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The speed of light is given as $c=\frac{1}{\sqrt{ε_0μ_0}}$ which is in terms of the electric and magnetic constants.

Hypothetically, another massless particle could exist which does not interact with the electronic or magnetic fields. Postulate that a "foo" field and a "bar" field exist which were mathematically analogous to the electric and magnetic fields, the "foobaron" particle might have a speed given by $c_{fb}=\frac{1}{\sqrt{f_0b_0}}$, derived in exactly the same way that $c$ is derived, but using different, independent fields.

If such a particle were to exist, must $c=c_{fb}$? Why?

Qmechanic
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spraff
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  • I guess it depends how wild you want to get. Suppose we theorize an "UberHiggs Field" with derived massless particles which travel at some speed $d = \pi * c$ . MIght be a few problems in figuring out how we'd ever detect either the field or the particles. – Carl Witthoft Oct 11 '21 at 14:49
  • It all depends on whether you're willing to sacrifice Lorentz invariance (i.e. relativity). – WillO Oct 11 '21 at 15:05
  • Possible duplicates: https://physics.stackexchange.com/q/651784/2451 and links therein. – Qmechanic Nov 29 '21 at 06:08

2 Answers2

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In special relativity, the relativistic factor

$$ \gamma = \frac1{\sqrt{1-v^2/c^2}} = \frac{E_\text{total}}{mc^2} $$

is just as closely related to the particle’s total energy as to the particle’s speed. Any object whose total energy is very much larger than its rest mass will be traveling near $c$. Objects whose rest mass is identically zero are a special case of this limit.

The speed limit $c$ is not a property of electromagnetism; it is a property of spacetime. If a massless particle were traveling at less than $c$, I could hop into my massive rocket ship and catch up to it. In its rest frame, it would not be a massless particle. A contradiction, because mass is invariant under Lorentz boosts.

rob
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    maybe one should add that all measurements up to now validate Lorenz invariance. If it is discarded any science fiction goes. – anna v Oct 11 '21 at 14:18
  • How did you reach the conclusion that “in its rest frame, it would not be a massless particle”? Why wouldn’t the particle be able to remain massless in its rest frame? – HelloGoodbye Nov 29 '21 at 01:42
  • @HelloGoodbye Because the mass $m^2 = E^2-p^2$ is a relativistic invariant, the same in all reference frames. If it has a rest frame, it’s massive, and the mass is the same in all frames. And if it’s traveling at less than $c$, I can match its speed, so it has a rest frame. – rob Nov 29 '21 at 02:49
  • Yes, but my question was, how do you come to the conclusion that the particle cannot be massless in its rest frame? What prevents us from having $E=p=m=0$? – HelloGoodbye Nov 29 '21 at 08:53
  • @HelloGoodbye Oh, I see. It’s not immediately clear to me how you would distinguish a particle with $E=p=m=0$ from the vacuum. It might also be the case that you get some $0/0$ limit weirdness when describing such a particle; I would have to think about that carefully. That specific question might make a good follow-up. – rob Nov 29 '21 at 11:29
  • I think if you involve relativistic quantum mechanics what you say is true, because (as was pointed out in an answer to a similar question) with the Klein–Gordon equation for example (or the Dirac equation), any wave will travel with the speed $c$ if $m=0$. I was also curious to see if this is a conclusion you can make using only relativity. – HelloGoodbye Nov 29 '21 at 13:15
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c is the speed of cause and effect. See Do we know why there is a speed limit in our universe?

The speed of light is a special case of this. If you move a point charge, the "news" that it has moved will spread out at this speed. That is, changes to the electric and magnetic fields from the charge will spread out at this speed. Forces from the charge acting on another charge will not change until the "news" arrives. See In what medium are non-mechanical waves a disturbance? The aether? for more.

It is possible for waves and such to travel at slower speeds. For example sound does. This is because it is a pressure wave. Atoms bump into other atoms and push them. It takes a while for atoms to accelerate. The pushing arises from electric fields in the atoms' electrons. The rate at which neighboring atoms begin to feel the push spreads out at the speed of light.

mmesser314
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