[Doctordick]

AnssiH said:

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.

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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 to a point with the varying velocity v, its trajectory is that actual curve which yields a stationary value for the action integral among all paths connecting and which can be run through with the same constant energy of the particle. We may express this principal in the obvious variational formula




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




and formulate the trajectory condition as




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.

AnssiH said:

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 radians. That makes arc lengths easy to specify. If the angle is measured in radians the arc length is just . Angular velocity is generally represented by . Thus the velocity along the circle of that rock out there on the end of the string is given by . 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 so the change in velocity in the time dt is (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 .

AnssiH said:

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.

AnssiH said:

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).

AnssiH said:

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

[AnssiH]

Just a short reply for now...

Doctordick said:

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.

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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.

Quote

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 radians. That makes arc lengths easy to specify. If the angle is measured in radians the arc length is just . Angular velocity is generally represented by . Thus the velocity along the circle of that rock out there on the end of the string is given by . 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 so the change in velocity in the time dt is (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 .

Ah, I see, pretty clever :)

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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.

Quote

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.

Quote

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

[Bombadil]

Doctordick said:

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 ). If m is a constant, the expression is identical to 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 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?

Doctordick said:

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.

Doctordick said:

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 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).

Doctordick said:

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

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.

[AnssiH]

Doctordick said:

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

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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 -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.

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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.

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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 (at fixed speed).

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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...?

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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...

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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

[Doctordick]

AnssiH said:

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.

I would say that the "supposed position of a defined object" can be called "an event".

AnssiH said:

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?

A collision between a virtual particle and a fermion can be seen as "an event", as can the simultaneous change in the form of the probability wave. Remember, the probability wave yields our expectations and, if a collision occurs, certainly our expectations change. In modern quantum mechanics that change is called "collapse of the wave function".

AnssiH said:

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...

If the exchanged particle is "virtual", it means, from a colloquial perspective, that it "really isn't there". From a quanta mechanical perspective, it means that it has insufficient energy to exist at that particular point in space. Since at this point we are talking about photons (which have no rest energy) the energy is given by its momentum and we can apply the uncertainty principal.




Essentially, so long as we can't detect it (to detect it would violate the uncertainty principal) and it qualifies as "virtual". Thus, so long as the exchange is virtual, the larger (the distance over which it is exchanged) the smaller (the momentum being exchanged) must be.

AnssiH said:

So I suppose when you refer to the speed of an element, you must be referring to its speed in -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.

That is exactly correct.

AnssiH said:

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

Yes. It seems to me to be quite reasonable that the momentum transfer rate would be proportional to the density of the sources that provide that exchange: i.e., the greater the total momentum change can be over the same change in tau (the reading on a clock traveling with the object being influenced).

AnssiH said:

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

The issue was brought up in the last six paragraphs of the "Anybody interested in Dirac's equation?" thread. I really do little with it beyond suggesting that such an interaction certainly appears to be required by my fundamental equation. By the way, the purpose of displaying these relationships to modern physics which I have uncovered was not to prove modern physics was a tautology. The purpose was to generate interest in the proof of my equation among people who had the education to subject that proof to hard analysis. I have been totally confounded by the fact that those who could analyze that proof absolutely refuse to look at it.

Personally I have been quite astonished by the fact that so many of modern physics theories can be mapped into approximate solutions to my equation. That equation is a constraint and I certainly would not expect it to be the only constraint on modern physics as that in itself would imply physics is a tautology (the conclusion I have come to after many years). What I would expect (and have been unable to discover) is a solution to my equation which could not be found in modern physics: i.e., that some additional constraint would be required by physics. Such an event would give physics an independent status above and beyond a tautology, but I haven't found it and it seems that no one else is interested in looking.

AnssiH said:

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.

Ah yes, parity is an interesting issue all by itself. Parity is often referred to as "mirror" symmetry. If you look in a mirror, you will find that the image is different from the image you would see if a duplicate "you" stood in front of you. That difference can be seen in three specific different ways.

The most common way to see the difference is to note the fact that a duplicate of you would have to turn around to face you. When he did that, his right arm would be on your left and his left arm would be on your right: i.e., everything along the "right left" axis would appear to be inverted. All other relationships remain the same.

