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The Final Piece Of The Puzzle!


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#18 modest

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Posted 22 April 2010 - 03:38 AM

In such a case, I can write [imath]\Phi(\vec{x})[/imath] in a very simple form consistent with standard notation:


[math]\Phi(\vec{x})=-\frac{\kappa M}{r}[/math]


where [imath]\kappa[/imath] is the proper proportionality constant to yield the correct potential generated by the mass M.


I was just reading the OP and it's getting late and I had a question, so figured it would be a good place to stop for the night.

If your wave equation propagates in 4 dimensional Euclidean space (an impression I got from the special relativity thread) then the vector field causing the refractive index, n, would presumably be an inverse (n-1)th conservative force (F [math]\propto[/math] 1/r3). That force, I would think, would be the derivative of Phi making Phi proportional to 1/r2 (in contrast to normal Newtonian gravitational potential) so I'm not quite following how or why you deduced Phi [math]\propto[/math] 1/r.

Are ds, c, and the geodesic in four dimensions? If so, how have I gone wrong?

Thank you,

~modest

#19 coldcreation

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Posted 22 April 2010 - 12:08 PM

Coldcreation, I really wish I knew why you feel compelled to post to this thread. You obviously have not the first hint of what I am talking about.


There's another erroneous presumption I guess we should add to the list.

The reason why I feel compelled to post to this thread is simply that it intrigues me why someone, with the background you have, would see the need to overhaul an aspect of a theory—Einstein's concept of gravity as a curved spacetime phenomenon—by replacing it with a concept (Euclidean space) that has shown not to be tenable. Indeed, it is precisely the concept of gravity as curvature that did away with the notions erroneous within that which it replaced (Newtonian gravitation), and the idea that space was Euclidean, independent of time, and absolute (in the sense that there existed a unique gods-eye reference frame).

Besides quantum theory there have been few theories produced since the beginning of modern science that have a better track record, i.e., there is excellent corroboration with empirical observations and experimental data related to phenomena such as gravitational time dilation and frequency shift, light deflection and gravitational time delay, orbital effects and the relativity of direction (precession of apsides, orbital decay, geodetic precession and frame-dragging). Nearly being the key word, it seems that a modification, if at all, rather than a complete overhaul, would be more appropriate, in order to clear up the discrepancies with GR and that which is observed and theorized to be operational on meso- and microscopic scales. What makes you think that gravity is not due to a curved spacetime manifold?


That “almost” is there for honesty. There exist aspects of modern physics which I have not yet proved are indeed implied by my fundamental equation. It is not an easy equation to solve. [...] and finally, (as the last piece of the puzzle of identifying constraints implied by my equation with supposed conclusions of modern physics) showing that Einstein's theory of General Relativity is essentially an approximate solution to my fundamental equation. I have not yet achieved “complete corroboration with all empirical evidence” but I have certainly deduced enough to convince me that modern physics is a tautology (as it is quite clear that my fundamental equation is certainly a tautology). As I said, it is not exactly an easy equation to solve. [...] Yes, and I will make a prediction: there probably exists other empirical evidence which can be shown to be approximate solutions to my fundamental equation.


I was referring to your statements:

"The fact that my result is not exactly the same as that obtained from Einstein's theory is not too troubling.


And:

"It appears (at least to me at the moment) that the effect of that extra term in my solution is to make the gravitational field appear to be slightly stronger than estimated via Einstein's field theory. If that conclusion is correct, then it could also explain the “dark matter” problem."


If there are indeed differences that you describe perhaps you should center your predictions around those.

For starters you could attempt to predict rotational curves without cold dark matter for galaxies more accurately than GR with CDM...


That would be one giant leap for mankind.




Of course you do because you have absolute faith in the validity of the standard “by guess and by golly” approach used by modern physics.


Actually I do not. I've been known on occasion to display a dislike for certain mainstream (and certain less mainstream) physical theories (e.g., inflation, with its false vacuum, strings with its 21 dimensions, branes, even the big bang theory with its CDM, DE and repulsive force: lambda).



For example, Einstein's general relativity tells us many things about the physical universe, 'reality' (and is thus not a religion).


Yeah, I know! That is exactly what you are convinced of and you would like the rest of us to believe. I say it is no more than another consequence of internal self consistency of the argument (but as I get something a little different, either I have made a deductive error, which is certainly possible, or Einstein's theory is not internally consistent). The fact that no one has succeeded in presenting a quantized version of that theory is somewhat indicative of its failure to be internally consistent.


Each is entitled to 'believe' that which pleases him or her. As it stands, GR is in empirical agreement with observations That is what makes it compelling (nothing to do with belief).

Again, if you can make a prediction to within a more accurate decimal place, please do so.

Your tau axis seems more akin to some belief system. You did after all write of it; "tau is a complete fabrication of our imagination"


Your hypothesis is not solely that "an explanation must be internally self consistent!" There is more involved than just that.


You clearly have never read the proof as, if you had, you could explicitly point out the other hypotheses.


Maybe there's a better word than hypothesis: contentions, or presumptions for example. One example is that you feel Euclidean space is more consistent a schema than curved spacetime. Your definition, or interpretation, of time is another. The idea that gravity is a pseudo-force is yet another. Not to mention tau again. These are all part of your hypothesis, or contention(s).


As I have commented earlier, you seem to have utterly no comprehension of what puzzles are under discussion. And, as an aside, if you were at all aware of my proof you would clearly comprehend the nature of that tau axis. [snip]


That would be a good place to start clearing up what people find confusing about your proof. To what extent do you feel tau is relevant to the natural world. How can the concept of tau be tested? Why do you need to include tau in your equations? What would be the consequences if it were to be removed?

If you like, you can hyperlink to a thread where you've discussed tau already. But that explanation clearly belongs in this thread.


PS. Your OP would have been easier to follow had you started with an abstract, and ended with a conclusion (at least for those who have not sifted through all your other threads).



CC

#20 Doctordick

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Posted 22 April 2010 - 07:46 PM

Thank you modest for taking the trouble to think about what I have said.

If your wave equation propagates in 4 dimensional Euclidean space (an impression I got from the special relativity thread) then the vector field causing the refractive index, n, would presumably be an inverse (n-1)th conservative force (F [math]\propto[/math] 1/r3). That force, I would think, would be the derivative of Phi making Phi proportional to 1/r2 (in contrast to normal Newtonian gravitational potential) so I'm not quite following how or why you deduced Phi [math]\propto[/math] 1/r.

Your assertion concerning the radial form of a conservative force is mathematically true in a Euclidean geometry in the Newtonian picture as it is intimately related to the way the surface area changes with respect to the radius. The effect is easily seen in a simplified notion of photon exchange forces.

Exchange forces are attributed to momentum exchanges mediated by the exchange of virtual particles. The force (i.e., that change in momentum) is directly related to the probability that a given virtual particle will be exchanged and that probability is in turn proportional to the cross section of the interaction as seen by the interacting bodies: i.e., the same spacial density of interacting virtual particles from a given source, as seen from a more distant object, declines as the area of the encompassing sphere increases. That area is proportional to r squared and therefore the magnitude of the force becomes inverse to r squared.

And you are correct. In a four dimensional Euclidean geometry in the Newtonian picture a conservative force would be proportional to the inverse of r cubed, yielding a potential proportional to the inverse of r squared. However, that is not what we have here. In this case, both interacting bodies are momentum quantized in the tau direction. The whole circumstance yields utterly no variation in the physical probability density of these interacting bodies (or the density of the virtual exchange particles) in the tau direction. This situation essentially projects out the tau dimension and all forces and potentials come back to the three dimensional dynamics (except for the subtle consequences of that momentum in the tau direction: i.e., what we call mass).

