electrical circuit containing /real/ resistors, capacities and inductances. Since we know that inductances and capacities induce a phaseshift of current vs tension of +- Pi/2, respectively, it is most convenient to complexify the description: we associate with the capacities and inductances imaginary, frequency dependent resistors and make electrical tension and current complex!

Nothing particular about this. People who do theory should simply know some math;-)...

Bye Fridger

PS: In Celestia we even use a generalization of imaginary numbers i^2=-1, the socalled quaternions, in order to conveniently represent /real/ rotations;-)

[wcomer]

Hi Fridger,

I had heard that the only practical use of quaternions was with rotation calculations, but I've never seen a description of how this is done. I assume it is a 3-d extension of how complex multiplication can represent a stretch and rotation in 2-d but I've never seen anything written on this.

As is well known for C:
let x1,x2 exist in C;
x1 = a1 + b1.i = c1.exp[theta1.i];
where c1=sqrt[a1^2 + b1^2] and theta = acos[a1/c1].
x2 = a2 + b2.i = c2.exp[theta2.i];
where c2=sqrt[a2^2 + b2^2] and theta = acos[a2/c2].
So x1.x2= c1.c2.exp[(theta1+theta2).i].
Hence stretch and rotation.

However the analog doesn't work for quaternions.
For those unfamiliar with quaternion algebra see Appendix I.
let q1,q2 exist in Q;
q1 = a1 + b1.i + c1.j + d1.k = e1.exp[f1(b1.i + c1.j + d1.k)] = e1.exp[q3];
where e1=sqrt[a1^2 + b1^2 + c1^2 + d1^2] and
f1 = acos[a1/e1]/sqrt[b1^2 + c1^2 + d1^2].
q2 = a2 + b21.i + c21.j + d2.k = e2.exp[f2(b2.i + c2.j + d2.k)] = e2.exp[q4];
where e2=sqrt[a2^2 + b2^2 + c2^2 + d2^2] and
f2 = acos[a2/e2]/sqrt[b2^2 + c2^2 + d2^2].
(or something similar, as that's off the top of my head.)

Unlike the case of complex numbers,
q1.q2 = e1.e2.exp[q3].exp[q4] != e1.e2.exp[q3+q4];
unless q1 commutes with q2.
This is analogous to matrix exponentials.

Thus trivial multiplication in Q does not seem to represent 3-d rotations. Could someone elaborate on how Celestia uses quaternions for rotation calculations.

cheers,
Walton

Appendix I:
quaternion multiplication:
q1.q2=(a1.a2-b1.b2-c1.c2-d1.d2) + (a1.b2+b1.a2+c1.d2-d1.c2).i + (a1.c2-b1.d2+c1.a2+d1.b2).j + (a1.d2+a2.c2-c1.a2+d1.a2).k
quaternion addition:
q1+q2=(a1+a2) + (b1+b2).i + (c1+c2).j + (d1+d2).k

[chris]

Thus trivial multiplication in Q does not seem to represent 3-d rotations. Could someone elaborate on how Celestia uses quaternions for rotation calculations.

It's convenient to write the quaternion as a pair--a real, scalar part together with the imaginary 3-vector:

q1 = w1 + x1.i + y1.j + z1.k = ( w1, v1 ), where v1 = (x1.i y1.j z1.j)

In vector form, the product of two quaternions q1 and q2 is:

q1.q2 = ( w1, v1 ) . ( w2, v2 ) = ( w1.w2 - v1.v2, s1.v1 + s2.v2 + v1 x v2 )

(which can be derived from the fundamental identities: i.i = j.j = k.k = -1, i.j = -j.i = k, j.k = -k.j = i, k.i = -i.k = j)

A rotation of theta about a unit axis u is represented by the quaternion q = (cos(theta/2), sin(theta/2).u) . . . Note that (cos(theta), sin(theta).u) is a unit quaternion, since |q| = (cos(theta).cos(theta)+sin(theta).sin(theta).(u.u)) = cos^2(theta) + sin^2(theta).(1) = 1

A point p can be represented as the purely imaginary quaternion (0, p). To rotate point p by q, we compute q.p.q*, where q* is the conjugate of q, or (w, -v). In practice, quaternions are usually converted into rotation matrices rather than used to directly compute rotations.

