The Walker-Dual experiment; phase, group, and signal velocities; superluminal phase velocity and superluminal gravitational propagation have all cropped up in a couple different threads. I'd like to draw the discussion together under one heading, and invite Tim Thompson, Agora Basta, David Hall, JS Princeton, Wiley, and all the others who have engaged the topic to explore it some more here.

The story so far:

Tom Van Flandern contends that the 1997 Walker-Dual experiment is evidence for superluminal propagation of gravity. The W-D paper did demonstrate velocities for electrical and gravitational phase propagation in excess of the speed of light (references and links at the bottom of this post). This was predicted by Feynman's 1963 calculations, and does not contradict conventional theory. Walker and Dual also tentatively reported superluminal group electrical wave propagation, and wrote that this may violate causality. However, conventional explanations note that both phase and group velocity may exceed c without violating special relativity or causality--see the reference to Mathpages, provided by Wiley, below.

In 2000 William Walker wrote a paper describing superluminal electrical wave propagation in the near-field realm, ie one wavelength or less from a dipole emitter. He then proposed that magnetic and gravitational fields would behave similarly. Near the end of his paper he made some of the same claims as Van Flandern:

"Light from the sun is not observed to be collinear with the sun's gravitational force. Astronomical studies indicate that the earth's acceleration is toward the gravitational center of the sun even though it is moving around the sun, whereas light from the sun is observed to be aberated (sic). If the gravitational force between the sun and the earth were aberated then gravitational forces tangent to the earth's orbit would result, causing the earth to spiral away from the sun, due to conservation of angular momentum. Current astronomical observations estimate the phase speed of gravity to be greater than 2X10¹⁰ c. Arguments against the superluminal interpretation have appeared in the literature [9, 10]." (emphasis added)

(The last sentence references a paper by Steve Carlip, and one by Ibison, Puthoff, and Little--see links below.)

And onward to new stuff...

I emailed Thomas Chen about the following enigmatic paragraph in the Walker-Dual paper, which seems to be the only place they challenge conventional physics:

"The analysis of the group speed of a longitudinally oscillating electrical field is currently inconclusive. The group speed is commonly thought to be equal to the speed of light, but preliminary analysis indicates that the group speed is much faster than light which is not thought possible due to causality violation."

Dr. Chen kindly replied as follows (quoted by permission):

"The sentence you ask me to comment on is correct in saying that the group speed is larger than the speed of light, if one takes the definition of group speed literally. But on the other hand, it is wrong by concluding a violation of causality."

"Please note that a superluminal group speed does not imply a superluminal transmission of information. The concept of group speed is artificial; by definition, it should not be applied to the 'near zone', but only in a spatial region where there is wave propagation in the usual sense. Otherwise, an artificial definition is being taken too seriously, and ad absurdum."

"My take on unravelling the logical structure of this seeming paradox is as follows."

"In the near zone, there is a superposition of a lot of outgoing and incoming waves, all propagating with speed of light. Due to these superpositions, it is not possible to observe an individual wave. If the source is modulated, it is in all real experiments done so with a frequency so small that the particles in motion (mass in GR, charges in electromagnetic radiation) exhibit a velocity much lower than the speed of light."

"Since the outgoing and incoming waves are all modulated by a low frequency signal, but propagate extremely fast, all one can measure is the average phase difference between a massive number of in- and outgoing waves. The averaged phase differences amount to the modulation signal of the source."

"Therefore, there is at this stage no clash with causality, because all waves in question do indeed travel with the speed of light."

"Now, the key question is whether the modulation signal travels faster than the speed of light. This is a little more tricky. The simplest modulation signal is a sine curve. One will see that the phase difference expected to be measured in a distance away from the source suggests superluminal propagation, as proposed by Walker and Dual."

"The quintessence of the problem, however, is that there is no way one can transmit information by single sine modulation. Any true information content must possess some decently complicated Fourier spectrum. We should thus in fact define 'information' as a signal with some decently elaborate Fourier spectrum."

"Hence, if we now assume a signal with some fairly sophisticated Fourier spectrum (think of a radio broadcast), the question becomes: Does that information travel superluminally in the near zone?"

"The answer is: It is never possible to tell."

"Why? It's because in order to decode the information, one must Fourier transform the measured signal. But in order to do that, one must require a certain amount of time, it's not possible to perform a Fourier transform by measuring a signal for only one moment."

"So, it is necessary to 'listen to the signal' for an extended time interval before one can determine its Fourier decomposition, and thus decipher the information it contains (the 'information' is the same as the Fourier spectrum of the modulation, and as I said above, a single spike in the spectrum doesn't count as information). How long does one have to listen? The time necessary for doing the Fourier transform is approximately the inverse of the 'typical
frequency' of the information signal."