Are ds, c, and the geodesic in four dimensions? If so, how have I gone wrong?

Yes, ds and c must both be evaluated in the four dimensional space as they are exactly that subtle consequence of momentum in the tau direction of which I spoke. Mass, the quantization of momentum in the tau direction, and likewise the tau component of its path and the tau component of its velocity do not change in the tau direction thus the existence of the tau dimension can not be ignored with respect to these terms.

I hope that makes things a little clearer. As I have commented elsewhere, I am currently working on a post of my proof of my fundamental equation and I think that post will clear up a lot of these kinds of questions.

The reason why I feel compelled to post to this thread is simply that it intrigues me why someone, with the background you have, would see the need to overhaul an aspect of a theory—Einstein's concept of gravity as a curved spacetime phenomenon—by replacing it with a concept (Euclidean space) that has shown not to be tenable.

Again you demonstrate that you do not have even the slightest idea as to what I am doing. I have no compulsion to overhaul any aspect of Einstein's theory; my only purpose is to endeavor to find the consequences of my proof. Since you have utterly no idea as to why my fundamental equation must be valid, your comments are totally off the subject. I won't put you on my ignore list because you appear to have a little education; however, I don't think I will respond to your posts until I have a little more evidence that you are trying to understand this stuff.

Have fun -- Dick

#21 IDMclean

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Posted 22 April 2010 - 08:23 PM

To CC,
One of the points DD's consistently made in this thread and in others is that his system and proofs show that physics is a logical tautology deriving from his fundamental equation. As far as I can tell, DD's perspective holds that this is a sort of disproof of physics as anything other than a mental construction of the human kind having little to do with the machinations of physical world beyond our interpretation of it.

Seems, DD has different aims in this respect than your average scientist. He's out to disprove the independence of physics and scientific method as it stands from philosophical and logical concerns by showing that it is in fact equivalent to any other axiomatic system we're familiar with. As such, it would constitute a logical argument against science as independent of the concerns of philosophy.

"There is no such thing as philosophy-free science; there is only science whose philosophical baggage is taken on board without examination." —Daniel Dennett, Darwin's Dangerous Idea, 1995.

DD, do correct me if I'm wrong in my interpretation of your argument, but this is the gist I get from what I've read so far of your work.

#22 Doctordick

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Posted 23 April 2010 - 12:00 AM

Well Mclean, I am impressed. I was kind of unimpressed by the character “KickAssClown”, but your recent posts have led me to believe there are possibilities there.

Your post number 21 to this thread seems to indicate a rather intelligent approach. I have no real argument with that post at all. All I have is a few subtle adjustments to your perspective.

One of the points DD's consistently made in this thread and in others is that his system and proofs show that physics is a logical tautology deriving from his fundamental equation.

Absolutely and incontrovertibly true.

As far as I can tell, DD's perspective holds that this is a sort of disproof of physics as anything other than a mental construction of the human kind having little to do with the machinations of physical world beyond our interpretation of it.

Essentially true; however, there are a number of conclusions which might be drawn from that statement that really kind of misrepresent the situation.

Seems, DD has different aims in this respect than your average scientist. He's out to disprove the independence of physics and scientific method as it stands from philosophical and logical concerns by showing that it is in fact equivalent to any other axiomatic system we're familiar with. As such, it would constitute a logical argument against science as independent of the concerns of philosophy.

Again an essentially correct statement; however, it seems to imply an attitude which is somewhat askew of what I actually have in mind. To quote post #14 I made to this thread

I discovered the proof of my fundamental equation when I was still a graduate student back in the sixties but it seemed pretty worthless because I couldn't solve the equation. I discovered the first solution around 1983 and after some work, I attempted to publish my proof about twenty years ago. I don't think it made it past any of the referees. The physicists said it was philosophy, the philosophers said it was mathematics and the mathematicians said it was physics.

I have since come to the conclusion that it is indeed philosophy, that is why I am posting to the “Philosophy of Science” forum. Certainly it has absolutely nothing to say about mathematics as nothing in my work yields anything new to the field of mathematics. And, essentially, my work has nothing to say about physics (except perhaps the fact that Einstein's theory of General Relativity has some problems) as, for all practical circumstances, it essentially confirms most all of modern physics.

Just as an aside with regard to the confirmation of modern physics, Newtons orbital calculations essentially (except for a few exceptions) confirmed the charts prepared through the “cycle and epicycle” theory of Claudius Ptolomy (which, today, is clearly seen as no more than a mathematical mechanism for cataloging cosmic positions). Newton admitted the possibility of error in his calculations; however, as it turned out the differences were errors on the part of the astronomers citing the actual data. Likewise, Einstein's theory could still be correct as it is entirely possible that I have made an error in my algebra. On the other hand, perhaps Einstein is wrong. Time will tell.

The reason I bring this up is the fact that I really have no complaint with physics. Considering the problems they have solved and the foundations they have to work with, they have done an excellent job. It is all based upon the presumption that the “by guess and by golly” attack is the only attack available to them.

My real complaint is with the field of philosophy. It is the philosophers who have dropped the ball here. They have taken the issues underlying the fundamental questions of interest and lathered them over with gobs and gobs of esoteric bullshit. It is time that philosophers became a little more exact in their analyses. I think I have kind of thrown out the ball there but, except for Anssi, have seen no rational reaction.

Philosophy was once considered the Queen of all scientific investigations. But no more. Today philosophy is considered to be a field totally concerned only with stirring bullshit and that is a sad report on their efforts over the last three thousand years.

"There is no such thing as philosophy-free science; there is only science whose philosophical baggage is taken on board without examination." —Daniel Dennett, Darwin's Dangerous Idea, 1995.

I couldn't put it any better myself (though I don't think I would call evolution a “dangerous idea”).

For what it is worth, consider yourself corrected. :confused:

Have fun -- Dick

#23 modest

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Posted 23 April 2010 - 12:36 AM

In this case, both interacting bodies are momentum quantized in the tau direction. The whole circumstance yields utterly no variation in the physical probability density of these interacting bodies (or the density of the virtual exchange particles) in the tau direction. This situation essentially projects out the tau dimension and all forces and potentials come back to the three dimensional dynamics (except for the subtle consequences of that momentum in the tau direction: i.e., what we call mass).


That makes sense. I think I had read previously, but forgot, the consequences of uncertainty in tau being infinite. Thank you for the explanation. I'll pick up where I left off when the girlfriend eases up on the whip (metaphorically speaking :confused:)

~modest

#24 Rade

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Posted 23 April 2010 - 01:31 PM

I have a comment and a question about tau dimension.

I have since come to the conclusion that it [the Fundamental Equation] is indeed philosophy, that is why I am posting to the “Philosophy of Science” forum

Well, this is exactly what I have been trying to explain to DD for over one year now. And, I have presented why his Fundamental Equation is important to philosophy. When DD indicated a few posts ago that he agrees with the comment that ....physics is a logical tautology deriving from his fundamental equation...I have suggested that the reason is because his Fundamental Equation is derived from "tautology itself"---that is, from the philosophic Law of Identity, which was presented by Aristotle > 2000 years ago as the ultimate Fundamental Equation A = A ! This is why I agree with DD that his Fundamental Equation has great philosophic importance. It shows how A = A (which is a statement about ontology) can be transformed into an isomorphic equation [the DD Fundamental Equation] that is a statement about epistemology (explanation itself). Imo, this is the final piece of the "philosophic puzzle"--the DD Fundamental Equation completes the thinking of Aristotle about the "mathematical" relationship between existence and knowledge--it puts philosophy on a sound mathematical basis. But of course--only if one accepts the Law of Identity as a valid premise.

