KJW

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).

Thanks. I hadn't come across time-limited editing before.

So there is a time limit to editing a post? I noticed the "Edit" item in the post above but not in another post of mine that I wish to edit.

How do I edit a post?

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.

It appears to me that you are referring to the energy of the electromagnetic field surrounding the positron-electron pair. The Wikipedia article "Electromagnetic mass" may interest you. It doesn't entirely answer your question because it is about a classical notion. But the notion that the mass of an electron includes the mass equivalent of the energy of the surrounding electromagnetic field is an intriguing one and is connected to the modern notion of renormalization.

Bear in mind that when you go out into the sunshine, you don't feel the force of the light being exerted on you, so it is evidently quite small in magnitude.

As for why a photon is massless, (IIRC) this has something to do with the Higgs field and the symmetry-breaking of the four electroweak bosons.

In Minkowski spacetime, using natural units where c = 1, there is the four-dimensional energy-momentum vector. Energy is the time-component and momentum is the spatial components of this vector. The invariant mass is the magnitude of this vector. That is: m² = E² – p² Just as a tangent vector of a lightlike trajectory in spacetime is a null vector, so is the energy-momentum vector of a photon. That is, E² = p² and thus m = 0. As mass is an invariant, the mass of a photon is zero in every frame of reference.

I'm not aware of any problems with special relativity. If you think that quantum entanglement and experiments such as the delayed-choice quantum eraser experiment are a problem for special relativity, then it would seem that you do not correctly understand the nature of quantum entanglement. Quantum entanglement is often presented as the communication of the result of a measurement of one particle to its distant quantum entangled partner. But this is incorrect. According to the no-communication theorem, it is not possible to use quantum entanglement to communicate information. In other words, there is nothing that a person can do to their quantum state that will have any effect on the result of any measurement performed by another person on the corresponding entangled partner state. This absence of communication means that the correlation between the entangled particles, even if it appears to be faster than light or backwards in time, does not violate special relativity.

It's my experience that a lot of people think Special Relativity is about light, with c being called "the speed of light in a vacuum". I prefer to think of c as a ratio of a distance in space over an equivalent distance in time. What I'm offering is an in-principle method of determining c without measuring the speed of light in a vacuum.

Fizeau performed the measurements of the speed of light in moving water compared to the speed of light in stationary water, but my purpose is to obtain the value of c without measuring the speed of light in a vacuum, thus disconnecting c from the properties of light. If the dispersion of light in water is ignored, the relativistic velocity-addition formula can be rearranged to express c in terms of the two velocities of light in water and the velocity of the water.

Special Relativity is embedded in Maxwell's equations. Maxwell's equations indicate that electromagnetic waves propagate at the speed of light, expressed in terms of the permittivity of free space and the permeability of free space. Maxwell's equations are invariant to Lorentz transformations, indicating that the speed of light is also invariant to Lorentz transformations. That is, the speed of light is the same for all inertial observers regardless of their velocity relative to some specified frame of reference. Thus, the notion of an absolute frame of reference is denied by Maxwell's equations. However, Special Relativity does go beyond Maxwell's equations and electromagnetism in general. Indeed, the speed of light in Special Relativity should not be regarded as specifically about light, but rather about the relationship between space and time. In principle, underwater creatures could determine the speed of light in a vacuum without measuring the speed of light in a vacuum by measuring the speed of light in both stationary and moving water, applying the relativistic velocity-addition formula. In this case, the speed of light in water is an "ordinary" speed, distinct from the c that appears in relativistic formulae.

Good day, Markus. Yes, it is me from The Science Forum. It seems that The Science Forum has finally died. It did seem like an inevitability for quite a long time. But I did stick with it till the very end. Anyway, I joined this site earlier today, posted some stuff. I notice you are still using the generalised Stokes theorem as your signature. 😉

You are correct. There is the question of "invariant to what?". For example, Special Relativity is invariant to Poincaré transformations, whereas General Relativity is invariant to all coordinate transformations, a much bigger group. But if the group of transformations for which the laws of physics are invariant is limited, and that the laws of physics are not invariant to transformations outside this group, then it becomes impossible to state what these varying laws of physics are because one doesn't have a reference against which one can state which laws of physics apply to which location in the space to which the transformations apply. But if the laws of physics are invariant, then they can be stated unequivocally due to that invariance. For example, if the laws of physics are expressed entirely in terms of partial derivatives, then they will vary according to the coordinate system. But the coordinate system is not actually known, so one can't know which laws of physics apply. But if the laws of physics are expressed entirely in terms of covariant derivatives, then they will be invariant to coordinate transformations, and therefore one no longer needs to know the coordinate system because the same laws of physics apply to all of them. I often see the question "What is time?". The answer I give is "Time is what a clock measures.". But what is a clock? A clock is defined by the instructions that are used to construct it. While this may not be particularly helpful in understanding the nature of time, time is nevertheless well defined by the instructions.

In what way are these different devices clocks? In what way are these measurements that disagree with the proper time of general relativity time measurements? I'm not discarding these devices as false, only your assertion that they measure time. It is my belief that it is the most fundamental principle of reality that the laws of physics are invariant. For without invariance, there are no laws of physics.