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Guest post: Carl Brannen, “Four Magnificent Papers by Authors Who Think I’m a Complete Idiot” *October 30, 2007*

*Posted by dorigo in Blogroll, books, internet, physics, science.*

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*Carl Brannen is an electronics engineer with a penchant for theoretical physics, and speculations on alternative theories to mainstream physics describing what our world is made of. He has a master’s degree in Math and in Physics, and is quite skilled with programming. He also owns * a blog* where he discusses topics of his liking, especially physics. Let us see what Carl has to tell…*

It’s quite an honor to be allowed to provide a guest post for Tommaso Dorigo. My working title for this post was “Four magnificent Papers by Authors Who Think I’m a Complete Idiot”. This was in preparation for my book, “A Complete Idiot’s Guide to Elementary Particle Physics.” The first was David Hestenes, of Geometric Algebra fame. The second was Lubos Motl, and his magnificent paper is the one that contains “tripled Pauli statistics”. The third was the somewhat obscure Mark Hadley, who has published a series of papers on a GR based theory of elementary particles. The fourth, David Bohm of Bohmian Mechanics, died too early to be provide an opinion on me, but one of his students says I’ve misunderstood Bohm’s easily understood opinions on relativity. I should link in an opinion of Lubos Motl, and a somewhat erroneous comment on Koide’s coincidences from Mark Hadley.

As if I were in a liferaft adrift on the ocean, I find myself wafting in the direction of the most recent wind. Louise Riofrio has kindly assembled together a series of posts, beginning with this one, discussing the evidence for a slow cosmological change in the speed of light, and promising a post for October, that is the direction I find that this post has written itself in.

The foundations of physics aren’t taught in grad school so much as picked up along the way, as one learns the techniques of calculation. Without classes devoted to the subject, it is easy to find that one has absorbed a certainty about the foundations which those who concentrate on the subject do not possess. This is a universal sociological fact pointed out by the author of Gravity’s Shadow. In bringing light to that peculiar form of blindness which is accompanied by belief that one already knows all that one can about the subject, one finds that the recipient has very little time, and less brain power available to analyze your effort. Accordingly, subjects that require time and effort to understand are off limits. Otherwise I’d be inclined to discuss Baylis’s paper on the problems that arise when one uses geometric algebra to geometrize spinors, and why I prefer density matrices.

So instead I am going to write a polemic against what I see as tendency of modern physics to misuse symmetry. To me, symmetry is a method that one uses to solve a set of equations. Symmetry cannot be an underlying principle in itself. Nor, as we show here, does a symmetry in observations necessarily imply the complete symmetry in the underlying physics.

While physics has been quite successful in abusing symmetry by worshipping it, that in itself is not evidence that symmetry is all there is to it. Nor is self-consistency and beauty proof against disproof. One remembers the elegant theory that the world is a flat plate, and rests on the back of a turtle. “And what is undernearth the turtle?” Another turtle of course. “And underneath that?” Yet another turtle. “And underneath the third turtle?” From there it’s turtles all the way down.

The most successful symmetry theory of physics is the special theory of relativity, from which we know that there can be no preferred reference frame. Newton out, Einstein in. Accordingly, let us derive the special theory of relativity from the very pracitical Newtonian engineering theory of Wave Motion in Elastic Solids, pp 274-281, by Karl F. Graff and kindly printed by Dover at a bargain price of $21.95.

We will assume that space-time is an isotropic elastic solid in 3 classical Newtonian dimensions. Such a media has a definite preferred reference frame, the media itself, and is the last thing one might suppose might lead to Lorentz symmetry, especially given the extreme efforts used to obtain Poincare invariance in the recent literature.

Let u(x,y,z;t) be the strain, that is, the deformation of the point (x,y,z) at time t. A strain in a material sets up a stress, that is, a force that reacts to undo the strain. For an elastic isotropic infinite solid media undergoing small linear deformations, there are two degrees of freedom available to characterize the media. We will use the Lamé parameters, lambda and mu. We will also assume a constant density, rho. Practical engineers need to apply external forces to media, but for our purposes we will leave these off and look only at waves propagating in the media itself. Then, from equation (5.1.3) of the above reference, we have the elastic equations of motion:

A very useful method of simplifying equations is to rewrite them in terms of a potential. For an arbitrary vector field like u, one requires a combination of scalar and vector potentials, the Helmholtz resolution. We write:

The scalar(vector) potential function is arbitrary in that one can add a constant (constant vector) to it without changing u. Of the two, the vector potential is even more arbitrary. Speaking in the physics language, we can make various gauge assumptions about it. The text assumes that the divergence of the vector potential is zero.

Substituting our potentials into the elastic equations of motion, we find that they are satisfied if:

The above equations are massless examples of the Klein Gordon equation, the relativistic generalization of Schroedinger’s equation. In a source-free region, the components of Maxwell’s equation satisfy the Klein Gordon equation, as do the components of the Dirac equation. A slight difference is that the wave speeds depend on the Lamé parameters and the density, rather than being the speed of light. Moreover, the wave speeds for the two wave types are different.

The upper, scalar potential, wave equation corresponds to longitudinal waves and has the faster wave speed. The lower, vector potential, wave equation has the slower wave speed and corresponds to transverse waves.

Suppose elastic creatures made of such a media wish to determine the preferred reference frame. They can find it by looking at the speed difference between the two types of elastic waves; the preferred reference frame is the only one where space appears isotropic. If, however, the creatures in the media are restricted to only measure one of the two types of waves, there will be only one wave speed, and, as with the situation with Maxwell’s equations, they will be unable to distinguish a preferred reference frame. An elastic creature might notice this, and become the elastic Einstein by promoting the idea that contrary to intuition, there can be no preferred reference frame.

The other day at the Crossroads Shopping Center chess club, a friend told me that he had great difficulty understanding how it could be that matter could have so much energy built into it that one could manipulate it into a nuclear explosion. I thought about it for a move or two. I told him that from the point of view of elementary particle physics it was not at all surprising. What was surprising is not that hydrogen bombs are so hot, but instead that the world as we see it, is so cold.

Our experiments in physics are restricted to particles with energies many orders of magnitude less than the Planck energy. The Planck energy is about the amount that a citizen of a developed country uses in electrical power in two weeks. In particle physics, it is difficult to explain why most particles don’t have this much energy. That’s right all that energy in just one electron. Maybe one explanation for the cold temperature of the world as we see it is that the energy per particle dropped due to inflation.

