# Speed of Light in Refractive Media

Discussion in 'Physics & Math' started by tsmid, Jul 25, 2006.

1. ### tsmidRegistered Senior Member

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346
I am presently looking into the issue of the 'Fresnel Drag' of light in moving media (see for instance http://home.att.net/~numericana/answer/relativity.htm#fizeau ).
As is evident from the link, the drag coefficient is explained by the relativistic addition of velocities

w=(u+v) / (1 + u*v/c^2).

However, c here is the speed of light in vacuum, and for light propagating in a medium it should be replaced by c/n, with n the refractive index. With this replacement, the formula becomes

W=(u+v) / (1 + u*v*n^2/c^2).

Following the further steps in the link (i.e. replacing u with c/n), this however results in

W= (c/n +v) / (1 + v*n/c) = c/n ,

i.e. the speed of light should be independent of the velocity v of the observer also in a refractive medium. So contrary to the claim, the relativistic addition of velocities, if done correctly, seems to actually to contradict the observed dependence on v rather than confirm it.

Thomas Smid

Last edited: Jul 31, 2006

3. ### przyksquishyValued Senior Member

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The c in the Lorentz velocity addition formula is the c that is referred to in the postulates of relativity and appears in the Lorentz transforms - ie. the speed of light in a vacuum, not the speed of a given photon in the medium you happen to be considering.
Only if you assume c/n is invariant to begin with. If the universe posesses an invariant speed which happens to be called the "speed of light", it does not follow that the speed of real photons travelling in some medium is also invariant.

5. ### tsmidRegistered Senior Member

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346
Yes, but that's excatly the point I am making. You have to remember where the c in the Lorentz transformation comes from: essentially it comes from the equation x=c*t, i.e. a light signal travels with speed c. Now this equation holds in a vacuum but obviously not in a medium where the speed of light is not c but c/n (i.e. we would have in the latter case x=c/n*t) . It is surely not consistent to assume that both equations hold simultaneously, but this is effectively the case if you use c in the velocity addition formula whilst assuming on the other hand that the light travels through a medium.

The invariance of c in a vacuum is not really an assumption, but an observational fact. On the other hand, in Fizeau's experiment the speed of light in a medium (i.e. the time it takes to get from one point to another) is not really measured but merely the phase difference between two light waves. So this would still leave the possibility that the speed of light in a medium is actually invariant as well (unless you know of other experiments where actually the speed as such was measured). Even if it isn't invariant, this still does not resolve the inconsistency with the velocity addition addressed above.

Thomas Smid

Last edited: Jul 31, 2006

7. ### przyksquishyValued Senior Member

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I don't see a link between this and relativity. The invariance postulate explicity claims that the speed at which light travels in a vacuum is invariant - not the speed in the current medium. In other words, the invariance is taken to be a general property of the speed c=299,792,458 m/s, rather than a property specific to the propagation of light.
Only if you assume that the c appearing in the Lorentz velocity addition formula tells you something about the motion of light in the medium you're considering. There's no reason to believe this.
It's also a postulate of SRT. You can argue that the postulates of the theory should be different, but the only velocity addition formula you can derive using SRT as it is currently formulated is (u+v)/(1+uv/c<sup>2</sup>).
Not really. First off, the index of refraction is dependent on the frequency of the light, so which would you use as "n" in your formula? You aren't claiming multiple invariant speeds, are you? Also, it'd have some very interesting consequences: suppose I have a block of transparent material with a refractive index so high that light comes to a virtual standstill in it. What would happen in a reference frame in which the block were moving?

Last edited: Jul 27, 2006
8. ### tsmidRegistered Senior Member

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346
Whether the speed of light in the medium is also invariant is not the crucial point here. Consider the observer at rest relatively to the medium. The light propagates then relatively to the observer according to the equation x=c/n*t. However, the equations on which the velocity addition formula is based imply x=c*t. This is thus inconsistent unless n=1 (vacuum conditions).

Thomas

9. ### przyksquishyValued Senior Member

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3,130
It is. The c appearing in the formula is the universe's invariant speed - it has nothing to do with the local motion of light.
No they don't - c is just the speed that is taken to be invariant. It's appearance in the Lorentz transformations implies nothing, other than that the speed c is the same in all inertial reference frames (hardly a surprise, considering that's the whole point of using them in relativity).

10. ### tsmidRegistered Senior Member

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346
If the local speed of light in a medium with refractive index n is c/n, then it appears to have a lot to do with the 'invariant speed' c.