==

My question is about the tau dimension used by DD.

Now, DD indicates that "mass" is derived from his tau dimension--yet he also indicated (I do believe) that this dimension is "abstract". So, I recall a similar situation with use of the Schroedinger Equation as relates to shell structure in the atomic nucleus for nucleons. This model predicts (using the Schroedinger Equation) the interactions of independent nucleons as if they are being "acted on" by an energy potential (V) that is an abstract dimension related to a central harmonic energy well. That is, the "mass" of the nucleons is a function of this interaction with an abstract concept.

So my question--is this what DD is saying with the relationship between "mass" and abstract "tau" ? Is his claim about mass & tau the same philosophic relationship as between the mass of a nucleon and abstract central energy well within the nucleus as a whole as predicted by quantum mechanics via use of Schroedinger Equation ? If yes, then what DD is claiming makes perfect sense to me. If no, please explain why.

#25 AnssiH

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Posted 29 April 2010 - 05:23 PM

Hi, I had not noticed you had posted this thing (Didn't pick up on the title I guess :offtopic:

I'll try to get around to start a walk through soon, probably this weekend.

But first one comment. It would be very nice if the discussion about the underlying issues would be held here;

http://hypography.co...ng-about-9.html

Simply because it will make it easier to follow and backtrack this thread if the responses are more strictly about the derivation of general relativity.

It is of course okay to make small comments, but long conversations about the underlying issues just make it time consuming for me to find information about the actual derivation later.

Following that note, I'll post a little comment to Coldcreation to that thread.

-Anssi

#26 lawcat

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Posted 30 April 2010 - 03:01 PM

To CC,
DD's perspective holds that this is a sort of disproof of physics as anything other than a mental construction of the human kind having little to do with the machinations of physical world beyond our interpretation of it.
.


I agree that indeed that is the DD's conclusion. However, it is a rather dangerous conclusion. DD negates experience. All that exists is sets and logic. DD concludes that physics is nothing but entertainment. in other words, physics is math which is entertainment of the mind. Only sets and relationships between sets exist, only math exist. Experience may exist but we know nothing of it.

His central premise is we know nothing other than that which we create in our mind. While enticing, this is rather illogical imo. Because the very sets and logic that give rise to mathematics are the product of experience of ourselves. If we do not trust ourselves, our experience, then we can not even begin to trust the sets. Only nothing would be valid.

I find DD's equation scientific, but his conclusions dangerous because they negate experience.

From the bottom up, DD's equation is scientific as a synthesis; not an analysis. It is no different than mathematically synthesizing any other set of sums into a single expression. To that end I agree with DD that it is falsifiable as a mathematical expression. As has been posted by others, if there exists one valid theory inconsistent with DD's equation, then the equation is falsifiable.

From the top down, his analysis of the equation is purely logical. He sets up some definitions and analyzes those to come up with the equation. To the extent that the specified elements are elements of DD's space, they appear valid--no different then any other valid set.

But what of this equation? In essence, so what? Of course we invented math. We certainly did not mine it in a cave.

#27 AnssiH

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Posted 01 May 2010 - 01:11 PM

Okay, let's get to it...

Relativity is the mathematical transformation between two different geometric coordinate systems. Back in Newton's day, such transformations were quite straight forward as Euclidean coordinate systems were assumed applicable to reality. If the origin of a Euclidean coordinate system (coordinate system “b”) was at point (x0,y0,z0) in the original coordinate system (coordinate system “a”) then any point in coordinate system “a”, say point (xa,ya,za) was simply represented by the point (xa-x0,ya-y0,za-z0) in coordinate system “b”.

This transformation was exactly the same even if the point being referred to as (x0,y0,z0) was moving in any arbitrary manner.


I.e. even if the coordinate system b was moving inside the coordinate system a in an arbitrary manner.

The difference between my system and Euclid's original system is that tau axis. Momentum quantization along that tau axis (mass) introduces some very subtle consequences when it comes to physical measurements such that the simple transformation above does not yield the measurements as taken by a person at rest in the moving system: i.e., special relativity is a necessary part of such a transformation in order to compensate for the effects of momentum quantization along the tau axis.


Since you mention momentum quantization along the tau axis, I stopped to think about that a bit. I am not sure why was it quantized instead of being a continuous variable. I suppose it had got something to do with the uncertainty in the position of tau being infinite (since the momentum is constant/known?), but my understanding is quite shaky when it comes to the details of how the uncertaintly principle plays out here.

Also, I suspect this issue related to the use of dirac constant in the definition of mass operator [imath]-i\frac{\hbar}{c}\frac{\partial}{\partial \tau}[/imath]...?

Nevertheless, yes I remember how special relativistic transformation came into play.

.
.
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The question then arises, how is it that Einstein's theory appears to circumvent Maupertuis' proof? The answer revolves around the principal of “least action” he invented as a means of calculating paths consistent with Newtonian physics (essentially minimizing the energy with respect to the path). Maupertuis showed that the problem was a consequence of the fact that, when it came to gravitational paths, different velocities led to different paths: i.e., two different objects behavior could not be reduced to geodesic motion in the same reference frame, something which must be true in the proper inertial frame. The inclusion of time in Einstein's space-time continuum allows this critical variation to be achieved; however, it turns out that this is exactly that same issue which creates the critical problems when it comes to “quantizing the gravitational field”. Thus it is that there are very real problems bringing quantum mechanics into Einstein's General Relativity theory. To date all attempts, that I am aware of, have resulted in failure.


Yup, thank you about that whole explanation. It might be interesting to take a quick look at that Maupertuis' proof but I did not find it... Nevertheless, I have pretty good suspicion about how the inclusion of time axis and its relativistic transformation can get around proofs that rely on newtonian definition of time.

I don't have very good idea about the problems with the traditional attempts of gravity quantization. Wikipedia only vaguely mentions problems with re-normalization.

Compare this to my presentation which is totally consistent with quantum mechanics from the very beginning. Beyond that, in my presentation, mass is defined to be momentum in the tau direction of a four dimensional Euclidean geometry. As a consequence, that hypothetical (see as unexaminable) tau dimension can be simply scaled to make the velocity of every elemental entity through that four dimensional geometry look exactly the same.


i.e. whatever velocity they are "missing" in the [imath]x,y,z[/imath]-axes, is to be attributed to their velocity along the [imath]\tau[/imath]-axis...

That final fact totally circumvents Maupertuis' proof. It seems certainly reasonable to once again look at the consequences of general transformations and perhaps find that “non-inertial” geometry which yields gravity as a pseudo force. So let us proceed to examine some aspects of that circumstance.

To begin with, my fundamental equation does indeed require a specific frame of reference: that frame of reference being at rest with respect to the entire universe. In that particular frame, (remember, that frame is a standard Euclidean frame) any object (any collection of fundamental elements forming a stable pattern) which can be seen as essentially not interacting with the rest of the universe (i.e., those interactions may be ignored and we are looking at a “free” object) will obey Newton's laws of motion in the absence of a force (it will not accelerate in any way). That is, the only forces which appear in such a circumstance are those pseudo forces we want to examine in this analysis: i.e., apparent forces which are entirely due to the fact that our coordinate system is not at rest with respect to the universe.

True, we have created a situation which obviously circumvents Maupertuis' proof...


Hmmm, well not very obvious to me, as I don't know how his proof worked exactly...