But that is because we commonly think of people turning around by rotating around their vertical axis. If we are going to actually analyze the phenomena itself, we could just as well presume he could have turned around by rotating around that horizontal "right left" axis. In that case we would expect to see his head on the bottom and his feet on the top: i.e., in the image, everything along the vertical axis would be inverted. (Note that his right arm is on the same side as your right arm; wave at yourself and think about it.)

The third way to see it is to see everything along the axis between the two of you inverted: i.e., you see the "front" of the image instead of the back you would expect to see if a duplicate "you" stood in front of you. In this case everything else is as expected. His head and your head are both on top and your right arm and his right arm are on the same side but his face is towards you and not away as it should be.

What I am getting at is the fact that a mirror image can be seen as accomplished by the inversion of any one axis. Now, consider the inversion of all three axes. If you invert all three axes, you get a mirror image: i.e., three parity swaps are identical to one parity swap. That brings up an interesting conundrum. Why should the physics of such an inverted universe be different from the original: i.e., when you consider the fact that all the information about the universe is contained in "the information about the universe", why should plotting the three axes in a particular direction make a difference. How does one know "the correct direction". It seems entirely reasonable that mirror symmetry should be a symmetry of the universe.

Things are a bit different in a four dimensional Euclidean universe. In that case, inversion of all the axes bring one back to the original configuration so inversion of all the axes have to be a symmetry of the universe. However, if we invert only the three axes of our perceived space (the tau axis not being inverted) then we have a parity inversion (a true inversion of all the axes would include inverting tau). This results in a rather strange circumstance when applied to massive boson exchange. If the velocity in the tau direction of an entity being exchanged is in the same direction as the source, one might very well expect the interaction to be subtly different from the case where they are traveling in opposite direction.

If that is a second order effect, it might very well be on the same order as the differential effect which would suggest that the weak nuclear force might well display parity consequences.

If you take a look at the fundamental equation, note that multiplying through by minus one changes the sign of all the terms. Changing the sign of all the momentum terms changes nothing as the total momentum is still zero and changing the sign of the energy term can be absorbed into the definition of "t" (which is unmeasurable anyway), thus the only real change is a change in the sign of the Dirac interaction term. That would change all attractive terms into repulsive terms and vice versa. (Kind of sets the direction of time in our explanations doesn't it?)

Inversion of tau is essentially changing particles into antiparticles so the above suggests that the interaction differences showing parity effects should be related to which version of the exchange pair is involved. What is above is really speculation as I have never honestly worked out all the details of the thing and there are a couple of things about the deduction which bother me. That is another reason I would like some well educated people to look at it.

AnssiH said:

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

What one person finds satisfying may bother another. I will direct you to Erasmus as he is apparently one hundred percent satisfied with the modern physics version of such things.

AnssiH said:

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.

What you seem to be missing here is the realization that my fundamental equation is essentially unsolvable. The best one can do is to find approximate solutions. In order to do that one needs to know exactly what terms can be ignored and what approximations for the terms which can not be ignored are reasonable: i.e., in essence, you need to know the mathematical solution before you start making your approximations. In other words, my equation is not a means of deducing a solution but is rather a means of eliminating proposed solutions. That is exactly the mistake Bombadil insists on making.

I read your reference and I have no idea as to exactly what the nature of the Allais effect is; that article does not give the details of the differences as compared to the expectations. It could be as simple as some kind of gravitational focusing effect due to the intermediate body or it may be some kind of local error not being taken into account. The Foucault pendulum is a very sensitive experiment and requires careful elimination of interfering forces. I was involved in setting one up in graduate school and we had some problems getting it to work properly. At any rate, my equation really doesn't provide any help on the thing that I can see. Sorry about that.

Just as an aside, I appear to have upset modest so I wouldn't expect any help from him. Sorry about that too.

Have fun -- Dick

[Rade]

DD said:

Inversion of tau is essentially changing particles into antiparticles so the above suggests that the interaction differences showing parity effects should be related to which version of the exchange pair is involved. What is above is really speculation as I have never honestly worked out all the details of the thing and there are a couple of things about the deduction which bother me.