Among the particles that we CAN experiment with, it seems that Lorentz symmetry is exact. Is this because Lorentz symmetry is an exact principle of nature? Or is it because we do not have the resources to excite the higher velocity elastic deformations of space-time?

As far as a unified field theory goes, the elastic equations of motion discussed above are missing a few key details. The most obvious one is that elastic deformations are not quantized. They can come in any size and any energy. In quantizing them, it would be natural to find that the minimum excitation energy for a quanta is on the order of the Planck mass. In our very cold condition, we are interested only in excitations that have energies far far below the Planck mass. Among the elastic deformations, we can eliminate one of the two branches, say the faster one, by assuming that its quantum excitations are all of the order of the Planck energy, and hence are not observed in the cold universe that we see. The remaining excitations will all satisfy the same Klein Gordon equation, and so will satisfy Lorentz symmetry.

Of course there are several other defects in the elastic proposal. The number of deformations is far too few, so the known elementary particles would have to be composites. Accordingly, researchers pursuing these sorts of ideas work on preon theories. But that is another story. What we intend to point out by this post is that Lorentz symmetry is a very slippery rock to stand on, if one’s highest energy experimental measurements for its exactness are 10 orders of magnitude below the natural energy scale. One should not be too surprised if advances in physics are made by people who do not cripple themselves with a slavish devotion to symmetry principles “all the way down”.

Light is a vector, or transverse wave rather than a scalar, or longitudinal wave. In the Standard Model of elementary particles, only the Higgs is a scalar particle, but the Higgs has never been observed. All the observed elementary particles are, like light, vector particles. Well, technically the fermions, such as an electron or quark, are Dirac spinors. A spinor is more or less the square root of a vector. I reject Dirac spinors, preferring the density matrix form. The density matrix squares the spinors, again returning them, more or less, to vector form.

In addition to scalar particles being absent from the experimenter’s observations, research on the interaction between black holes and elementary particles suggest that scalar particles would be rather stranger than is currently expected. See the section on “tripled Pauli statistics” in the above linked paper by Lubos Motl.

The elastic equations of motion were defined under the assumption that the density of the media, rho, is constant. For the usual engineering problems, this is a good approximation for both longitudinal and transverse waves. But of the two types of waves, it is only the longitudinal (scalar) deformations that change the density of the media, the transverse waves preserve density to first order.

Let’s get back to the subject of the constancy of the speed of light. Since the big bang, the universe has considerably thinned out. If we were to naively model space-time as a classical isotropic media, this will result in a decrease in the density. But the spreading of an elastic media is also accompanied by changes to its elastic parameters. The speeds of the longitudinal and transverse (scalar and vector) wave speeds depend on these parameters as follows:

As the universe expands, presumably its lambda, mu, and rho change. And this returns us to Louise Riofrio’s equations for the changing speed of light.

## Comments

Sorry comments are closed for this entry

Fantastic! Thanks. I wouldn’t mind if Tommaso let you do another post on the

Papers by Authors who think you’re a complete idiot. Is there a simple way to clarify the connection between the last two relations (c_s and c_v) and a quantisation of these quantities via, eg., soliton solutions.“Suppose elastic creatures made of such a media wish to determine the preferred reference frame. They can find it by looking at the speed difference between the two types of elastic waves; the preferred reference frame is the only one where space appears isotropic. If, however, the creatures in the media are restricted to only measure one of the two types of waves, there will be only one wave speed, and, as with the situation with Maxwell’s equations, they will be unable to distinguish a preferred reference frame. An elastic creature might notice this, and become the elastic Einstein by promoting the idea that contrary to intuition, there can be no preferred reference frame.”

Yes, but not all elastic creatures need to stay in the media, these type of creatures can see both types of waves, right just before they have to do a rumpelstiltskin! Or is it Jack and the Beanstalk or a Schroedinger’s cat even. Then they have their own preferred reference frame. But of course such frames can be faked using tricks of entropy, but all said and done; such a balancing act is bound to fail! You could try bypassing a frame, but if there is nowhere to go even if there is somewhere for billions and billions of years the potential will collapse, but at least the creature can take a breath!

Carl Brannen said “… Light is a vector, or transverse wave rather than a scalar, or longitudinal wave …”.

I don’t think that the following comment contradicts some of Carl’s arguments about varying c, etc,

but

I do think that it might be useful in understanding light.

If you identify light with QED photons, then consider that Sakurai, in his book Advanced Quantum Mechanics (Benjamin 10th printing 1984, at pages 254-256), said:

“… take all four components of A_mu to be four independent Hermitian Klein-Gordon fields of zero mass … the elecron-electron intereaction is visualized as taking place via the exchange of four types of photons –

two transverse,

one longitudinal,

and one timelike.

… we obtain the covariant photon propagator.

… a virtual phton can be visualized as having four states of polarization.

On the other hand, we know that a real photon, or a free photon, has only two [transverse] states of polarization …[because]… the exchange of a timelike photon, tends to cancle … the exchange of a longitudinal photon; this cancellation becomes complete when the photon becomes real …

one may say that the photon always has four states of polarization … but that whenever it is real, the timelike photon and the longitudinal photon always give rise to contributions which are equal in magnitude but opposite in sign. …”.

That cancellation is known as the Gupta-Bleuler mechanism.

Roughly speaking, the real photons live in the far-field which is far (with respect to wavelength scale) from source and sink,

while virtual photons (with longitudinal (scalar) components) live in the near-field which is near (with respect to wavelength scale) to source and sink.

With quantum tunneling microscopes, etc, the longitudinal (scalar) photon is not “… absent from the experimenter’s observations …”.

As Paesler and Moyer said in their book Near-Field Optics (Wiley1996, at pages 7-8),

“… This [ optical microscope resolution ] limit has come to be known as … the far-field diffraction limit. …

near-field microscopes … surpass this limit and have been shown to provide resolution of structures down to the atomic scale …”.

A historical note:

“… Heisenberg’s oral examen rigorosum … July 23, 1923 … In Munich … Since Munich physics was split between Wien [experimenter] and Sommerfeld [theorist], both attended the orals and both had to agree on a single grade. … the 21-year old Werner … fell flat on his face when confronted by Wien … Heisenberg proved unable to derive the resolving power … even of the telescope or microscope … resulting in Heisenberg’s receiving the poor grade of III for physics, an average of Solmmerfeld’s I and Wien’s V … he had to accept the overall grade of III (the equivalent of a C) for his doctorate – the second-lowest passing grade …”. (quote from David C. Cassidy’s book Uncertainty (Freeman 1992 at pages 151-152))

It is interesting that the quantum theory to be developed by Heisenberg, applied to near-field quantum tunneling microscopy, would show that Wien himself did not understand the real answer to the resolving power question that he asked Heisenberg.

