Does the photon have a 4-velocity in a medium?

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1. Nov 15, 2015

PFfan01

From classical electrodynamics textbooks, we know that the Fizeau experiment supports relativistic 4-velocity addition rule. But a recently-published paper says that the photon does not have a 4-velocity. See: "Self-consistent theory for a plane wave in a moving medium and light-momentum criterion", http://www.nrcresearchpress.com/doi/10.1139/cjp-2015-0167#.Vki8xGzovIU

I wonder who's right?

2. Nov 15, 2015

PWiz

I thought this was a well known fact.

3. Nov 15, 2015

PFfan01

What do you mean? Do you mean the Fizeau experiment is not a support in relativistic 4-velocity addition rule?

4. Nov 15, 2015

PWiz

No, I'm saying that the four-velocity is not defined for a photon.

5. Nov 15, 2015

PFfan01

There is a definition of four-velocity of light in the book by W. Pauli, Theory of relativity, (Pergamon Press, London, 1958), Eq. (14), p. 18, Sec. 6. Seems the definition is the same as that for a matter particle.

6. Nov 15, 2015

PWiz

I don't have the book with me. Can you post the definition?

If the definition is the same as that for a massive particle, then it can't be right. Photons move on null lines and experience 0 proper time. Since four velocity is the proper time derivative of four position, it is not defined for a photon.

7. Nov 15, 2015

Staff: Mentor

I agree with PWiz. Another way to think of it is that the four velocity is the four momentum divided by the mass, which is 0. Or that the four velocity is the unit tangent to the worldline and a null worldline can only have null tangents, not unit tangents.

8. Nov 15, 2015

PFfan01

If in free space, you are right. In a medium, the light speed is less than the vacuum light speed. In the book by Pauli, Fizeau running water experiment is used as a support in relativistic 4-velocity addition rule.

9. Nov 15, 2015

PWiz

But a photon always moves at $c$ regardless of the medium (it interacts with other particles in the medium and on a macroscopic scale you can say that the average speed of light reduces, but microscopically individual photons always move at the same speed). The photon still experiences 0 proper time, and you still cannot define its four velocity.

10. Nov 15, 2015

Staff: Mentor

I think that you probably want to ask about classical light waves rather than photons.

In a medium a plane wave will have a phase velocity which is less than c. You can definitely use the relativistic velocity addition formula on the phase velocity, so I assume that you could make a phase four velocity. Although I don't recall seeing anyone do that before.

11. Nov 16, 2015

PWiz

But then there is no contradiction. It might be possible to define a four velocity if you deal with light classically in a medium, and still have an undefined four velocity for the photon treatment.

So addressing the OP, I guess both statements can be right. I would still like to see the four velocity definition for the classical treatment of light in a medium though.

Last edited: Nov 16, 2015
12. Nov 16, 2015

Staff: Mentor

Yes. I agree.

13. Nov 16, 2015

PFfan01

Very interesting argument, but could you please show any references for your argument? Thanks a lot.

PS: The paper by Leonhardt, Ulf (2006), "Momentum in an uncertain light", Nature 444 (7121): 823, doi:https://dx.doi.org/10.1038%2F444823a [Broken] , says that the photon in a dielectric medium moves at the dielectric light speed, but the author did not tell why.

Last edited by a moderator: May 7, 2017
14. Nov 16, 2015

PWiz

References for the 2nd postulate of relativity or for atomic spacing?

15. Nov 16, 2015

PFfan01

The references for your statement that "a photon always moves at c regardless of the medium".

PS: In my understanding, Einstein's second hypothesis is the constancy of light speed in free space.

16. Nov 16, 2015

PWiz

So are you saying that there is no atomic spacing in a medium? Think about it. Between electron interactions in a medium, what does a photon move in? Reading post #9 again might help.

17. Nov 16, 2015

PFfan01

1. I never said "there is no atomic spacing in a medium".
2. I am just asking you to give any references for your statement that "a photon always moves at c regardless of the medium".
3. In fact, it is enough for you to tell me whether your statement is your reasoning from Einstein's second hypothesis or there are any references to support it.

Even if your statement is your reasoning, I am not able to judge whether it is correct or not, because it is far beyond my knowledge.
Sorry.

18. Nov 16, 2015

PWiz

A photon is always moving through empty space (when a photon is moving in a medium it's actually moving through the atomic spaces, which is nothing but empty space) or interacting with other particles. The 2nd postulate says that light always moves at $c$ in empty space (as measured in an inertial frame).
I just restated two well known facts. From these two facts, it follows that a photon always moves at $c$. It's just that the interactions of the photon with other particles in a medium "delay" the photon, so the effective speed of light seems to reduce in any particular medium compared to a vacuum (but the photon moves at $c$ between the interactions). That's all I'm saying.
P.S. I'm not trying to be confrontational here.

Last edited: Nov 16, 2015
19. Nov 16, 2015

PFfan01

So your statement that "a photon always moves at c regardless of the medium" is just your reasoning, without any references to support it. Right?

20. Nov 16, 2015

Staff: Mentor

Really? There are hard boundaries around the atoms that delimit them from "empty space"? And the photons never cross those boundaries? See further comments below.

Not really. You restated a common model for photon propagation in a medium, but that's a lot different from "well-known fact".

In fact, although it's a common model, it's not actually correct. For example:

The interactions you are talking about here are the absorption and emission of photons by atoms in the medium. These interactions do not "delay" one photon; they destroy one photon (when it's absorbed) and create a second photon (when it's emitted). (Note, btw, that the absorption and emission is actually done by electrons in the orbitals of the atom, which means that the photons do in fact have to cross the "boundary" of the atom--the electrons aren't all sitting on the boundary, they are in the interior.)

It is true that, in this somewhat more accurate model, the photons move at $c$ between interactions. However, the model is, as I just said, only somewhat more accurate. We don't actually measure the speed of the photon between interactions; we can't. And if we make our model more accurate still, by bringing in more quantum mechanical details, we will find that the concept of the "speed" of the photon between interactions isn't even well-defined; the quantum amplitudes will have contributions from off shell virtual photons.

The moral is to be very careful what you think of as a "well-known fact".