Hi AnssiH. Given that DD not longer views my posts, I look to you to serve as middle person in possible dialog. I find this comment of DD to be of great interest. In his response to you, DD indicated that the purpose of his fundamental equation is to serve as "a means of eliminating proposed solutions". OK. Here is a "proposed solution" to the fundamental equation, and I would like to know the mathematics of how the fundamental equation can be used to "eliminate it". It involves what I can best explain as a combined real+virtual interaction of matter and antimatter along the tau dimension (that is, momentum in both the positive tau direction and what DD calls the inversion of tau direction).

I would like to know how the fundamental equation would eliminate the possibility of predicting the "mass" for the proposed solution for the event of a matter object of 9 mass units (as quarks) as a coordinated fermion interacting with an antimatter object of 6 mass units (as antimatter quarks) as a coordinating boson along the tau dimension. It is my opinion that, if the fundamental equation CANNOT eliminate this possible event as a proposed solution to formation of mass along the tau dimension, then a new physics understanding of quark interactions within nucleons (protons, neutrons) will result. I hope this question will show DD I have a true interest in his fundamental equation.

[AnssiH]

Doctordick said:

I would say that the "supposed position of a defined object" can be called "an event".

...

A collision between a virtual particle and a fermion can be seen as "an event", as can the simultaneous change in the form of the probability wave. Remember, the probability wave yields our expectations and, if a collision occurs, certainly our expectations change. In modern quantum mechanics that change is called "collapse of the wave function".

Right, of course. It represents the probability of actually detecting something.

Quote

If the exchanged particle is "virtual", it means, from a colloquial perspective, that it "really isn't there". From a quanta mechanical perspective, it means that it has insufficient energy to exist at that particular point in space. Since at this point we are talking about photons (which have no rest energy) the energy is given by its momentum and we can apply the uncertainty principal.




Essentially, so long as we can't detect it (to detect it would violate the uncertainty principal) and it qualifies as "virtual". Thus, so long as the exchange is virtual, the larger (the distance over which it is exchanged) the smaller (the momentum being exchanged) must be.

Hmm, okay, I don't really understand that...

I suppose means uncertainty in momentum (/energy in this case), and means uncertainty in position. I understand that idea about "insufficient energy to exist at that particular point in space", but I don't understand why "less uncertainty in energy" means "less energy". (At least that's the implication I got from what you wrote)

Likewise, I don't understand why "uncertainty in position" is related to the "actual distance over which the exchange occurs".

I suppose this is traditional physics terminology, but it's a bit confusing/funny/odd to someone like me, that a virtual exchange, i.e. undetected exchange, is said to be the kind of exchange that has got "least uncertainty" to it; undetected exchange is considered to be the most certain kind of exchange :D

Anyway, I do understand that as long as we are limited to staying below the minimal uncertainty limit, the larger uncertainty to position means smaller uncertainty to momentum, but again I do not understand why less "uncertainty to momentum" also means less "momentum".

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The issue was brought up in the last six paragraphs of the "Anybody interested in Dirac's equation?" thread. I really do little with it beyond suggesting that such an interaction certainly appears to be required by my fundamental equation.

Okay, now I remember. Yeah that seems like valid speculation.

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[parity symmetry]
...
Inversion of tau is essentially changing particles into antiparticles so the above suggests that the interaction differences showing parity effects should be related to which version of the exchange pair is involved. What is above is really speculation as I have never honestly worked out all the details of the thing and there are a couple of things about the deduction which bother me. That is another reason I would like some well educated people to look at it.

Hmmm, yes... That is interesting speculation too.

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What you seem to be missing here is the realization that my fundamental equation is essentially unsolvable. The best one can do is to find approximate solutions. In order to do that one needs to know exactly what terms can be ignored and what approximations for the terms which can not be ignored are reasonable: i.e., in essence, you need to know the mathematical solution before you start making your approximations. In other words, my equation is not a means of deducing a solution but is rather a means of eliminating proposed solutions.

Actually the reason it popped up into my mind is because of how you arrive at "gravity" via virtual particle exchange. More about that below.

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I read your reference and I have no idea as to exactly what the nature of the Allais effect is; that article does not give the details of the differences as compared to the expectations.

Yeah, I just picked up pretty much the first article that came up saying anything about the Allais effect, because I thought you were probably aware of it.