Tony Smith

Hi Carl,

I cannot deny that you may be correct.

However, if c is treated as a constant extrema (MAX) rather than as an ordinary constant then the difference is subtle.

Many things may to be able to perturb the direction or speed of light during a voyage form source to observer, such as magnetic fields, dust or water, but if c is an extema then c does not change for c defined in a vacuum.

Tommaso, nice choice of photo. The worried look is because I’m standing on a 50 foot tall ethanol distillation tower that sways horribly with every movement. I’m gradually getting accustomed to these things. It would be better if I didn’t know as much as I do about how engineers measure soil characteristics when specifying the depth of concrete that keeps these things upright.

Tony, perhaps the lesson from Heisenberg failing to understand optics is that to break new ground in physics, it helps to be a bit of an idiot. The more intelligent students were too careful. They color only inside the lines of their coloring books, and you can’t set up anything new that way. And, uh, in labeling the near field electromagnetic fields as “longitudinal waves”, could you tell us what their wave speed is?

Kea, the value of c_s and c_v are typically related by a factor of sqrt(3) to 1, around 1.7 or so. For example, see the relative speeds of the two types of earthquake waves. The factor of sqrt(3) also shows up in the Feynman checkerboard when it is generalized to 3 dimensions.

When you apply a force in one direction on a cube, it shortens the cube, but most materials will also get wider in the other dimensions so its cross section is not preserved. Neither is the volume preserved. In defining the ratio of stress to strain, there are two degrees of freedom, the force required to change the length, and the force required to change the volume. These are related to the Lame constants, and then to the two wave speeds.

But I don’t mean to put forth the proposal that space-time is an elastic solid in n-dimensions. I just meant to show that the Klein Gordon equation is ubiquitous in differential equations. I’m not proposing a theory, just taking pot shots at the one that people worship relativity just because a few experiments done at extraordinarly low energies happened to show that it was correct to a rather sloppy level of precision. (By contrast, Nature knows the mass of the electron to more than 100 decimal places accuracy.)

As you know, my real efforts are in waves that take values from a Clifford algebra, that is, the geometric algebra or geometric calculus. But I think the subject is too difficult to put into a guest post here. The example stuff is just trash talking.

On the other hand, if someone could do it, I’d love to see a way of getting from the elastic equations in n-dimensions to a Dirac operator. My intuition says that there is a way of doing it. That is, if you choose the right manifold, probably something very very simple like the circle R^4 x S1, and compute the elastic equation on it, you might be able to get a quantization effect from the feedback around the circle, and then be able to classify various sorts of vibrations according to a Clifford algebra.

At any given time I’ve got about a dozen things that I’m working on. I am very slow and it helps to have lots of things to work on because it gives my subconscious time to come up with ideas. So maybe one of these days I’ll have something cool to say; some way of deriving Clifford algebra from elasticity. But I haven’t put much effort into it, and it’s quite possible that David Hestenes has already done this in one of his $160 books that I’m too poor to buy.

Great post. Thank you. Maybe one day you can write about the original problem elementary particle physics is trying to answer.

Carl, about “… a way of getting from the elastic equations in n-dimensions to a Dirac operator …”,

here are some remarks about 3+1 dimensions:

In November 1870, Maxwell wrote a Manuscript on the Application of Quaternions to Electromagnetism, which is reprinted in Volume II of Maxwell’s Scientific Papers at pages 570-576. In it Maxwell says:

“… The invention of the Calculus of Quaternions by Hamilton is a step towards the knowledge of quantities related to space which can only be compared for its importance with the invention of triple coordinates by Descartes. The limited use which has up to the present time been made of Quaternions must be attributed partly to the repugnance of most mature minds to new methods involving the expenditure of thought …”.

Note that Quaternions are a 4-dimensional Clifford Algebra, Cl(2,R) the real Clifford algebra over a 2-dim vector space, with graded structure

1 2 1

At this time, Maxwell had a clear idea that waves should have Scalar and Vector parts, and used the following terms in his Quaternionic formulation of the equations of Electromagnetism:

Slope = what we call Grad ( represented by Nabla )

Convergence = what we call Div

Curl = what we call Curl

Concentration = what we call Laplacian

Since Maxwell then had both the concept of waves in an elastic medium and the concepts of Grad, Div, Curl, and Laplacian, he had everything you need to write the equations for Longitudinal Waves in an elastic medium as described, for example (as Jack Sarfatti pointed out some time ago) on pages 142-151 of Methods of Theoretical Physics by Morse and Feshbach (McGraw-Hill 1953). Explicit connection to the Dirac equation is discussed on pages 260-267.

Since (as shown in Morse and Feshbach) the Longitudinal Waves are faster than the Transverse Waves, and the Transverse Waves travel at the Speed of Light, the Longitudinal Waves are Superluminal if the Aether is a general elastic medium.

The question of the existence or non-existence of Longitudinal/Scalar Waves is then the question of whether or not the Aether, regarded as an elastic medium, is compressible:

If not, there are no Longitudinal/Scalar Waves.

If so, then there are Superluminal Longitudinal/Scalar Waves.

Tony Smith

PS – Further, about the “… wave speed ..[of]… the near field electromagnetic fields … “longitudinal waves” …”, William D. Walker in physics/0001063 (see also physics/0702166) has shown

“… that electromagnetic near-field waves and wave groups, generated by an oscillating electric dipole, propagate much faster than the speed of light as they are generated near the source, and reduce to the speed of light at about one wavelength from the source. …”.

PPS – The 2 volumes of Morse and Feshbach are wonderful, especially since they have neat little 3-dim stereograms (for example, Fig. 2.8 on page 145).

PPPS – As to generalizing such wave equation stuff to n dimensions, you need material such as found in the books

Clifford Algebras and Dirac Operators in Harmonic Analysis, by Gilbert and Murray (Cambridge 1991)

and

Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, by Hua (AMS, third printing, revised 1979, which is an English translation of the Russian version (Moskva 1959) which in turn is a translation of the original Chinese text published by Science Press, Beijing 1958).

Such material formed the basis of Armand Wyler’s calculation of the Fine Structure Constant around the 1970s, which was bitterly attacked by the USA physics establishment (see for example Physics Today issues of November 1971 and December 1989). It seems that much current work related to Wyler’s approach is blacklisted by the Cornell arXiv.