I'm guessing what you are referring to is that, since Maupertuis' proof had something to do with the inability to "reduce two different objects' behaviour to geodesic motion in the same reference frame", this presentation form with a [imath]\tau[/imath]-axis (that gets "projected out"), will allow two different objects with different velocities in [imath]x,y,z[/imath] space to have the same total velocities when the velocity along [imath]\tau[/imath] is included.

Not really sure yet how that will circumvent the proof.

I understand though, that in the standard GR presentation form, the objects in gravitational free fall are following geodesic paths in the general relativistic coordinate system.

but, since energy is now not a function of velocity,


...i.e. not a function of the total velocity in the [imath]x,y,z,\tau[/imath]-space...? Are you saying that because, for instance, an object is considered to gain (kinetic) energy when it gains velocity in the [imath]x,y,z[/imath] directions, while its total velocity remains unchanged?

we have also made the original formation of his principal of action into an unusable procedure (his relationship related velocities and we now have no relationship to minimize);


...since velocities never change in the [imath]x,y,z,\tau[/imath]-space...?

however, there is another attack (actually the same attack but somewhat subtly different). If gravity is to be a mere pseudo force, we can use the fact that our model must reproduce the exactly the same classical pseudo forces produced by Newton mechanics. This is true as these forces are no more than a direct consequence of expressing the path in a non-inertial frame or, in my case, a reference frame not at rest with respect to the universe.


Yup.

It is interesting to look at centrifugal force as a well understood Newtonian pseudo force. From the perspective of an observer at the center of the rotation, a string exerting a force equal to the centrifugal force will appear to maintain the object under the influence of that pseudo force at rest: i.e., a rock at the end of a string swinging in a circle will appear to be at rest in a reference frame rotating with that rock. We can then see the object as a test probe into the force field describing that specific pseudo force. Since my total interest is in explaining gravity as a pseudo force, I want to examine this force as seen from the perspective of being m times the negative gradient of a gravitational potential (which is the typical way of defining a gravitational potential). In this case [imath]\vec{F}=-m\vec{\nabla}\Phi [/imath] where [imath]\Phi(\vec{x})[/imath] is the gravitational potential.


Had a little adventure in Wikipedia about the definition of gravitational potential etc, and yes that seems to make perfect sense to me.

I gathered that the "gradient of a gravitational potential" is essentially the radial derivative of the function describing gravitational potential... I.e. the "slope" of a graph describing that function.

Also I gathered that the definition of gravitational potential is the "potential energy" PER unit mass, so that's why you are including the "m" there in your equation (I was getting a bit confused at first).

So in other words, [imath]\vec{F}=-m\vec{\nabla}\Phi [/imath] refers to the strength of the "fictional force" that affects an object with given mass in a given location in a gravitational field. (Just thought I'll say that out loud to benefit other people who are starting with almsot zero physics knowledge, like me :D)

Any freshman physics text will provide an excellent derivation of centrifugal force. The result is the quite simple form, [imath]\;\vec{F}=m\omega^2\hat{r}\;[/imath] where omega is the angular velocity and [imath]\hat{r}[/imath] is a unit vector in the radial direction.


I'll take that on faith for now.

It follows that the analogous gravitational representation implies that the required [imath]\Phi(\vec{x})[/imath] (which, by the way, must by symmetry be a radial function) must obey the relationship


[math]-\frac{\partial}{\partial r}\Phi(r)=r\omega^2[/math].


I.e. the radial derivative, or the "slope" of the gravitational potential, is related to the angular velocity somehow?

Hmmm, you must be referring to a situation where some object is orbiting the center of the gravitational field, so then its angular velocity and the distance from the center are related to the strength of the associated "fictional force", which is keeping it in stable orbit.

Intuitively, that makes sense, but I am not entirely sure how you got that exact expression (specifically, the "r" on the right side).

I mean, I understand that [imath]\vec{\nabla}\Phi = \frac{\partial}{\partial r}\Phi(r)[/imath] since its a radial function, and I guess we are essentially associating gravitational force with a centrifugal force capable of negating it...?

[math]-m\frac{\partial}{\partial r}\Phi(r) = m\omega^2\hat{r}[/math]

...I guess... :shrug: ?

And you have removed the "m" from both sides, but I am not sure how the "r" finds its way to the right side the way it did. I mean, I understand the associated fictional force is a function of the radius, but how do we know the [imath]\omega^2[/imath] is simply multiplied by the radius?

I'll pause here as I am getting the feeling I may well be interpreting you wrong already...

-Anssi

#28 Doctordick

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Posted 02 May 2010 - 12:35 AM

Hi Anssi, I am sorry I hadn't let you know about the post. There is another post you should be aware of, I am speaking of “Laying out the representation to be solved”. I would appreciate any comments you might wish to make on the clarity of that post. I don't think you will have any problems understanding what I said, but, if you do, we can try and clear them up.

Meanwhile, I will attempt to clarify this thread.

It would be very nice if the discussion about the underlying issues would be held here;

http://hypography.co...ng-about-9.html

Simply because it will make it easier to follow and backtrack this thread if the responses are more strictly about the derivation of general relativity.

I wouldn't argue with you but I suspect you will not find many takers. There are a lot of people here who's sole purpose is to create confusion by making it difficult to backtrack threads. Rade long ago made his intentions quite clear and at the moment I am seriously suspicious of lawcat. I am considering placing lawcat on my ignore list. If he keeps up the kind of comments he has been making I will do so. I didn't see the “ignore list” as a proper response to ignorance but I am beginning to realize that intentional ignorance is a bothersome issue which should be ignored.

Since you mention momentum quantization along the tau axis, I stopped to think about that a bit. I am not sure why was it quantized instead of being a continuous variable.

It is a direct consequence of the manner and purpose with which tau was introduced. That issue will show up (hopefully more clearly) in my further posts regarding my proof of my fundamental equation. For the moment, let's not worry about it as discussion at this point will probably generate nothing but further confusion. Just take it as a proved aspect of tau.

Also, I suspect this issue related to the use of dirac constant in the definition of mass operator [imath]-i\frac{\hbar}{c}\frac{\partial}{\partial \tau}[/imath]...?

That issue you seem to have somewhat backwards; however, again I suggest you not worry about it for the moment. It will be clarified in the actual proof which I am currently trying to restate.

Yup, thank you about that whole explanation. It might be interesting to take a quick look at that Maupertuis' proof but I did not find it... Nevertheless, I have pretty good suspicion about how the inclusion of time axis and its relativistic transformation can get around proofs that rely on newtonian definition of time.

As I said, Maupertuis' proof has to do with the fact that objects with different velocities follow different paths in a gravitational field. This is totally counter to the proposition that “pseudo forces” create paths which are mirror images of the path of the non-inertial frame. That idea means that the path certainly can not depend upon the velocity of the object; not in a Euclidean geometry anyway.

i.e. whatever velocity they are "missing" in the [imath]x,y,z[/imath]-axes, is to be attributed to their velocity along the [imath]\tau[/imath]-axis...

Absolutely correct!

Hmmm, well not very obvious to me, as I don't know how his proof worked exactly...

Well, if everything in the universe is traveling through my four dimensional Euclidean geometry at exactly the same velocity, then velocity differences don't exist and the actual paths can once more be mirror images of the path of the non-inertial frame. Path dependence on the velocity of the objects vanishes absolutely.

Are you saying that because, for instance, an object is considered to gain (kinetic) energy when it gains velocity in the [imath]x,y,z[/imath] directions, while its total velocity remains unchanged?