Anyway, the original observation was, that the rate of change to the azimuthal angle changed abruptly during the solar eclipse. Before the solar eclipse, Allais had his swinging pendulum rotating at the steady rate of 0.19 degrees per minute, clockwise, as was expected at his latitude. At the beginning of the eclipse, the rotation changed direction to counter-clockwise and was changing an average of ~13 degrees per minute. In the middle of the eclipse the rotation changed direction again, and in the end of the eclipse the rate of rotation resumed to normal rate of 0.19 degrees per minute, clockwise.

Corresponding anomalies to gravity during a solar eclipse have been observed since, with different kinds of pendulums, gravitometers and atomic clocks. Some experiments have come out negative though.

There's a lot of information around the net, I just dug up couple that seem to contain little bit more information;
http://arxiv.org/ftp...408/0408023.pdf
Review on Gravitational Anomalies

At any rate, there are two reason why that popped into my mind. One, it's one of the gravitation-related effects that current models of gravitation don't explain directly (in addition to the "pioneer anomaly" and the anomalous rotation speed of galaxies Galaxy rotation curve - Wikipedia, the free encyclopedia ).

And second, since you are viewing gravity as an expected differential effect (inside our solution to reality), and since you are arriving at it via the idea of virtual photon exchange, it would not surprise me at all if that also necessitates one to expect disturbances to the total momentum exchange, due to the expected interference caused by the moon between the sun and the location of the pendulum on the surface of earth. The way the foucalt pendulum reacts in the very beginning and in the very end of solar eclipse just reminds me of interference patterns a lot.

But as I said, I do not have the skills to work that out, and my knowledge of virtual photon exchange is far too limited to figure out if anything can be made out of it (I don't know what sort of effect a "middle body" puts to virtual photon exchange, if any...).

But let it be said, that if someone can show that Allais effect is an expected effect of gravitation directly, in terms of your paradigm, I think it should generate some more interest towards your work. Just like the possibility of it explaining the pioneer anomaly and the rotation speed of large galaxies (and other things explained via dark matter), as you mention in the end of the OP.

-Anssi

[AnssiH]

Rade said:

Hi AnssiH. Given that DD not longer views my posts, I look to you to serve as middle person in possible dialog. I find this comment of DD to be of great interest. In his response to you, DD indicated that the purpose of his fundamental equation is to serve as "a means of eliminating proposed solutions". OK. Here is a "proposed solution" to the fundamental equation, and I would like to know the mathematics of how the fundamental equation can be used to "eliminate it". It involves what I can best explain as a combined real+virtual interaction of matter and antimatter along the tau dimension (that is, momentum in both the positive tau direction and what DD calls the inversion of tau direction).

I would like to know how the fundamental equation would eliminate the possibility of predicting the "mass" for the proposed solution for the event of a matter object of 9 mass units (as quarks) as a coordinated fermion interacting with an antimatter object of 6 mass units (as antimatter quarks) as a coordinating boson along the tau dimension. It is my opinion that, if the fundamental equation CANNOT eliminate this possible event as a proposed solution to formation of mass along the tau dimension, then a new physics understanding of quark interactions within nucleons (protons, neutrons) will result. I hope this question will show DD I have a true interest in his fundamental equation.

I'm sorry but I'm not entirely sure what you are asking about, and I suspect even if I did understand, I would not have the necessary know-how to actually analyze it properly...

-Anssi

[Doctordick]

AnssiH said:

Hmm, okay, I don't really understand that...

Well the real problem here is that you cannot solve the relevant equations. Without mathematics the issue becomes rather meaningless. That being the case, I will try to explain it from a pseudo classical perspective.

Let's start with a real photon and two entities interacting with that photon. If one of the entities emits a photon and the other absorbs that emitted photon what do you expect? The photon starts in one place and ends up in another. That would be taken to imply that the photon had momentum in the direction it traveled. Conservation of momentum implies the emitter gains a momentum in the opposite direction and the absorber ends up gaining the photons momentum. The end result is that the two entities gain momentum in the opposite directions: i.e., their change in momentum is always away from one another. This case implies a repulsive force due to the exchange.