Tony, thanks for the references, I’ll look around for Morse and Feshbach.

Since I’m very interested in superluminal effects (and believe that we will eventually measure them), (you might see my earlier post on the speed of gravity waves), I was aware of Walker’s earlier paper, but I don’t think that either have anything to say about longitudinal wave speeds, as discussed in this post, because they are not plane waves and do not travel at any particular speed.

Furthermore, regarding his latest paper, I think it’s a done deal that Maxwell’s equations are fully compatible with relativity. When you boost E&M fields, it changes E to M and M to E, and it does it in a fully relativistic matter. In arguing otherwise, I think that Walker is putting a great deal of effort into something that will get nowhere. As far as E&M goes, Einstein won that round.

I don’t believe in relativity either, (and believe that gravity waves travel faster than c) but I find Walker’s arugment completely unconvincing. Maxwell’s equations are relativistically invariant. Period. Every physicist knows that static charge generates a field identical to Newton’s gravity, and that Newton’s gravity propagates with speed infinity and that Newton’s gravity is Galilean. So you can also put a Galilean interpretation on static E&M. It convinces no one.

Every now and then I get the urge to include a first slide at a physics conference that says “Einstine was Wrong”, preferably in a green crayon. So far I’ve suppressed it. I think that eventually we will find plane waves that travel faster than c, and that the evidence will be undeniable. Until then, I really don’t think that there is any way anyone can be convinced by these sorts of arguments. On the other hand, I didn’t think my guest post would convince anyone either, LOL.

Physicists have grown very complacent about these things because paradoxical theories are now the only theories left standing. I think this will eventually change, but it will have to change by experimental evidence disproving the standing theories. Universities drill into us: “The king is naked. So what? Shut up and calculate.”

And by the way, I think you should start a blog on WordPress.

Carl, before you dismiss the example of spacetime as an elastic solid as “trash talking”, you may want to (re)read

http://arxiv.org/abs/gr-qc/0510015

🙂

Hi all,

this is Carl’s post and so it is appropriate if he is the one answering comments, however I wanted to contribute a little:

Kea (and Carl): if Carl feels like to, he is welcome to make an update to the post above, and I will be glad to add it. For a new post I would say I’d wait a week or two.

Tony, your comments are always very appreciated. I knew that Heisenberg had a shaky career in his studies, but had never read the account you mention. Thanks!

Pioneer, maybe you should know that the real problem is asking the right question. Not trivial at all. So your “original problem” might have to be agreed upon before we start discussing how to solve it…

Cheers all,

T.

Carl, thanks for the interesting post, but I am not convinced. However, if anyone finds experimental evidence that you mention, I’ll definitely go back and re-read your posts in more detail. But don’t give up!

Best.

Guess Who, that is a nice link. The next time the spirit wills me to study this subject more, I’ll reread that more carefully and chase down some of the references. My main conern is elementary particles, though.

Changcho, I am not in any danger of giving up. Physics is a lot of hard work, and to devote oneself to it requires a certain degree of insanity, a presumption that one understands some part of the subject better than the billions who have studied it before. Believe me, I’ve got what it takes. Ask any of my friends, they will tell you that I’m nutty as a fruit cake.

As far as evidence, you can find little hints and dribbles, but you have to know where to look. The place to measure the speed of gravity waves is with gravity wave detectors. The way to compare their speed with that of light is to look for coincidences. If they find coincidences, then my ideas on gravity running faster than c are dead.

According to Marni, this issue came up at the most recent GR convention. I haven’t found the paper that came out, here is a slightly older one: So far, no coincidences. Unfortunately, it will be some time before they begin detecting gravity waves outside coincidence windows, given the nasty statistics of g-wave detectors at low amplitudes.

Hi Carl,

You know David Hestenes.

I have only read [or misread] some of his papers.

I wonder why you do not examine Maxwell’s equations from the IEEE phasor equation perspective?

Phasor equations certainly appear to be a form of Grassmann Algebra.

D Hestenes, ‘GRASSMANN’S VISION’ discusses the angle operator and the relation with Clifford Algebra.

I would agree that light in motion is a vector, but c is a scalar, perhaps because c is an extrema constant?

Since light is an electromagnetic wave, isn’t Hestenes’ idea of zitterbewegung or oscillatory motion of the helix a possible, if not optimal, wave form?

D Hestenes, ‘THE KINEMATIC ORIGIN OF COMPLEX WAVE FUNCTIONS’

Recall that Paul AM Dirac had no degree in physics, but a BS in electrical engineering and PhD in mathematics.

http://nobelprize.org/nobel_prizes/physics/laureates/1933/dirac-bio.html

…the real problem is asking the right question.I agree, you are right. I was trying to ask if Carl Brannen could also include experimental basis of his more theoretical considerations. For instance, in astronomy, we have observations and we predict the positions. I understand how astronomical database is compiled. But in the field of elementary particles it is not clear to me (because of my ignorance) where the database comes from. For instance in your recent post you were talking about pseudo experiments used now routinely by physicists. That may be pseudo in name only, but still, I don’t understand what is measured or how. In any case, your article about the color of astronomical objects was super. Thanks.

Pioneer1;

If I can answer for Tommaso my own version. Unfortunately, very few people, if any, understand quantum measurement. It’s debated at great length in obscure and unimportant papers by a small number of authors. The rest of the physics industry makes use of QM as engineers are supposed to use engineering equations.

Basically, the experimental particle people look for particle tracks. From that they work up probabilities for certain things happening.

A short way of describing “Hilbert Space” is that physicists assume that you can describe the inital conditions of an experiment as a linear collection of parameters. And you describe a possible final condition as another linear collection of parameters. QM gives a way of computing the probability that you get the final condition if you begin with the initial condition. Particle experiments set up the inital condition and measure the final conditions. The result of the experiment is a table of “transition probabilities”, the probability of getting the final condition if you set up the initial condition.

This is a rather general way of describing particle experiments and it works rather more generally than the way that initial and final conditions are actually specified in practice. That is, the universe is simpler than what one would need if the Nature could put together any sort of transititon probabilities whatsoever.

A more elegant way of describing this is also my favorite introduction to QM, a book on quantum measurement by Julian Schwinger called “measurement algebra“. It’s rather obscure, but I think elegant and gets to the heart of QM better than any other book. Like with any QM book, it gets into a lot of math.