Not really. The issue here is that nothing can depend upon the velocity if the velocity of every object in the universe is the same.

...since velocities never change in the [imath]x,y,z,\tau[/imath]-space...?

Exactly!

I gathered that the "gradient of a gravitational potential" is essentially the radial derivative of the function describing gravitational potential... I.e. the "slope" of a graph describing that function.

That is correct.

So in other words, [imath]\vec{F}=-m\vec{\nabla}\Phi [/imath] refers to the strength of the "fictional force" that affects an object with given mass in a given location in a gravitational field. (Just thought I'll say that out loud to benefit other people who are starting with almsot zero physics knowledge, like me :D)

I think you have that pretty straight.

I.e. the radial derivative, or the "slope" of the gravitational potential, is related to the angular velocity somehow?

As I stated, centrifugal force is given by


[math]\vec{F}=mr \omega^2\hat{r}[/math]


(Ah, Anssi you have once again caught an error in a post of mine. I omitted that “r” in there. I have edited the original post to correct the error. I thank you once again. You are clearly the only person who reads my stuff carefully.) Sorry you took it on faith to be correct. You shouldn't do that with my stuff; I do certainly make errors.

At any rate, the force due to the gravitational potential is given by [imath]\vec{F}=-m\vec{\nabla}\Phi[/imath]. Setting these two forces to be identical (together with the fact that [imath]\vec{\nabla}\Phi=\frac{\partial}{\partial r}\Phi(r) \hat{r}[/imath] because there is no change in the potential outside a change in r) and, setting these two forces to be identical one has immediately the fact that


[math]r\omega^2=-\frac{\partial}{\partial r}\Phi(r)[/math]


which I am sure you would have accepted had it not been for my gross error.

Hmmm, you must be referring to a situation where some object is orbiting the center of the gravitational field, so then its angular velocity and the distance from the center are related to the strength of the associated "fictional force", which is keeping it in stable orbit.

No, I am talking about an object restrained by a string to the center of the coordinate system. The coordinate system is not an inertial system (or, in my case, at rest with respect to the universe) but is rather rotating with an angular velocity of omega. Because I do not know the coordinate system is rotating, I presume the force is caused by a gravitational potential. Thus, what I am calculating is the consequence of the rotation and then attributing it to gravitational effects; essentially casting the supposed gravitational force as a "pseudo" force.

Intuitively, that makes sense, but I am not entirely sure how you got that exact expression (specifically, the "r" on the right side).

Yeah, I omitted that r in my specification of the centrifugal force.

I'll pause here as I am getting the feeling I may well be interpreting you wrong already...

Nope, you are doing beautifully. As per usual, you just caught me in a serious error and I thank you for that.

Know that I appreciate you beyond belief.

Thanks -- Dick

#29 AnssiH

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Posted 02 May 2010 - 08:52 AM

Hi Anssi, I am sorry I hadn't let you know about the post. There is another post you should be aware of, I am speaking of “Laying out the representation to be solved”. I would appreciate any comments you might wish to make on the clarity of that post. I don't think you will have any problems understanding what I said, but, if you do, we can try and clear them up.


Okay, yeah, I saw it yesterday, but didn't read it through yet.

As I said, Maupertuis' proof has to do with the fact that objects with different velocities follow different paths in a gravitational field. This is totally counter to the proposition that “pseudo forces” create paths which are mirror images of the path of the non-inertial frame. That idea means that the path certainly can not depend upon the velocity of the object; not in a Euclidean geometry anyway.


Hmm, there's something I'm definitely missing here... :I
I suppose you are referring to, for instance, a cannon ball being shot at different velocities, and consequently following different trajectories... I'm just thinking that, since only the rate of descent would be attributed to the fictional force, the different trajectories shouldn't be a problem, at least not very obviously so.

I'm also thinking that with a coriolis effect the path obviously depends on the velocity of the object too, i.e. the final path is not exactly a mirror image of the path of the non-inertial frame... Hmmm, unfortunately I just can't find any information about that Maupertuis' proof, I don't know why is it so hard to find. I'm just finding all sort of material about his expidition to lapland :D

Well, if everything in the universe is traveling through my four dimensional Euclidean geometry at exactly the same velocity, then velocity differences don't exist and the actual paths can once more be mirror images of the path of the non-inertial frame. Path dependence on the velocity of the objects vanishes absolutely.


No comments, as I don't yet even understand the problem exposed by Maupertuis... :P

(Ah, Anssi you have once again caught an error in a post of mine. I omitted that “r” in there. I have edited the original post to correct the error...


Oh okay, now [imath]r\omega^2=-\frac{\partial}{\partial r}\Phi(r)[/imath] makes sense to me :D

I guess I should have expected to see an "r" somewhere in there, but I had already absorbed too much information at one sitting, and was starting to feel cloudy :)

No, I am talking about an object restrained by a string to the center of the coordinate system. The coordinate system is not an inertial system (or, in my case, at rest with respect to the universe) but is rather rotating with an angular velocity of omega. Because I do not know the coordinate system is rotating, I presume the force is caused by a gravitational potential. Thus, what I am calculating is the consequence of the rotation and then attributing it to gravitational effects; essentially casting the supposed gravitational force as a "pseudo" force.


Yup.

Back to OP;

It follows that the analogous gravitational representation implies that the required [imath]\Phi(\vec{x})[/imath] (which, by the way, must by symmetry be a radial function) must obey the relationship


[math]-\frac{\partial}{\partial r}\Phi(r)=r\omega^2[/math].


This fact directly implies that [imath]-\Phi=\frac{1}{2}(r\omega)^2=\frac{1}{2}|\vec{v}|^2[/imath]


At first I did not realize at all how you got to that expression (so unfamiliar with math), but after toying around with wolfram alpha a bit, I realized that [imath]\frac{\partial}{\partial r}\frac{1}{2}(r\omega)^2 = r\omega^2[/imath], so that's why you are saying [imath]-\Phi=\frac{1}{2}(r\omega)^2[/imath].

Then, regarding [imath]\frac{1}{2}(r\omega)^2=\frac{1}{2}|\vec{v}|^2[/imath] we are in unbelievable luck; my eyes just accidentally landed on a wikipedia text mentioning that the alternative to [imath]F = m\omega^2r[/imath] is [imath]F = \frac{mv^2}{r}[/imath]

So that implies;

[math]\frac{v^2}{r} = r\omega^2[/math]

i.e.

[math]v^2 = (r\omega)^2[/math]

Hence; [math]\frac{1}{2}(r\omega)^2=\frac{1}{2}|\vec{v}|^2[/math]

And even I can't believe I figured that out, like I said, complete accident since I don't know the derivation of the centrifugal force :D hehe

Anyway, looks valid to me.

or, multiplying by [imath]\frac{2}{c^2}[/imath], one can conclude that


[math]\frac{2}{c^2}\Phi(r)=-\left(\frac{|\vec{v}|}{c}\right)^2[/math].


Yup.

In other words, the gravitational potential (as seen in the frame where the object being observed appears to be at rest) seems to be directly related to the actual velocity as seen from the correct frame (the frame at rest with the universe).


Wait a minute... About the "velocity as seen from the correct frame"; are we now talking about a situation where an object is orbiting the center of a gravitational field? I.e, the frame where the object being observed appears to be rest, is essentially a rotating frame?

Or a situation where an object is basically just sitting on the ground?

Or both?