Now, let's look at a “virtual” photon exchange: i.e., it doesn't really exist but does nonetheless exchange momentum. In this case, which is emitting and which is absorbing is unknown. Not only that but the time of these events us unknown. In fact, from a quantum mechanical perspective (and that would be a solution of the dynamic equations of quantum mechanics) it could even be absorbed before it was emitted (has to do with the uncertainty in time). Its momentum could be in the opposite direction of its motion (remember it isn't really there so we can't check it, this is a sort of “let's look at this strange possibility from a classical perspective”). If that were the case (the momentum of the photon could be opposite to its travel direction), the two entities could then gain momentum towards one another: i.e., one can obtain an attractive force. (That is the real value of "virtual exchange forces".)

This type of thing can only occur if the uncertainty principal yields . From the classical perspective this thing just can't happen. If the actual real distance between the interacting entities is known to be x, the uncertainty in position of the photon must be at least x (it has to be able to get from one entity to the other somehow). Divide through the uncertainty relationship above by the factor and you obtain . If follows that, if , it's a classical exchange and the momentum of the photon is a measurable thing. In order to keep it in the realm of “unknowable”, the transferred momentum must fall off by more than one over x.

AnssiH said:

I don't understand why "less uncertainty in energy" means "less energy". (At least that's the implication I got from what you wrote)

Because the thing can only violate conservation of energy if it does it quickly. If it took long enough to observe (i.e., obeyed the uncertainty principal) it would be detectable from the classical perspective and it wouldn't be “virtual” and wouldn't violate conservation of energy.

AnssiH said:

Likewise, I don't understand why "uncertainty in position" is related to the "actual distance over which the exchange occurs".

Again if it doesn't violate the uncertainty principal, it is detectable as a classical entity: i.e., it's not virtual.

AnssiH said:

I suppose this is traditional physics terminology, but it's a bit confusing/funny/odd to someone like me, that a virtual exchange, i.e. undetected exchange, is said to be the kind of exchange that has got "least uncertainty" to it; undetected exchange is considered to be the most certain kind of exchange :D

It isn't “undetected” which is important here, it's rather “undetectable”. Photon exchange, real or virtual, is essentially undetectable by classical means except by detecting the energy and/or momentum exchange.

I read a few of the references you gave for the Allias effect and got the impression that the effect has not really been verified yet and there are people who believe it's bogus. As I said earlier, my equation doesn't produce solutions, it only points out flaws in solutions where you know what can and can not be ignored. If someone hands me a solution to the Allias effect then, knowing what kinds of terms they are ignoring, I can determine if or if not that solution does indeed obey my equation (it often takes some deep thought but it can generally be done). But no one has offered a solution that I can understand.

AnssiH said:

But as I said, I do not have the skills to work that out, and my knowledge of virtual photon exchange is far too limited to figure out if anything can be made out of it (I don't know what sort of effect a "middle body" puts to virtual photon exchange, if any...).

The problem here is that gravity is such a small effect that only the grossest exchange is detectable (two body exchange). If one adds in a third body the equation becomes quite difficult to solve and is probably solvable only for very special cases.

AnssiH said:

But let it be said, that if someone can show that Allais effect is an expected effect of gravitation directly, in terms of your paradigm, I think it should generate some more interest towards your work.

I doubt that very much. All it would do is add further support to the “crackpot” designation, a title I have already well earned. :lol: :lol: :lol:

Have fun -- Dick

[AnssiH]

Doctordick said:

Well the real problem here is that you cannot solve the relevant equations. Without mathematics the issue becomes rather meaningless. That being the case, I will try to explain it from a pseudo classical perspective.
.
.
.
...If the actual real distance between the interacting entities is known to be x, the uncertainty in position of the photon must be at least x (it has to be able to get from one entity to the other somehow).

Ah, okay, now that's clear. I was thinking of the emission and absorption as two different positions, both having their own uncertainty.

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Quote

I don't understand why "less uncertainty in energy" means "less energy". (At least that's the implication I got from what you wrote)

Because the thing can only violate conservation of energy if it does it quickly. If it took long enough to observe (i.e., obeyed the uncertainty principal) it would be detectable from the classical perspective and it wouldn't be “virtual” and wouldn't violate conservation of energy.

Hmm, I'm still confused...