For an introduction to QM that requires no math at all, but gives a good flavor of the theory (but not the experiment), see Feynman’s book, “QED: the strange theory of light and matter”, or something close to that. The advantage that Schwinger’s book has over Feynman’s is that Schwinger’s is more closely connected to the experimental apparatus that is actually used in particle experiments. Schwinger’s model quantum apparatus is the Stern-Gerlach experiment. He treats everything as a sort of S-G experiment.

It should be noted that it is easy to exaggerate the importance of quantum measurement; it was already an issue in divulgative texts after the experiments of Alain Aspect , and now it is even more exaggerated with the advent of quantum computing. But more people in more areas does not worry about a single electron trajectory, but about a single property of a huge collective. For instance the electrical current coming from a solar cell, or the light coming from a sodium or a mercurium gas tube. In such case, safety theorems are in action and we can observe quantum effects but classical measurement process.

For particle physics I agree it is bit more frustrating because a single particle, or a few ones, emerge from the collision. I’d say that the measure is done classical by the interaction with a huge quantity of particles in the detector; this is safe to claim for instance in calorimeters, and more or less the same in he later. But when the detector keeps tracking the trajectory of a single particle, it still sort of amazing; I guess someone somewhere in these obscure journals have put the numbers and assured deltax deltap is still safe over h.

The simplest quantum measurement is that of spin and the Stern-Gerlach apparatus. This is discussed in the Schwinger book and articles. You arrange for a region to have a sort of a pointy magnetic field. Particles (say silver atoms) with spin get deflected by the magnetic field in different ways.

You put a piece of photographic film as a target. When a silver atom hits the photographic film, it leaves a spot. Each atom that comes through makes a new spot (or hits an old one). Eventually you get a pattern. The pattern turns out to show that “spin is quantized”, which means that there are separate spots instead of one big wide spot.

Check out this summary report at:

USA Today, ‘New spin on how stars are born’

By Ker Than, SPACE.com

Copyright 2007, SPACE.com Inc. ALL RIGHTS RESERVED.

http://www.usatoday.com/tech/

science/space/2007-11-01-star-birth_N.htm

Original Letter:

Nature 450, 71-73 (1 November 2007) | doi:10.1038/nature06220; Received 8 June 2007;

Accepted 4 September 2007

http://www.nature.com/nature/

journal/v450/n7166/abs/

nature06220.html

Antonio Chrysostomou, Philip W Lucas and James H Hough,

‘Circular polarimetry reveals helical magnetic fields in the young stellar object HH 135–136’.

If this Nature Letter is confirmed, then David Hestenes may have great insight into the value of the helix.

Doug referred to “… helical magnetic fields …” in young stars.

According to an ESO web page at http://www.eso.org/public/outreach/press-rel/pr-2006/phot-45-06.html

“… Spatial, three-dimensional distribution of galaxies in a slice of the Universe as it was 7 billion years ago, based on the VVDS study … the galaxy distribution – the ‘building blocks’ of the large scale structure – takes the shape of a helix at this primordial epoch …”,

so

it may be that helical magnetic fields might be involved in galaxy formation as well as in star formation.

Further, Battaner et al, in in astro-ph/9801276, astro-ph/9802009, and astro-ph/9911423, suggest that the simplest network pattern for distribution of superclusters of galaxies that is compatible with magnetic field constraints is made up of octahedra contacting at their vertexes,

which is related to a tiling of 3-dim space by cuboctahedra and octahedra,

and also to the heptaverton of Arthur Young( http://www.arthuryoung.com/ ) in his book The Reflexive Universe (Robert Briggs Associates 1978), and octonionic structures of Onar Aam.

Tony Smith

If, however, the creatures in the media are restricted to only measure one of the two types of waves, there will be only one wave speed, and, as with the situation with Maxwell’s equations, they will be unable to distinguish a preferred reference frame.Michelson-Morley doesn’t work in an isotropic, elastic, infinite medium? Why on earth were they doing the experiment then?

If necessary, let us say somehow lambda = – mu.

I think there is a mistake, and that is to take a wave equation derived in the rest frame of the isotropic elastic solid, note its Klein-Gordon form, and **assume** that the derivation of the wave equation works in a moving frame. Elastic media people do have a preferred frame and that is one in which the wave equation holds.

carlbrannen wrote:

The rest of the physics industry makes use of QM as engineers are supposed to use engineering equations.In this sense does Quantum Mechanics qualify as a data based model and not a rule based model? Or is it some other type of model? I was trying to classify model types used in physics in this post.

If you are saying that physicists working, for instance, in LHC, use Quantum Mechanics only in an engineering sense, I have a question about that.

I was thinking about this after reading Monica Dunford’s recent post in US/LHC blogs where she writes about observing cosmic ray events. She writes that “the muon track can be clearly seen, entering at the top of the calorimeter, exiting through the bottom.” I don’t see that in the figure, but that’s not important. She then gives a plot showing “the photomultiplier tube pulse shapes for one of the yellow cells. The blue points are the data, the red curve is the reconstructed pulse shape.”

To me the data points look incredibly clean for a real experiment. How did she really obtain that data? Is there anything more involved in this measurement than electonics?

Naively speaking, LHC appears to work on the same principle as an oscilloscope used to measure sine waves. To me it seems that the experimental set up involves only practical electronics. How are the big theoretical frameworks such as QM involved in the measurement? Does Quantum Mechanics enter here even in an engineering context? How do you explain the relationship between theory and experiment in big detectors? Is there a disconnect between what is measured and the theoretical frameworks? Where does observation start and where does computer simulation end? How can we know this?

Basically, the experimental particle people look for particle tracks. From that they work up probabilities for certain things happening.Is this what Monica Dunford was looking for, particle tracks of muons? And then she worked probabilities to obtain the plot in her blog?

. . . Particle experiments set up the inital condition and measure the final conditions . . .Great explanation, thanks. Are physicists really using these theoretical methodologies or, in practice, are they using computer code and simulation?

Something similar happens in astronomy. NASA uses their practical Orbit Determination Program, which is devoid of any theory, to determine orbits but if you read their websites, they claim to use Newtonian mechanics to place satellite into orbits. Is something like this happenning in detectors experiments?

. . . That is, the universe is simpler than what one would need if the Nature could put together any sort of transititon probabilities whatsoever.Sorry, I didn’t understand what you mean here.

He treats everything as a sort of S-G experiment.My Stern-Gerlach is limited to Wikipedia. I hope some day when I am done with the Cavendish epxeriment 🙂 I’ll be able to read about this stuff. So please include some initial historical and experimental motivation in your book. Personally I understand things much better if there is a historical continuity.