I've omitted the proof that the quaternion rotation formula actually performs a rotation, because ASCII is cumbersome for equations, and the proof is available elsewhere anyhow:

http://www.cs.berkeley.edu/~laura/cs184/quat/quatproof.html

One important thing to note is that q and -q represent the same rotation--the set of unit quaternions is a double cover of SO(3), the set of orientations in 3-space. This is different than unit complex numbers, which are isomorphic to orientations in 2D. Another thing: to undo a rotation q, use the conjugate q* (just as for complex numbers.) It's trivial to verify these properties of quaternions . . .

The quaternion exponential is defined as:

exp(q) = ( exp(w).cos(|v|), exp(w).sin(|v|).(v/|v|) )

Now, consider the exponential of a purely imaginary quaternion q0 = theta.(0, u), where u is a unit vector. In the above formula, w = 0, |v| = theta, and v/|v| = theta.u/theta = u, so . . .

exp(q0) = ( exp(0).cos(theta), exp(0).sin(theta).u )

Simplifying . . .

exp(q0) = ( cos(theta), sin(theta).u )

So, the exponential of the pure imaginary quaternion q0 is a rotation, with the angle of rotation equal to twice the magnitude of q0.

--Chris

[t00fri]

Hi Fridger,

I had heard that the only practical use of quaternions was with rotation calculations, ...

Walton,

there are actually many applications of quaternions in theoretical physics.

1) Since one representation of quaternions is through the well-known 2x2 Pauli spin matrices and the 2x2 unit matrix (see below), there are naturally many applications in quantum mechanics, where the Pauli matrices are at home.

2) Certain extended, topologically nontrivial solutions of classical (Euclidean) field theory, instantons and solitons, are most elegantly described in terms of a quaternionic formalism.

3) The Lorentz group in Euclidean space (imaginary time!) is isomorphic to the 4d rotation group SO(4) and thus ready for a quaternionic description. Also in Minkowsky space some people use quaternions, see e.g. the nice WEB site from MIT,
http://world.std.com/~sweetser/quaternions/qindex/qindex.html

4) Finally, they are excellently suited to parametrize the elements of the 15 parametric conformal symmetry group in Euclidean space. Conformal symmetry has really many applications in theoretical physics, notably also in String theory! The conformal symmetry group consists of a generalization of the 10 parametric Poincare group of inhomogeneous Lorentz transformations to including in addition the dilatations, x -> x'= lambda*x (1 parameter) and space-time inversion x -> x' = b/x (4 parameters).
Quaternions q represented as 2x2 matrices, read

Code: Select all

q= q_mu  sigma_mu,   mu=1..4, sigma_mu <=> (-i tau_1,-i tau_2, -i tau_3, 1) ,


where sigma_mu is to denote a special 4-vector composed the 3 familiar Pauli matrices tau_j and the 2x2 unit matrix

So we represent the Euclidean space-time 4-vector x_mu also as x= x_mu sigma_mu, i.e. in quaternionic form and consider the conformal (Euclidean) space-time transformations x -> x',

Code: Select all

x' = (alpha x + beta)(gamma x + delta)^(-1),


with alpha, beta, gamma, delta being arbitrary quaternions, involving 4x4=16 parameters.
Since scaling these 4 quaternions by a real number has no effect, we arrive at 15 degrees of freedom, i.e. the correct number of parameters for the conformal group. Hence

Code: Select all

k^2= det (alpha beta / gamma delta)


may be fixed at unity if desired.

Bye Fridger

[Chey]

well, i will need few days to go through several calculations, so don't go away, it may be a lengthy one

i have come accross several of equations which have been modified to fit into the TR even tho it was unnecessary, this is not just my opinion but several people have come to agree that TR is in an essence a signpost, not a theory just a direction tool, which is why in itself is modifying values attempting to be a theory which it is not.

I myself am not a great mathematician but i know few who will make sure that they are valid before i present them, do not get me wrong, i have believed for the longest time that TR is the greatest thing ever to have happened to physics, but with time i started to see it as a doorway through which one has to walk through and dispense with it. Just as the door cannot describe what is in the room neither can TR.
I do know that TR is a form of representation of the real but to truly see the real for what it is I believe we will have to go back and start at the begining. it is not a matter of discovery but a mater of understanding how to gain valid results with standard mechanics. This may give us valid values which are not imaginary and results will therefore equally be valid.

I will not jump into it like last time but will do it one step at the time