I now understand how larged distance between the bodies implies larger uncertainty in position (of the exchange particle), and also I understand why that implies smaller uncertainty in total momentum/energy being exchanged.

I still don't understand why small uncertainty to the total energy transfer means the amount of energy transfer is also small. From your response, I'm guessing virtual energy exchange is always seen as something violating conservation of energy, by definition(?)

And I gather that the time it takes for the energy to transfer (i.e. the time between emission and absorption events?), is somehow related to the amount of energy transfer?

But I still have no idea why that is...

Quote

I read a few of the references you gave for the Allias effect and got the impression that the effect has not really been verified yet and there are people who believe it's bogus.

Yeah, it's not easy to setup experiments because you have to have a solar eclipse for them. Still it's generated enough interest that people have performed different types of experiments, and actually appears to be generally accepted as a real phenomena. At least real enough for mainstream press to write articles about its existence without getting frowned upon. It may be that most physicists believe that a conventional explanation will be found.

Okay, back to the OP. In it, I just spotted something that looks like an editing error of some kind:

Doctordick said:

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.

I don't think you meant to say that if the influence of the distant object is ignored, then the influenced element will appear to proceed slower.

Doctordick said:

But let us get back to gravitational effects. It is my position that gravity is a direct consequence of the fact that the presence of an object with momentum in the tau direction yields a secondary differential effect which causes the speed of elements through my four dimensional Euclidean space to slow: i.e., refraction of the wave solution to my fundamental equation occurs. In order to check my proposition, I need to be able to work out the geodesic paths implied by such a notion. To begin with, my assertion is that the four dimensional velocity of all elements can be seen as slowed in the presence of a gravitational potential. Thus it is that the “index of refraction” of a gravitational field is given by the instantaneous value of evaluated a specific point in that field (where c is the velocity of light far removed from any massive influence). Geodesic paths can be obtained by minimizing nds where “ds” is the distance differential along that geodesic path in my four dimensional Euclidean geometry.

Right I think I understand that. If there was no gravitational field in the situation, the "n" would be taken to be "1" everywhere, and in that case minimizing the nds would just yield a straight line in the coordinate system. In the case that there is a gravitational field, which yields larger "n" as you move closer to the "source" of that field, minimizing the nds means the path must curve away from the source little bit.

Quote

In order to accomplish that result, we need to use what is called “the calculus of variations”. We want the variation of the path integral along the geodesic to vanish: i.e.,

.


Thankfully this is a problem already solved by the physics community. The vanishing of the path integral occurs when the function being integrated satisfies the Euler-Lagrange equation.

Okay, I started up by reading your references, and wikipedia article about the calculus of variations. The wikipedia article is much more useful for me because it contains more extensive explanations.

I didn't yet read it all too carefully, I think I'll have to spend little bit more time on this. I did not yet understand what it means that "the variation of the path integral along the geodesic vanishes". I can sort of guess what it's about, but I would just like to be sure I understand what "variation of an integral" refers to exactly.

Overall, my understanding of this is very shady right now. I'm just picking up that the calculus of variations is a method for finding the function which corresponds to some minimum (or other extreme) of some integral of a given set of functions. Like the functions corresponding to the geodesic path in the case there are some constraints that yield non-linear paths, just like our problem with the the refraction index.

I'll continue from here later...

-Anssi

[Doctordick]

AnssiH said:

I was thinking of the emission and absorption as two different positions, both having their own uncertainty.

Generally speaking, the uncertainty in the positions of the two entities being influenced by virtual exchange is not a significant factor. In many cases those entities (and their motion) can be expressed via classical mechanics.

AnssiH said:

I don't think you meant to say that if the influence of the distant object is ignored, then the influenced element will appear to proceed slower.

If the object being observed lies between the observer and the distant object being ignored then, yes, the influenced object will appear to proceed slower than expected as it lies in a lower gravitational potential than the observer. If the distant object is being ignored, one’s calculations would essentially be presuming the two were at the same gravitational potential.

AnssiH said:

I can sort of guess what it's about, but I would just like to be sure I understand what "variation of an integral" refers to exactly.

In this case we are talking about the integral of a function defined along a specific path through a space. The variation being spoken of is a variation in that defined path. We are looking for the minimum in the length of that path: i.e., if the path is varied in any way, the integral will increase. This is analogous to “a straight line is the shortest distance between two points” used in Euclidean geometry.