Thanks again for the nice post and the comment.

A null result in the Michelson-Morley experiment eliminates a lot of possibilities, the experiment does “work”. There were still some other things on the table, like “dragged aether”, that were eliminated in other, later, experiments. My point is that there are other possible explanations, in the context of classical mechanics and non relativistic quantum mechanics, that are compatible with an undetected preferred reference frame.

If you read the usual introductions to relativity, it is easy to get the impression that Michelson-Morley was a definitive experiment with only one possible interpretation. My point is that one can also interpret the null result as indicating that the longitudinal waves are of an energy too high for us to excite. This possibility isn’t discussed in the books partly because quantum mechanics (and the notion of frozen degrees of freedom at low temperatures) was not well developed at the time that the Michelson-Morley experiment was discussed. And it is partly not discussed because it would require an excursion into elastic equations of motion. And it is not discussed partly because it would unnecessarily confuse students, very few of whom need to understand the true complexity of the foundations of physics.

“I think there is a mistake, and that is to take a wave equation derived in the rest frame of the isotropic elastic solid, note its Klein-Gordon form, and **assume** that the derivation of the wave equation works in a moving frame.”

No such assumption need be made. It’s well known that the Klein Gordon equation is Lorentz invariant. So as soon as you derive that equation, no matter whether you derived it from a model of an elastic solid or a universe made from interacting wind-up cat toys, you automatically have that you cannot use that equation to define a preferred reference frame. Even though the equation came from an assumption of a preferred reference frame, the reference frame drops out of the results.

This is not true with certain other wave equations. In particular, it is not true for the elastic wave equations themselves. To rephrase the argument of the blog post, it is only when you separate the elastic wave equations into longitudinal and transverse parts that it reduces to two Klein Gordon equations. The assumption of “no detectable preferred reference frame” comes from assuming that one of those Klein Gordon equations is inaccessible because of energy concerns.

Carl says, about Michelson-Morley, “… one can also interpret the null result as indicating that the longitudinal waves are of an energy too high for us to excite … it is not discussed partly because it would unnecessarily confuse students, very few of whom need to understand the true complexity of the foundations of physics …”.

At the risk of causing confusion by asking unconventional conjectural questions:

If the Higgs is really a Tquark condensate (something that can be explored at LHC and any successors that might be built),

then

could it be that the Higgs VEV of about 250 GeV could be regarded as a measure of elasticity/compressibility of an Aether with respect to far-field longitudinal wave propagation ?

In other words:

If mass comes from the Higgs,

and

if our sub-250 GeV experiments only see far-field longitudinal components in massive particles,

then

could it be that experiments over 250 GeV might see a regime in which the Higgs (as a Tquark condensate) breaks down

and

in which no particles get Higgs mass (and therefore are all massless),

so

might it not be interesting to try to see by high-energy experiment whether Aether elasticity might be observable in terms of far-field longitudinal components of some massless particles ?

In still other words:

Could Carl’s Elastic Aether point of view be useful with respect to designing experiments at the LHC and possible future colliders ?

OK, you might say it is an unlikely conjecture to explore, but it seems to me to be to be at least as likely an avenue of research as the widely discussed searches for conventional supersymmetry or for extra dimensions of conventional superstring theory,

with respect to which theorists get funding and jobs and collider experimenters agree to set up experiments.

Tony Smith

Hi all,

first of all, thanks to Carl who has served well as a guest here – one of the requests in the stipulation is that the guest has to take care of answering comments. Please continue doing so, Carl!

Second, I apologize for having been away for a few days of vacation. I wish to answer to a few things here myself though…

Pioneer1 #14: what is measured in particle physics experiments are, ultimately, positions in the detector where a particle has left ionization. From these “hits”, either tracks of charged particles are reconstructed, or energy deposits are sized up. In both cases, one ends up with a list of particle trajectories and four-momenta, which one proceeds to interpret as the effect of a particular quantum process, the hard scattering of the colliding bodies. Pseudoexperiments may start from distribution of energy or momentum of the resulting final state particles, or just as well from the variety of hard subprocesses, each weighted by its probability. In both cases, quantum mechanics has little impact, because it is already encoded in a model of the observable effects (say, a 50 GeV gluon produces a jet of particles: one does not care any more of the mechanism by which color recombines to produce the various particle species, but only cares to reproduce the correct spectrum of possible momenta).

And #21: yes, cosmic rays are clean things to study – they leave only a few hits in the detector, from which reconstructing what has happened is really as straightforward as finding rat c**p in the living room and a hole in a pack of spaghetti in the kitchen. Real proton-proton collisions when LHC starts up will be much, much tougher – imagine having to find the pack of spaghetti inside a silos full of cow d**g….

Tony, your comment above is really intriguing, but I am unqualified to elaborate on it. I only venture to say that “Elastic Aether point of view be useful with respect to designing experiments at the LHC and possible future colliders ?” sounds far fetched to me, not too differently from the infamous paper on checking reverse causality by turning LHC off on the flip of a coin 🙂 Anyway, at very high energy as you know all particle masses indeed stop playing a role. Whether that is connected to your speculation I am however unable to say.

Cheers all,

T.

Wow, Carl, your confusion is profound, and unfortunately it seems you’ve managed to confuse more people here. Arun spotted your mistake — your equations for the elastic waves only holds in the preferred frame. In other words, these equations are not invariant under a transform to a moving frame. The whole point of relativity was originally about that Maxwell’s equations, properly written, are invariant under the Lorentz boost, while elastic wave equations are not.

Zheng-zheng, you seem to be missing the point. “your equations for the elastic waves only holds in the preferred frame”

Yes, the elastic equation does imply a preferred reference frame. We all know this already. What I showed is that the elastic equation splits into two Klein Gordon equations which, taken individually, are relativistically invariant with different “speeds of ligth”. If only one of these branches is observable, then, as with any other Klein Gordon equation, it will appear to have no preferred reference frame. That is, the preferred reference frame melts away when you drop longitudinal (or transverse) waves and consider only the other sort.

Taken together, the two Klein Gordon equations do imply a preferred reference frame because they do not agree on the speed of light. But taken individually, they are indistinguishable from the usual massless Klein Gordon equation, and therefore are relativistically invariant. They’re the same equation.

The math is very clear and simple, and I certainly cannot use it to “confuse more people” around here since most of them have PhDs in physics and understand these equations and principles fairly well. If you disagree with the majority, then you should consider the possibility that you haven’t thought about the subject as much as they do and your understanding is defective. Try rereading the post and commentary a few times and you might understand it better.