This result is very interesting. As the observed object is actually moving in the correct frame, we should expect a clock (or any temporal physical process moving with that object) to proceed in accordance with special relativity. This implies that the correct relativistic transformation of the instantaneous time differentials should be given by


[math]dt'=dt\sqrt{1-\left(\frac{|\vec{v}|}{c}\right)^2}\equiv dt\sqrt{1+\frac{2\Phi}{c^2}}[/math]


Indeed. I realize that is essentially the standard lorentz factor, but for the benefit of lurkers, in this analysis that relationship was derived in the thread about special relativity, and the same conclusion can be seen in the end of the little animation related to the analysis:

YouTube - Presentation of a moving clock

[math]Cycles_m = Cycles_r \sqrt{1 - sin^2(\theta)}[/math]

I.e, if you backtrack from there a bit;

[math]Cycles_m = Cycles_r \sqrt{1 - \left (\frac{v}{c} \right )^2}[/math]

which happens to be exactly the standard gravitational red shift.


Looked at the wikipedia page for gravitational red shift and gravitational time dilation, and yes looks very familiar, albeit I did not find that exact expression.

This implies that any geometry which yields gravity as a pseudo force must also yield the standard gravitational red shift...
.
.
.
......i.e., instead of seeing the speed of light as slower in a gravitational field we could just as well see the speed as unchanged and the distances as increased. After all, once time is defined, distances are reckoned via the speed of light. Though that satisfies the original goal expressed above, the idea of refraction (the speed of light being slowed in a gravitational field) is a much simpler expression of the solution. It is certainly most convenient method of finding the proper geodesics. In fact, there is a very simple view of the situation which will yield exactly that result.


Got all the way to that paragraph and did not have problems with it. I'll have a rest here again.

-Anssi

#30 modest

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Posted 02 May 2010 - 09:15 AM


[math]dt'=dt\sqrt{1-\left(\frac{|\vec{v}|}{c}\right)^2}\equiv dt\sqrt{1+\frac{2\Phi}{c^2}}[/math]


which happens to be exactly the standard gravitational red shift.


Looked at the wikipedia page for gravitational red shift and gravitational time dilation, and yes looks very familiar, albeit I did not find that exact expression.


Substitute [math]\Phi = -GM/r[/math] (from the definition of gravitational potential) and that's probably how wiki has it expressed. It is sometimes expressed as a function of potential in the literature. Eq. 4.4:

http://books.google....lation"&f=false

Eq. 17.18:

http://books.google....tential&f=false

~modest

#31 Doctordick

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Posted 02 May 2010 - 08:56 PM

Hmm, there's something I'm definitely missing here... :I
I suppose you are referring to, for instance, a cannon ball being shot at different velocities, and consequently following different trajectories... I'm just thinking that, since only the rate of descent would be attributed to the fictional force, the different trajectories shouldn't be a problem, at least not very obviously so.

I'm also thinking that with a coriolis effect the path obviously depends on the velocity of the object too, i.e. the final path is not exactly a mirror image of the path of the non-inertial frame... Hmmm, unfortunately I just can't find any information about that Maupertuis' proof, I don't know why is it so hard to find. I'm just finding all sort of material about his expidition to lapland :D

Sorry about that. I also googled Maupertuis and found no reference whatsoever to the proof. The central problem here is that I studied physics so long ago that I suspect many things were given quite different spins back then. It could also be that my approach might be colored by idiosyncrasies of of individual professors. Whatever the cause, I have been out of contact with the academy for many many years. I seem to remember Maupertuis getting credit back then.

In my original paper, I had referenced that issue to page 6 of the 1965 edition of “Introduction to General Relativity” by Adler, Bazin and Schiffer.
The following is an exact quote of what is presented on page 6 of the 1965 edition.

Why has Einstein's idea of geometrizing the gravitational field of force not been conceived before? To answer this question let us look at the most geometrical of all variational principles of mechanics, namely, the principle of Maupertuis. In its simplest form it states the following: Let a particle move in a field of force with the potential V(x,y,z). If it travels from a point [imath]P_1[/imath] to a point [imath]P_2[/imath] with the varying velocity v, its trajectory is that actual curve which yields a stationary value for the action integral [imath]\int^{P_2}_{P_1}vds[/imath] among all paths connecting [imath]P_1[/imath] and [imath]P_2[/imath] which can be run through with the same constant energy [imath]E=\frac{1}{2}mv^2 +V[/imath] of the particle. We may express this principal in the obvious variational formula


[math]\delta\int^{P_2}_{P_1}\left(\frac{2}{m}(E-V)\right)ds=0[/math]


In the case of V=0, we obtain the rectilinear motion asserted by the law of inertia. In the case of a nonvanishing potential V(x,y,z), we can introduce a metric based on the line element


[math]dl^2=\frac{2}{m}[E-V(x,y,z)](dx^2_1+dx^2_2+dx^2_3)[/math]


and formulate the trajectory condition as


[math]\delta\int^{P_2}_{P_1}dl=0[/math]


In the new differential geometry with this line element dl, the trajectory would indeed be a geodesic. But observe that, for different particles in the same field and with different energies E, the geometry would have to be a different one, which is impossible. This fact precluded a geometrization of dynamics.

We can see the same difficulty from the following consideration. Suppose that the gravitational field of the sun creates a non-Euclidean geometry and that the planets have to move along the geodesics of this geometry. It is well known that, if we prescribe a point in space and a direction through this point, there exists exactly one geodesic passing through the point with the prescribed direction. On the other hand, two particles in a gravitational field fired from the same point in the same direction will move along the same trajectory only if their initial velocities are equal. Thus only one projectile could at most follow the corresponding geodesic. Indeed geometry deals with the space variables and directions, but velocity is a concept involving time, and it is the initial velocity which enters into the determination of a trajectory. In the theory of special relativity Einstein had shown that space and time variables are inextricably connected and transform among each other under Lorentz transformations. A reduction of gravitational theory to geodesic motion in an appropriate geometry could be carried out only in the four-dimensional space-time continuum of relativity theory. That this is indeed possible is the main thesis of this book. That a reduction of the theory of gravitation to geometry was hardly possible before the special theory of relativity should be clear from the preceding considerations.

Note that the whole issue boils down to “different velocities” for different objects.

Note that with regard to the centrifugal and Coriolis forces, there is a simple Euclidean transformation which yields a straight line path for a free particle: i.e., the transformation to a non-rotating system. The issue with gravity is that they were searching for a transformation to a non-Euclidean system. Since the orbits return upon themselves the geometry simply can not be Euclidean because, in a Euclidean system, straight lines can not yield closed paths. In other words to find a rational result, they had to examine much more complex systems.

And even I can't believe I figured that out, like I said, complete accident since I don't know the derivation of the centrifugal force :D hehe

I think you are beginning to learn some mathematics.

Regarding centrifugal force the issue is actually quite simple. Most people measure angles in degrees. Physicists tend to measure angles in “radians” (it makes a lot of angular mathematics easy). By definition, instead of 360 degrees in a circle, there are [imath]2\pi[/imath] radians. That makes arc lengths easy to specify. If the angle [imath]\theta[/imath] is measured in radians the arc length is just [imath]r\theta[/imath]. Angular velocity is generally represented by [imath]\frac{d\theta}{dt}=\omega[/imath]. Thus the velocity along the circle of that rock out there on the end of the string is given by [imath]r\omega[/imath]. Note that the speed of the rock does not change; what changes is its direction. The change in velocity is always perpendicular to the path; it is towards the center, the direction the string is pulling. So the change in velocity (the acceleration) is the component of its velocity parallel to the string after time dt. Well the distance the rock has moved along the path is [imath]vdt=r\omega dt[/imath] so the change in velocity in the time dt is [imath]dv=(vdt)d\theta=r\omega d\theta[/imath] (draw a picture, the two triangles, radius against distance moved and velocity against velocity change, involved here are similar). Divide that by dt and one has the acceleration [imath]\frac{dv}{dt}=r\omega \frac{d\theta}{dt}= r\omega^2[/imath].