Have fun -- Dick

[AnssiH]

Doctordick said:

If the object being observed lies between the observer and the distant object being ignored then, yes, the influenced object will appear to proceed slower than expected as it lies in a lower gravitational potential than the observer. If the distant object is being ignored, one’s calculations would essentially be presuming the two were at the same gravitational potential.

Oh that's how you meant it, actually existing in the situation but being ignored.... I thought you meant ignored as in "taken out of the situation being considered"... I think someone else might make the same mis-interpretation...

Quote

In this case we are talking about the integral of a function defined along a specific path through a space. The variation being spoken of is a variation in that defined path. We are looking for the minimum in the length of that path: i.e., if the path is varied in any way, the integral will increase. This is analogous to “a straight line is the shortest distance between two points” used in Euclidean geometry.

Ah, okay I see.

I'll try to get around to continue from here soon...

-Anssi

[AnssiH]

Doctordick said:

In order to accomplish that result, we need to use what is called “the calculus of variations”. We want the variation of the path integral along the geodesic to vanish: i.e.,

.


Thankfully this is a problem already solved by the physics community. The vanishing of the path integral occurs when the function being integrated satisfies the Euler-Lagrange equation.

Just in order to avoid making unfortunate false assumptions, that delta sign before the integration sign is simply to say "variation of..." right?

And the equation for "n", I see you got it simply from and established earlier in the OP.

Mathworld said:

...if is defined by an integral of the form



where



then has a stationary value if the Euler-Lagrange differential equation



is satisfied.

I take it the dot on is just to denote a derivative?

So... how I'm reading that is, that states an integration of some function over , and it's a function which takes the arguments , (which is a function of ), and the derivative of with respect to .

(I reckon is required to be a function of because it states means derivative of with respect to ...)

Then



looks to me like requiring the derivative of with respect to , to exactly negate the derivative of "the derivative of with respect to " with respect to (sic)

Like if they are seen as two different curves, they exactly negate each others. Sooo, if that's at all right, I think I'm starting to get some sort of grasp of this idea... Not very crystal clear yet though.

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In order to simplify the problem, I would like to turn to a spherically symmetric case. Essentially a point source with a very large momentum in the tau direction and negligible momentum in the other three dimensions. This is in order to look at geodesic paths in the vicinity of a single very massive object where the other properties of the object can be ignored. In such a case, I can write in a very simple form consistent with standard notation:




where is the proper proportionality constant to yield the correct potential generated by the mass M.

Um, right, so that is just to say that the gravitational potential is proportional to the mass, and inversely proportional to the distance... Hmmm, shouldn't it be the square of the distance?

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This expression can be further simplified by changing to cylindrical coordinates (omit z as by symmetry we need only consider motion in the x-y plane).

Yup.

Quote

Thus it is that the differential of the path along which our metric is to be minimized is given by .

Struggling here a bit...

I get the squared and , but I can't figure out what that last term means exactly and where you got it...

I'll have a rest here...

-Anssi

[Doctordick]

AnssiH said:

Just in order to avoid making unfortunate false assumptions, that delta sign before the integration sign is simply to say "variation of..." right?

And the equation for "n", I see you got it simply from and established earlier in the OP.

Right!

AnssiH said:

I take it the dot on is just to denote a derivative?

The symbol, “”, is commonly read as “equivalent” which essentially means “identical” so, yes, that is exactly what it means.

AnssiH said:

Like if they are seen as two different curves, they exactly negate each others. Sooo, if that's at all right, I think I'm starting to get some sort of grasp of this idea... Not very crystal clear yet though.

This is a rather deep and complex mathematical relationship and you really shouldn’t expect see it as an obvious idea. That is the essence of the power of mathematics: once one has proved a mathematical relationship one knows it is true and seeing it as something other than an expressible mathematical relationship is unnecessary. In fact, trying to see it in terms of a mental picture can be dangerous because it is always possible that your mental picture may either add aspects not contained in the mathematical expression or omit aspects contained in the mathematical expression.