Carl,

In acoustic media, where Lame mu is zero, only one mode exists, i.e. “only one of these branches is observable” in your words — are you saying then sound waves in air travel at constant speed c in all directions regardless of the frame of reference?

You have to work out how the potentials transform under a change of coordinates by first figuring out how the real observables, the particle velocities, transform. In EM, the E and B fields conspire to transform in such a way that the equations remain invariant under Lorentz boosts. This was first verified by experiments. From there, you then work out how the potentials transform, and that leads to more compact formulations but no new physics.

That the standard wave equation looks like the Klein Gordon Eq in some particular frame is not remarkable. Relativity is about how the physical observables transform to keep the equations invariant. In EM when this was first discovered, it was so remarkable that people were very confused and thought the elastic medium that carried light must have extraordinary properties unlike any known classes of elasticity. Relativity is the realization that this strange brand of elasticity is not elasticity at all but in fact the transform property of space-time itself.

If you identify the observable fields as the motion of an underlying medium, they will never transform in such a way that your equations remain invariant. The potentials must respect this lack of invariance because their only purpose is calculational and they must reproduce the field equations for the observables or they would be useless.

The uncritical responses to your stuff are disappointing.

Zheng, very good, I see what you are thinking. This is the heart of your comment:

In acoustic media, where Lame mu is zero, only one mode exists, i.e. “only one of these branches is observable” in your words — are you saying then sound waves in air travel at constant speed c in all directions regardless of the frame of reference?Yes! That is exactly what I am saying! When you transform to a moving reference frame in a media governed by the Klein Gordon equation you will have to adjust time the same way you learned to do it with relativity. Same Klein Gordon equation, same Lorentz symmetry, same physical results.

Let me put it in a more familiar way. Suppose you are upwind or downwind of a friend. You want to synchronize your watches by shouting the time back and forth to each other. You have no way of measuring wind speed and correcting for the time it takes for your voice to go each way, because you do not have access to experiments that define the “elastic media”; all you have are the waves.

What happens? Yes, the same thing as happens in the usual relativistic case you’ve already discussed in class and for precisely the same reasons.

Person A shouts to B and starts counting seconds. When B hears A he shouts back. A stops counting when he hears the shout back. In the rest frame of A and B, the distance between them is half the count, in units of “sound-seconds” (think “light-years”).

Go ahead and work it out algebraically. You will find that as the wind speed increases, your estimate of the distance between you and your friend also increases with the same formulas as that of relativity. Time dilation, length contraction, they all work the same way, but with the speed of sound replacing c.

You have to work out how the potentials transform under a change of coordinates by first figuring out how the real observables, the particle velocities, transform.As soon as you start talking about particles and velocities, you are no longer talking about elastic waves, and yes, you have a preferred reference frame; that is, that defined by the average velocity. But that’s not a part of the Klein Gordon equation. And for that matter, we cannot measure the velocity of spacetime, so we are in the same boat as the friends who cannot measure the velocity of the wind.

Carl,

You’ve just disproofed your own theory: if sound behaved like light, the shouters in the wind would have measured the same constant echo delay regardless how the wind blowed. Theshouting experiment is really a version of the M-M experiment discussed above. Do it with light, no seasonal variation. Do it with sound, there is. End of discussion.

Zheng-zheng, you’re missing the point: if the guys shouting, and their clocks, and everything else they can observe, if all that is made of quasiparticles emerging from an underlying “solid state” style theory with a preferred frame, then they have no observable “anchor” in that frame, and as far as they can see their world is fundamentally relativistic.

If you want to see a simple example of the emergence of a relativistic equation from a nonrelativistic system, see Chapter V.5 in Zee’s “QFT in a nutshell”:

http://www.amazon.com/Quantum-Field-Theory-Nutshell-Zee/dp/0691010196/ref=pd_bbs_sr_1/104-8886897-1217564?ie=UTF8&s=books&qid=1194407626&sr=1-1

There is a guy who’s written tons about Lorentz invariance, gravity and gauge symmetries emerging from a less symmetric underlying thoery, Grigori Volovik

http://ltl.tkk.fi/personnel/THEORY/volovik.html

You can download the final draft of his book, “The universe in a helium droplet”

http://www.amazon.com/Universe-Droplet-International-Monographs-Physics/dp/0198507828

from his site, or for a quicker read try this review:

http://arxiv.org/abs/gr-qc/0005091

Zheng-zheng,

If sound behaved like light, the shouters in the wind would have measured the same constant echo delay regardless how the wind blowedWhen you say this, you’re failing to apply the logic of relativity to the shouters. The shouters only way of measuring distance is by the passage of time between signals. When the wind changes direction, the shouters are convinced that the distance between them has changed.

On the other hand, if you use your knowledge of some other media, say electromagnetics, to measure the distance between the shouters and find that it did not, in fact, change, then you are using information from more than one branch of the elastic wave solutions.

Soon your arguments will become slightly more sophisticated and you will begin talking about watches and parallel transport and all that. What you need to take into account is that in sound world, clocks are made from sound waves. And, sure enough, their ticking is slowed when their velocity is increased through the mechanism of time dilation.

Arguing my side on this is very simple because like any PhD in physics, I know special relativity inside and out. I need simply take the appropriate argument from relativity and replace light with sound. If you name some complicated thought experiment that violates “sound relativity”, I need only find the place where you used some other mechanism than sound to make a measurement. It gets more interesting when you start talking about energy, LOL.

HI Carl: I appreciate your mention and your replies to the postings. A discussion like this erupted on Asymptotia when I was limited to a very primitive computer atop Kilauea.

People brought up the most elaborate mathematical “objections” which boiled down to one thing: It is impossible to “prove” experimentally that c is constant, but one can invent mathematics that treat it as constant just as Earth is centre of the Universe.

Talking to a brick wall with a PhD hanging from it is still talking to a brick wall.

Lol! Louise, your sentence above is hereby elected as the Say of the Week!

Cheers,

T.

Any new idea should and does receive a very critical response. Math is complicated, physics is complicated, philosophy is complicated, nature is complicated, and these are things that man can just barely understand to a small degree.

I don’t think we should be surprised or disheartened when our new theory (a) is given a tough hearing, (b) is ignored, or (c) turns out to be completey wrong. Even the comments of those completely against a new idea are useful in understanding that idea. And even a completely wrong idea probably helps to understand the correct ones.

And being human, we really cannot ever know what turns out to be the right idea. So for me, no brick walls. Just keep on moving forward.