Wait a minute... About the "velocity as seen from the correct frame"; are we now talking about a situation where an object is orbiting the center of a gravitational field?

No, I am still talking about the centrifugal force, the rock on a string. Seen from the rotating frame (where it is at rest) there is an apparent force holding it out there against the string. I am just calling that force a “gravitational force” because, looking at it from my rotating frame, I have no idea what is causing it to pull on that string. The “correct frame” is the one which is not rotating.

Looked at the wikipedia page for gravitational red shift and gravitational time dilation, and yes looks very familiar, albeit I did not find that exact expression.

I will plead senility on that one. I don't know exactly where I got the expression (that was a lot of years ago). I will go with modest on the validity of the expression; see his second link (equation 17.19).

Got all the way to that paragraph and did not have problems with it. I'll have a rest here again.

Thank you very much for your efforts. I won't post any more of my proof of the fundamental equation until you have proof read the “laying out the representation” post.

Have fun -- Dick

#32 AnssiH

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Posted 04 May 2010 - 02:05 PM

Just a short reply for now...

In my original paper, I had referenced that issue to page 6 of the 1965 edition of “Introduction to General Relativity” by Adler, Bazin and Schiffer.
The following is an exact quote of what is presented on page 6 of the 1965 edition.
Note that the whole issue boils down to “different velocities” for different objects.


Okay, I didn't understand all of that, but I have a faint idea of it having to do with getting inconsistent energies, and about relativistic time relationships adjusting them correctly... I think.

Note that with regard to the centrifugal and Coriolis forces, there is a simple Euclidean transformation which yields a straight line path for a free particle: i.e., the transformation to a non-rotating system. The issue with gravity is that they were searching for a transformation to a non-Euclidean system. Since the orbits return upon themselves the geometry simply can not be Euclidean because, in a Euclidean system, straight lines can not yield closed paths. In other words to find a rational result, they had to examine much more complex systems.


Right, okay.

Regarding centrifugal force the issue is actually quite simple. Most people measure angles in degrees. Physicists tend to measure angles in “radians” (it makes a lot of angular mathematics easy). By definition, instead of 360 degrees in a circle, there are [imath]2\pi[/imath] radians. That makes arc lengths easy to specify. If the angle [imath]\theta[/imath] is measured in radians the arc length is just [imath]r\theta[/imath]. Angular velocity is generally represented by [imath]\frac{d\theta}{dt}=\omega[/imath]. Thus the velocity along the circle of that rock out there on the end of the string is given by [imath]r\omega[/imath]. Note that the speed of the rock does not change; what changes is its direction. The change in velocity is always perpendicular to the path; it is towards the center, the direction the string is pulling. So the change in velocity (the acceleration) is the component of its velocity parallel to the string after time dt. Well the distance the rock has moved along the path is [imath]vdt=r\omega dt[/imath] so the change in velocity in the time dt is [imath]dv=(vdt)d\theta=r\omega d\theta[/imath] (draw a picture, the two triangles, radius against distance moved and velocity against velocity change, involved here are similar). Divide that by dt and one has the acceleration [imath]\frac{dv}{dt}=r\omega \frac{d\theta}{dt}= r\omega^2[/imath].


Ah, I see, pretty clever :)

No, I am still talking about the centrifugal force, the rock on a string. Seen from the rotating frame (where it is at rest) there is an apparent force holding it out there against the string. I am just calling that force a “gravitational force” because, looking at it from my rotating frame, I have no idea what is causing it to pull on that string. The “correct frame” is the one which is not rotating.


Right, okay.

I will plead senility on that one. I don't know exactly where I got the expression (that was a lot of years ago). I will go with modest on the validity of the expression; see his second link (equation 17.19).


Yes, looks like there it stands in exactly the same form. Thank you for digging that up Modest.

Thank you very much for your efforts. I won't post any more of my proof of the fundamental equation until you have proof read the “laying out the representation” post.


Yup, I'll take a look at it.

-Anssi

#33 Bombadil

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Posted 09 May 2010 - 12:00 PM

That idea together with differential calculus created an extremely powerful mathematical method of predicting the dynamic behavior of objects (an object being any suitable stable defined collection of information). Since acceleration is the time derivative of velocity (where velocity is the time derivative of position in that “inertial frame”) force can be thought of as the time derivative of momentum (momentum being given by [imath]m\vec{v}[/imath]). If m is a constant, the expression [imath]\vec{F}=\frac{d}{dt}(m \vec{v})[/imath] is identical to [imath]\vec{F}=m\vec{a}[/imath] and if m is allowed to change, that fact simply allows a rather simple mechanism to handle cases where identical forces cause different accelerations. As a consequence, m ends up being little more than a parameter allowing a more versatile definition of force.


So here we are using the equation [imath]\vec{F}=m\vec{a}[/imath] as the definition of force in any inertial frame. That is, any frame that is not accelerating, or equivalently one that has no force acting on it, which is sort of a circular definition so maybe we should avoid it. But doesn’t this bring up the issue that different frames will not agree on the force on a object as they may not agree on length or mass so they wont agree on the acceleration of an object?

Also, what about the possibility that the mass will be a function of t or is it also required by our choice of using an inertial reference frame that the mass is a constant?

That brings up the interesting question, “what does the dynamic behavior look if one is using a non-inertial frame”. Clearly, a non-inertial frame is a frame which is accelerating relative to an inertial frame. It should be clear to the reader that any object at rest in any inertial frame (which then, by definition, has no forces accelerating it) will appear to be accelerating if its position is represented by coordinates in a non-inertial frame representation. It should also be quite clear that it is not really accelerating at all, it is only the reference frame which is actually changing (accelerating). The apparent motion of any force free physical object who's position is being represented via the non-inertial frame will be an exact mirror image of that frames acceleration.


So is there a way to tell what frame we are in, is it as simple as saying that in a non-inertial reference frame we are experiencing a force and the transformation you are talking about is a purely mathematical means of making the force on a object vanish in our explanation.

That is, we are now interested in finding a reference frames where if an object is experiencing a force it no longer has a force acting on it in the new coordinate system. And the rest of the universe is now experiencing a force in the opposite direction that results in the same acceleration.


[math]dt'=dt\sqrt{1-\left(\frac{|\vec{v}|}{c}\right)^2}\equiv dt\sqrt{1+\frac{2\Phi}{c^2}}[/math]


which happens to be exactly the standard gravitational red shift. This implies that any geometry which yields gravity as a pseudo force must also yield the standard gravitational red shift; or, alternately, gravitational red shift is not really a valid test of Einstein's general theory of relativity. This really isn't very enlightening as the gravitational red shift can be shown to be required by conservation of energy, but it does nonetheless imply that the above analysis is valid.


That is, the only thing that a different force will imply is the actual form and value of [imath]\Phi[/imath] which is the equivalent velocity of an object. That is, it is the velocity that the object will have in a reference frame where the force on the object vanishes that will make the Lorenz transformation the correct transformation.

But I don’t understand how this results in red shift of a object as it looks like all that you have done is calculate the change of a clock in an accelerating reference frame (the left side of the equation) in comparison to what a clock in an inertial reference frame will measure (the right side of the equation).