I am not saying that the mental model you have in mind is erroneous; but rather that, when it comes to advanced mathematics, one should get in the habit of seeing such things as “required mathematical relations”. As I have said in that post about rational thought, mathematics is a powerful tool capable of extending logical thought far beyond what we can comprehend in our minds eye. Mathematicians are a very dependable as a group; once they accept that something has been proved, you can pretty well accept it as fact. To prove all the stuff yourself would take a lifetime. What is important is that you understand the various symbols used and you seem to be picking up on that just fine.

AnssiH said:

Um, right, so that is just to say that the gravitational potential is proportional to the mass, and inversely proportional to the distance... Hmmm, shouldn't it be the square of the distance?

You are thinking of the gravitational force field (almost all “fields” referred to in physics are “vector force fields”). The idea of a potential field arises from the fact that most force fields (remember, force is a vector) can be expressed via the expression




Or, if the function G has only a radial dependence,




In other words, the force is proportional to the derivative of the potential and the derivative of is : i.e., if the force is proportional to , the potential must be proportional to .

AnssiH said:

Struggling here a bit...

I get the squared and , but I can't figure out what that last term means exactly and where you got it...

This is nothing more than dl expressed in radial coordinates (radial coordinates for the x and y directions). See
Polar Coordinates -- from Wolfram MathWorld

The differentials dr and d theta are always orthogonal to one another so the distance dl can easily be expressed in either set. The only difficulty to be handled is the fact that the actual distance moved when theta is changed is proportional to “r”. I think you can figure the thing out. If you still have problems, let me know and I will clear it up in detail.

Have fun -- Dick

[Bombadil]

Doctordick said:

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

I’m somewhat puzzled by how we can measure the distance that the light travels and conclude that the light in the hole is traveling slower then the light outside of the hole because I don’t see how we can establish the horizontal measures and know that the distance at the top will be the same as that in the hole. Unless we can say that any scaling can only have an effect in the vertical direction, And then don’t we also have to consider the clock to be small enough that the scaling over the hole clock is in the same direction. For what we seem to be interested in it seems that this effect can be easily ignored.

Also won’t the person in the gravity well see a clock outside of the gravity well run fast? As a result they will see the speed of light outside of the hole to be faster? Again I don’t see how we know that the distance between the mirrors is not being scaled unless we say that the result of gravity is the same as an object that is accelerating and an accelerating object can only have a vertical scaling and can‘t have a scaling along the horizontal direction due to it being independent of the vertical direction.

Doctordick said:

Secondly, the faces of the mirrors cannot be perpendicular to the floor of the elevator; it they were, the light would miss the original mirror on the return (the component of the elevator vertical velocity has changed, the vertical component of the light has not). It must now have an additional component of its velocity in the direction the elevator is accelerating. As a result, both mirrors must lean slightly towards the outside such as to assure that the reflected light always picks up that required change in velocity to keep up with the elevator. A little thought about the situation should be enough to prove to the reader that, at the instant immediately prior to reflection (as seen by the rest observer), the path of the light must be parallel to the floor of the elevator and immediately after reflection the path must be towards the position the other mirror will be in when the light gets to the other side.

But, is the angling of the mirrors a consequence of the acceleration or is it a consequence of the design of the clock. While a clock could be designed that had this built into it, do we need to build it into it? If it is a consequence of it accelerating then wouldn’t this have to be taken into effect when we try to measure the distance between the two mirrors. If this is not an effect of the acceleration then are you in fact saying that the physics are different for an accelerating object and so the original clock must be redesigned?

Doctordick said:

Thus it is that the clock we have proposed will run even slower than what was predicted a few paragraphs ago. The error is clearly a function of the physical size of the clock. This effect can be eliminated by defining our clock's size to be small enough to make the error negligible. I point this out so that we can discuss further ramifications of those instantaneous relativistic effects.

We know that these are different since one is a consequence of the size that we build the clock while the other one is a differential consequence of the Lorenz transformation and being in a gravity well. Both seem to be a consequence of the laws of physics not being the same in a gravity well. So are we in fact looking for a transformation that will result in the objects being correctly scaled so that the fundamental equation will be unchanged by being in a gravity well? But won’t this scaling result in the faces of the mirror being parallel in this frame and so it would make sense that the clock is the same as one not in the gravity well?