I got reminded of all this the other day. In pursuing a flat space particle theory version of general relativity, I eventually came to the conclusion that the “right” way to slap coordinates on the Schwarzschild metric was the way Painleve did. So I put the GR equations of motion in Painleve coordinates into Newtonian form (so gravity would be a force like any particle force), and wrote a simulation, which html page includes a copy of the exact relativistic equations of motion around the black hole (no approximations, these are the exact post Newtonian correction to black hole orbits) in Painleve coordinates, but with proper time eliminated.

I got an email a few days ago from the a professor at MIT who said that he loved the simulation (which automatically steps through a series of comparisons between Newtonian gravity, Schwarzschild and Painleve coordiantes, but can be changed by the student to simulate other situations) would use my simulation the next time he taught GR.

So I got to thinking about Painleve coordinates again, which are fairly obscure. There is one paper on arXiv that stands out, Hamilton and Lisle, “River Model of Black Holes”. As far as I can tell, this excellent paper still hasn’t been published.

The fact that the paper still hasn’t been published reminded me of the amazing amount of crap I got when I used Physics Forums’ LaTeX to do the messy calculations for putting Painleve orbits into Newtonian form.

Carl,

“Even the comments of those completely against a new idea are useful in understanding that idea. And even a completely wrong idea probably helps to understand the correct ones.”

This is so very true, and it is at the heart of the reason why I try to host exotic ideas just as well as mainstream ones here.

Thank you for your wisdom, and keep thinking!

Cheers,

T.

Carl is right, but I still think that’s a

great quoteby Louise! Lol.Ok, so Carl your point boils down to that sound appears relativitic to other sound of the same mode. I have no objection to that. One would certainly perceive this illusion when sound is all there is. Introducing interactions and massive particles under this picture without destroying this illusion would be tricky, but I guess string theorists have shown that with enough math, you can always patch up a model. However, there is zero motivation to replace space-time with a medium made up of more fundamental stuff — that would just be the first turtle in the “turtles all the way down” model of the vacuum.

Zheng,

The motivation to replace space-time with something more fundamental is to reduce the number of arbitrary constants in the Standard Model. If one can posit a more fundamental object, one might hope that one can derive relationships between some of those arbitrary constants.

Of those arbitrary constants I’m (more or less) famous for finding a relationship between the neutrinos masses. My formula now has 4 citations in the peer reviewed literature and more in arXiv and elsewhere.

In looking for these sorts of things, the assumption that it is “turtles all the way down”, that is, the assumption that relativity is perfect all the way down, is very restrictive. This is due to a series of theorems started by Coleman and Mandula that basically say that you can’t mix internal symmetries with external symmetries, except trivially. By going outside that restriction, I can find solutions using the easy to use and familiar QM / QFT.

In an ideal “Einstein” world, we would have all attributes of elementary particles depend on geometric calculations. But the Coleman Mandula theorems say that under the assumption of perfect Lorentz symmetry, this cannot be done.

In short, it’s nothing personal. To save Einstein’s vision of a geometric basis for elementary particles, I have to stick a knife in Einstein’s special theory of relativity. Otherwise physics will be stuck chasing after mathematical will o’ the wisps, like string theory and link quantum gravity.

Nor is this a problem just with me. Garrett Lisi just put out a paper with a unified field theory with the same problem, he has to violate the Coleman Mandula theorem. It’s all over the web right now, and you can read what they (the old stick in the muds that insist on turtles all the way down) have to say about it.

[…] died a few years ago and so cannot be placed on my list of brilliant physicists who think I’m a complete idiot. And in addition he isn’t here to comment on the concept of modeling fermion bound states […]

Hi there everyone, great thread.

Regarding waves in mediums, would they be physical or etheric, the definite authority (?!) in this matter on the web right now is Gabriel Lafreniere @ glafreniere.com, more precisely

http://glafreniere.com/sa_Doppler.htm

Basically, SR is simply what he rightfully dubbed the Lorentz Doppler effect in which the frequency of the oscillator gets contracted along sqrt((C-V)/C)=g, to keep the transverse length unchanged. He wrote a small program, near the bottom of the page, called Doppler_Voigt_transformations.exe in FreeBASIC, which clearly shows how it works. The source is provided, its simplicity is baffling.

This approach isn’t widely known, and yet yields a concrete, mechanical understanding, e.g. clearly shows how kinetic energy is embodied as mass: sa_active.htm

About the speed of propagation of Gravity/Forces, one of the best (simple, clean, to the point, hinting at the relevance Davis Third-Order Mechanics, but against faster than light force transmission, sorry Carl :p) written account available is Jaume GINÉ’s on the precession of Mercury’s perihelion.

As a first side note, Gabriel Lafrenière writes that higher density at the core of a free-standing oscillator akin to his spherical standing wave electron (he refined this concept recently) imply faster speed, and an amplification/lensing effect. Look at the extended picture to see quasi-flat wavefronts emerge in the distance, propagating at C.

As a second side note, thinking in terms of E and B is outdated and flawed: the magnetic field is spherical, even though around a closed path of integration the longitudinal part cancels. Look at

Newelectromagnetism.com for clean and easy to grasp point to point force equations. From Assis’ “Weber’s electrodynamics” book, the second Distinti equation contains (p. 57) the relative radial velocity “overdot r”=v_target-v_source scalarprod hat r. College level electrodynamics that are in fact high-school level when properly debunked are indeed a fascinating subject, don’t you think? =) The sad part is that it should have been done a long time, i.e. 150+ years ago!

Turns out that the author of the “you’re never too old” blog wrote a thesis on gravity wave detection that happens to include the above calculations (but in reference to the vibrations of the gravity detector). See equations 3.4-3.7 of his PhD thesis (1997).

Some updates on the speed of gravity. Two arXiv papers talk about recent evidence that the speed of gravity is not c:

A Decisive test to confirm or rule out the existence of dark matter emulators using gravitational wave observations by E. O. Kahya, January 2008. And Effect of the Earth’s Time-Retarded Gravitational Field on Spacecraft Flybys by J. C. Hafele, April 2009. Both these papers suppose that the speed of light is different from c and use this to explain a couple of astrophysics puzzles.

The other way of looking at the problem is to suppose that gravity is carried by a flux of particles similar to how the photon carries electromagnetism. Then deviations form 1/r^2 force imply that the flux is interacting with itself to make more (or perhaps less though this would be difficult to believe). This sort of analysis can be done without specifying the speed of the flux. And this was the topic of a short essay I turned into the annual gravity essay contest.