More importantly, the above suggests an attack towards determining the geometry which will yield gravity as a pseudo force in our four dimensional Euclidean geometry. I have already shown how static structures appear as three dimensional objects in this geometry so let us examine what is commonly called “a gravitational well”. The gravitational well consists of a vertical hole where there is a gravitational field in the vertical direction. If an experimenter in a gravitational well sets up a clock via a light pulse traveling back and forth between two horizontally displaced mirrors, since we can establish horizontal measure (simple vertical lines carry those measures to different heights in the hole) and his clock must run slow, we must see the apparent velocity of light to be


[math]c'=c\sqrt{1+\frac{2\Phi}{c^2}}[/math]


So do we conclude that the passage of the clock in the gravitational well is running slower because just like in your example of centrifugal force the object must act as though it is moving in comparison to a inertial reference frame. That is, the Lorenz transformation is valid we just have to solve for the required function to substitute in for the velocity of the object in it’s inertial frame.

Won’t this result though, in the conclusion that the length of his clock is longer then our clock rather then a faster speed of light since we have defined length and speed to be interdependent of each other and so it would make sense to measure his clock by means of how long light takes to move from one of the mirrors to the other. As a result we would conclude that the lines that are extended up from the mirrors are not straight but are bent lines.

Or is this the very problem with general reactivity, it assumes that the speed of light is constant and as a result must generate a geometry where the lines are bent? And since the geometry that you are using is a Euclidean geometry those lines must be straight lines so we have no way to conclude that the speed of light is the same in a gravity well and must in fact conclude that it is slower. But if this is the case what will someone in a gravity well see when looking at a clock that is not in the gravity well? Will they conclude that our clock is running faster then one not in the gravity well.

Also, I don’t understand how you arrived at the above equation but I suspect that it is an issue of velocity rather then length being scaled by the Lorenz transformation.

#34 AnssiH

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Posted 12 May 2010 - 05:07 PM

With regard to the issue of refraction, my fundamental equation is a wave equation with Dirac delta function interactions. Clearly, in the absence of interactions, the probability wave representing an event will proceed at a fixed velocity. Any specific delta function interaction can be seen as an impact changing the direction and energy of that probability wave. What is important here is the fact that interaction will depend upon the distance between the two elements connected by that delta function interaction: i.e., the hypothetical element (which must be a boson) must carry the momentum and energy being transfered and the transfer must be consistent with the Heisenberg uncertainty principal: i.e., the uncertainty in momentum is directly related to the uncertainty in position. This implies that the further apart the interacting fermions are, the less momentum transfered must be (see virtual particle exchange).


I'm struggling here a bit.

First I am not exactly sure what do you mean by a probability wave representing an event. I was aligned to think of this in terms of probability waves representing the positions of defined objects. Do you just see them as essentially the same thing, or do you refer to something like a probability wave representing something like a collision between virtual particle and a fermion?

Anyway, I read that page about the virtual particle exchange, did not understand everything about it, but I picked up some idea of what you are talking about. I did not pick up though, why the Heisenberg uncertainty principle implies that the further apart the interaction fermions are, the less momentum transfer must be... I guess it has got something to do with how the wave functions interfere, but I am completely unfamiliar with that subject and didn't manage to figure it out... :P

Any physical object (any structure stable enough to be thought of as an object) must have internal forces maintaining that structure. Any interaction with another distant object must be via the virtual particle exchange I just commented about. Thus it is that one would expect the fundamental element of that physical object interacting via that delta function would have its momentum altered, not the whole object; however, that alteration would create a discrepancy in the structure of the object under discussion. Since that object must have internal forces maintaining its structure, it is to be expected that those internal interactions (which are also mediated by delta function exchange forces) will bring the trajectory that interacting fundamental element essentially back to its original path (at least on average).

Thus it is that the path of that fundamental element can be seen as crooked as compared to its path in the absence of that distant object. Of issue is the fact that, if the influence of the distant object is ignored, the influenced element will inexplicitly appear to be proceeding at a slightly slower velocity than it would if the distant object didn't exist. What is important here is that this effect decreases as the distance from the distant object increases. That means that the net effect is to yield a very slight change in the speed of the elements which make up that object as one moves across the object. The net effect of such an interaction is to refract the wave function of the object under examination.


Right I see, the point is that those sub-elements of the object that are closer to the distant object, are slowed down slightly more than those sub-elements that are further away, so the total average path would be expected to curve... I understand the analogy to refraction now.

So I suppose when you refer to the speed of an element, you must be referring to its speed in [imath]x,y,z,\tau[/imath]-space. I.e. it is because every element must move at the same fixed speed, that more wiggling means slower speed towards the average direction of the entire object.

I spent quite a while thinking about this, and yeah, it seems valid.

If the distant object and the object under observation are not moving with respect to one another (they are moving parallel to one another in the tau direction), the net effect of that refraction is to curve the paths of the two objects towards one another: i.e., there will be an apparent attraction between them.


Yup.

It is also evident that, since the mass of the source object (the source of these bosons external to the object of interest) is proportional to the total momentum of that object, one should expect the apparent density (as seen from the object of interest) should be proportional to its mass: i.e., one should expect the exchange forces to be proportional to mass.


Right... When you say "total momentum", you are essentially referring to the number and density of the individual elements moving along [imath]\tau[/imath] (at fixed speed).

Clearly the interaction just discussed arises from differential effects in the basic interactions thus it will amount to a force considerably less than the underlying force standing behind that differential effect. Thus it is that the two forces I have already discussed (the forces due to massless boson exchange: shown to yield electromagnetic effects and the forces due to massive boson exchange: shown to yield fundamental nuclear forces) will end up being split into four forces.


I may be forgetting something, but where and when the massive boson exchange/nuclear forces were brought up...?

Differential effects will yield a correction to both basic forces which correspond quite well with the forces observed in nature. The differential effect on massless boson exchange yields what appears to be a very weak gravitational force (weak when compared to the underlying electromagnetic effects) and the differential effect on massive boson exchange yields what appears to be a very weak nuclear force (weak compared to the underlying nuclear force).


Right, that makes perfect sense to me, if there's a force mediated by boson exchange, there must be a corresponding differential effect via refraction, due to the fixed speed of the elements...

What is interesting is that the “weak nuclear force” can be shown to violate parity symmetry whereas the “weak electrical force” (gravity) does not. This is a direct consequence of the fact that the nuclear exchange bosons are massive.


That is very interesting... Another very unexpected feature of modern physics turning out to be a very expected feature of the symmetry requirements. Should probably discuss that issue little bit at some point as well.

Apart from this analysis, does any satisfying explanation exist as to why weak nuclear force violates parity symmetry?

I'll have a pause here again, but first I still have one more comment. When I'm thinking about the differential effect caused by boson exchange forces, I can't help but think of the unexplained behaviour of foucault pendulum during a solar eclipse.

Decrypting the Eclipse - NASA Science

Current theories of gravity don't explain the Allais effect, and AFAIK the best attempt to explain it so far is that the shadow of the moon causes air to cool down and its density to increase, which causes a gravitational effect. That apparently causes too small gravitational effect to actually explain the Allais effect though (I don't have the skills to work it out, but certainly that explanation sounds dubious at best).

At any rate, someone more skilled than me could probably work out whether this paradigm explains Allais effect directly as a gravitational effect. I guess the moon should be expected to affect the virtual particle exchange between the sun and the pendulum one way or another. I can't work out what sort of interference one might expect between all the relevant bodies, but I'm just thinking if this explains the Allais effect as a gravitational effect, that would be remarkable. And another first. :shrug:

-Anssi