KJW

Is this some form of holography? I tend to agree with this. But ultimately, I was simply justifying the statement that phase velocity can be obtained from the measurements of wavelength and frequency.

It's definitely from x-ray crystallography. Note that only the carbon atoms are showing because hydrogen atoms are rather insensitive to x-rays. Neutron diffraction would be better if one needs to explore bonding to hydrogen atoms.

I don't think so. And? Yes, I did. The velocities of the two frames are relative to the third frame. Perhaps I should have simply said the velocity relative to the third frame of the second frame is equal and opposite that of the first frame. Why would I?

Correct. Although I haven't seen an electron density map of hexamethylbenzene before, I have seen electron density maps of other compounds, so I did know what I was looking at with your picture. Hmmm, perhaps I spoke too soon. The direct image from x-ray crystallography is an arrangement of spots of various intensities. A Fourier transformation of these intensities yields a Patterson function (a convolution of the electron density with its inverse) due to the "phase problem". So how does one obtain a photographic plate image of the electron density? Specifically, what is performing the Fourier transformation? In particular, how is the "phase problem" being solved? I didn't use those as examples as I'm not very knowledgeable of those fields.

There are two inertial frames of reference in which there is no acceleration. The acceleration in going from one inertial frame to the other inertial frame does not count. Otherwise, it would be one non-inertial frame of reference.

I did substantiate my claim. I said:

As I said, I'm familiar with chemistry.

I'm familiar with chemistry. That's why I chose an example from chemistry. It seems to me that you don't understand the point I'm making.

No, I don't have a diagram. I actually don't see the difficulty.

The same could be said about any quantum wavefunction because we are in fact talking about the quantum wavefunction of a free non-zero mass particle.

I don't think it would be too difficult to arrange the frames of reference and the rod orientations to give the same length contraction, thereby establishing that the rod is the same length in both frames of reference. The simplest would be for the rod to be oriented parallel to the change in velocity in both frames of reference, and for the third frame of reference to be collinear to the other two frames of reference.

One could measure the clock and the rod in both frames of reference from a third frame of reference relative to which the two frames of reference have equal speeds.

How is the above a direct measurement of electron density? I've already indicated above that x-ray crystallography is not a direct measurement of molecular structure. Producing a detailed image of the electron density of a molecule is not the same as being a direct measurement. That's exactly my point. If a direct measurement is unavailable, then an indirect measurement based on sound principles is acceptable. Thus, although the phase of a De Broglie wave cannot be measured, the phase velocity of a De Broglie wave can be indirectly measured by measuring the wavelength and frequency of the wave.

No, you have misunderstood what I said. If in some inertial frame of reference, I have a clock that ticks away seconds and a rod that is one meter long, and I accelerate to some other inertial frame of reference, then the clock will still tick away seconds and the rod will still be one meter long. This is the principle of relativity in action and neither time dilation nor length contraction would make sense without it.

A lot of science is based on indirect measurements based on some theory. For example, in chemistry, do we really have direct knowledge of chemical structure? Perhaps the most direct measurement of a molecule's structure is through x-ray crystallography, but even that involves the theory of x-ray diffraction along with dealing with Fourier transforms.

I don't know if the frequency is currently measurable, but in principle at least, one could use time crystals.

Or alternatively, by measuring the frequency and wavelength by some form of diffraction experiment.

You need to recheck this.

No, λf does not equal the group velocity. vg = dω/dk whereas λf = ω/k = vp.

It seems to me that the non-relativistic approach is simply incorrect. For De Broglie waves of a non-zero mass particle, one considers wavelength, but I do not recall anyone considering frequency in a non-relativistic context. The frequency of a De Broglie wave is directly proportional the energy of the particle, and this energy quite simply includes the energy-equivalent of the mass. Thus, the relativistic approach is the only correct approach.

But isn't it true that λf = vp? Are you suggesting that neither λ nor f are measurable? I do accept that the phase of a De Broglie wave is unobservable, but it is not clear to me why vp is not at least indirectly measurable.

I agree that the electromagnetic wave equation with the speed of sound instead of the speed of light is invariant to Lorentz transformations with the speed of sound instead of the speed of light. But we know that sound waves do not behave like light waves, and I would like to know precisely where the properties differ. The wave equation you have given agrees with https://en.wikipedia.org/wiki/Acoustic_wave_equation, but I'm not convinced that this wave equation applies to any coordinate system other than the one in which the medium is at rest. This would break the invariance. In the case of light, the Doppler effect depends only on the relative velocity between the source and the observer, whereas in the case of sound, the Doppler effect depends not only on the relative velocity between the source and the observer but also on the relative velocity of the medium. Note that whereas both the speed of sound and the speed of light are independent of the speed of the source, only the speed of light is independent of the speed of the observer. Elsewhere, I have said that c of relativity is not really about the speed of light but rather about the relationship between space and time. The relativistic velocity-addition formula, as measured by the Fizeau experiment, allows one to determine in principle the value of c without measuring the speed of light in a vacuum, thus highlighting the space and time aspect of c. The speed of sound cannot satisfy the space and time aspect of relativity. Only the Lorentz transformations with the speed of light are transformations between different frames of reference in which the space and time coordinates mean the same thing in both frames of reference.

In this case, actual Lorentz transformations (with the speed of light in a vacuum).

I see two problems with this: 1: The wave equation for electromagnetism but with the speed of sound replacing the speed of light may not be the correct wave equation for describing the propagation of sound. In this wave equation there is no dependency on the velocity of the medium, whereas sound waves have a fixed velocity relative to the medium, which may differ from the velocity relative to the observer if the medium is in motion relative to the observer. 2: If the Lorentz transformation but with the speed of sound replacing the speed of light is applied to space and time coordinates, the resulting new coordinates are no longer space and time coordinates. Indeed, it is not obvious what these new coordinates represent. Note that it is Minkowskian spacetime that is invariant to Lorentz transformations, not just the electromagnetic wave equation, so that applying a Lorentz transformation to space and time coordinates results in new coordinates that are also space and time coordinates.

The force between two charges corresponds to the increase or decrease in the total energy of the surrounding electromagnetic field as the distance between the charges is changed. I think what happens to the electromagnetic field of the electron and positron when they annihilate is quite relevant.