"But this implies that it is not possible to measure how fast the signal propagated if one is closer to the source than a wavelength of the TYPICAL MODULATION FREQUENCY (not the wavelength corresponding to the speed of light!!!). However, the wavelength of the modulation frequency is much larger than the diameter of the near zone that corresponds to the speed of light. So in a distance of some wavelengths (corresponding to the modulation frequency) away from the source, one is already in the radiation zone with respect to the speed of light, and one will measure a propagation velocity of the signal in agreement with the speed of light."

"This is in fact a hidden form of a Heisenberg uncertainty principle: The more spectral content a transmitted signal contains, the longer one has to measure it for an accurate Fourier analysis, and the farther away from the source one must move in order to determine the propagation speed of the signal."

"The one single most important concept to emphasize in the discussion of the seeming paradox arising with the Walker-Dual experiment is the Heisenberg uncertainty principle: It is not possible to state at the same time that a signal has a certain frequency, and that it is located at a sharp position in space."

"In quantum mechanics, it's the exact same statement: There, the frequency corresponds to quantum mechanical momentum, and the ususal Heisenberg principle says that momentum and position can't be measured with infinite precision at the same time. In quantum mechanics, the precision limit is bounded by Planck's constant h. In Fourier analysis, it's 1 instead of h. Most people believe that Heisenberg's uncertainty principle is a special feature of quantum mechanics."

"It is actually a very deep insight that in all natural phenomena where some form of wave propagation occurs, there is a kind of a Heisenberg uncertainty principle. Something that is not widely known except among mathematicians working in the field (I am quite sure, not even among physicists), is the following fact: The Heisenberg uncertainty principle is NOT a mysterious physical law special to quantum mechanics, and discovered by Prof. Heisenberg, which is inexplicable and mysterious. This is in contrast to the Schrodinger equation, which is truly as miraculous as nature itself."

"The truth is: The uncertainty principle is a generally valid mathematical theorem in the theory of Fourier analysis (this mathematical field is called harmonic analysis, a deep and beautiful area with many surprises). It states that the product of the standard deviations of any pair of conjugate Fourier variables (i.e. wave vector and position vector, or energy and time, or frequency and time, etc) is bounded below by 1 (depending on the normalization with factors pi etc)."

"Whenever you are given a physical system described by Fourier theory, of which wave propagation is a prominent example, you will ALWAYS have an uncertainty principle. The catch is to find out how it must be formulated. In the case of information transmission versus causality, it's as explained above."

--------

Now THAT'S an explanation! (Incidentally, BA, Dr. Chen delved into the forum before giving his permission to quote, and was very complimentary about the discussion: "I have looked at the web forum, and have enjoyed the constructive and engaged discussion, I very much encourage it to continue.")

A technical discussion of some of the same issues, with math, is given by Mohammad Mojahedi and Kevin J. Malloy in "Superluminal But Causal Wave Propagation" linked below.

So, from all this I would say that not only is the Walker-Dual experiment fully explained in the framework of conventional physics--and should be removed from Van Flandern's list of experiments supporting superluminal gravity--but the later work of William Walker on superluminal near-field propagation is easily explained as well: "In the near zone, there is a superposition of a lot of outgoing and incoming waves, all propagating with speed of light. Due to these superpositions, it is not possible to observe an individual wave."

Van Flandern is mistaken in using Walker-Dual as support for superluminal gravitation.

References:

Mathpages: Phase, group, and signal velocity and Lead-lag frequency response

Short description of retarded field theory in electromagnetism, with animated applet, courtesy of Wolfgang Christian: http://webphysics.davidson.edu/Apple...etard_FEL.html

"The Speed of Gravity: Repeal the Limit II" by Tom Van Flandern: Metaresearch org

"Gravitational Forces with Strongly Localized Retardation" by William Walker, Jurg Dual, and Thomas Chen: abstract in html, full paper in pdf

"Propagation Speed of Longitudinally Oscillating Gravitational and Electrical Fields" by William Walker and Jurg Dual: abstract in html,, full paper in pdf

"Experimental Evidence of Near-field Superluminally Propagating Electromagnetic Fields" by William D. Walker: abstract in html, full paper in pdf

"Aberration and the Speed of Gravity" by Steve Carlip: abstract in html, full paper in pdf

"The Speed of Gravity Revisited" by Ibison, Puthoff, and Little: abstract in html, full paper in pdf

"Superluminal but Causal Wave Propagation" by Mohammad Mojahedi and Kevin J. Malloy: full paper in pdf

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