The point is that the Fresnel drag coefficient (as derived in http://home.att.net/~numericana/answer/relativity.htm#fizeau ) is in fact a local quantity, and so should be the speed of light that appears in the velocity addition formula (i.e. the Lorentz transformation). If you are using the vacuum speed c for the latter when in fact the local speed of light is c/n, then this is in my view inconsistent. You can not simultaneously have two local speeds of light, one that takes the medium into account, and another which would hold if the medium would not be there. On the other hand, the speed of light in a completely different region of space (where there is a vacuum) can not possibly be relevant for the speed of light in the medium.

Thomas

11. ### przyksquishyValued Senior Member

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It does. The velocity u is substituted for the local speed c/n. It doesn't follow logically that c should also be substituted for c/n, though.
Actually, you can have a whole range of different speeds of light in a medium - the refractive index is dependent on the frequency of the light.
Why not? You are assuming that relativity is derived directly based on the properties of light in a vacuum. In reality, relativity holds that a particular speed, referred to in terms of the speed of light in a vacuum, is invariant. Of course it's more than just a coincidence that the two speeds are equal, but they're not quite the same thing.

12. ### tsmidRegistered Senior Member

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I was of course referring to a particular refractive index i.e. a particular frequency of light. The speed of light for this should be unique in a medium.

As already indicated above, the Lorentz transformation is derived on the basis of the postulate that the location x of a light pulse as a function of time t is given by the equation x=c*t (in any inertial reference frame). So c is clearly defined as the physical propagation speed of light here and nothing else. However, in case the locations x are not associated with a vacuum, a light pulse propagates in fact according to the equation x=c/n*t (relatively to the medium). This is an obvious mathematical contradiction which can not be rationalized away by metaphysical re-interpretations of either of the equations.

Thomas

13. ### przyksquishyValued Senior Member

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Yes, but which speed would you put in the Lorentz transformation? All of them?
No it's not. It's derived based on the postulate that anything travelling at the speed c in one inertial frame is also travelling at speed c in every inertial frame. Note also that the velocity of such a moving object is not invariant - the direction is frame dependent.
Yes, c is explicitly defined as the propagation speed of light in a vacuum. Given additional, independent assumptions that have been made about c (such as its invariance), other functionally equivalent (but not logically equivalent) definitions also exist:

&bull; The speed that is invariant with respect to a change in (inertial) reference frames.
&bull; The speed at which all bodies move through space-time (ie. the norm of the four-speed of any moving object).
c doesn't appear in the Lorentz transform because light travels at that speed locally. It appears because of the assumption that this speed is invariant. The only way you can conclude that it should be replaced by c/n is if you postulate that c/n is invariant - something that isn't a part of SRT. In this sense, SRT assumes that light travelling through a vacuum has a special property not shared by light travelling through a medium, which it is perfectly entitled to do. You may feel that giving c special treatment in this sense is inconsistent, but it is most certainly not a logical fallacy.

Also, while I'm not familiar with experiments directly related to the Fresnel drag, I can quickly indicate that another relativistic formula - namely the relativistic momentum formula - would fail if c were substituted for c/n. It would imply that nothing could travel faster than c/n in a medium, which is false. It is well known that particles can be forced beyond this speed. In the case of charged particles, this results in the emission of Cherenkov radiation.

14. ### DaleSpamTANSTAAFLRegistered Senior Member

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You cannot logically have more than one invariant speed, c.

-Dale

15. ### tsmidRegistered Senior Member

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I am not necessarily saying that c should be replaced by c/n in the Lorentz transformation. I am only saying that the speed of light in a medium is not c, so the basic postulate of relativity (which has been formulated for vacuum conditions) is per se not applicable here.

This circumstance does not remove the inconsistency addressed by me (especially as it does not involve the speed of light but of massive objects).

Thomas

16. ### tsmidRegistered Senior Member

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346
Why can there be only one invariant speed? You could only say this either on the basis of observational facts or as an ad-hoc postulate. I don't see any logical problems with the speed of light being also independent of the reference frame in a medium (this would seem to contradict the Fresnel drag of course, but as mentioned above already, at least with the original Fresnel drag experiment carried out by Fizeau, the speed of light was actually not measured here, but only the phase-difference of the light waves)

Thomas

17. ### przyksquishyValued Senior Member

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3,130
You can't argue that the velocity addition formula should be changed without the Lorentz transformation also being modified. You cannot have two velocity addition formulae resulting from a single transformation.
It wasn't formulated exclusively for vacuum conditions; it was assumed to hold generally. It does reference the speed at which light travels in a vacuum, but this in no way restricts it's use to vacuum conditions only.

Suppose I rewrite the invariance postulate like this: "The speed s is invariant."

Now it's got nothing directly to do with the motion of light at all. Suppose that it then follows from the complete theory that massless particles (like photons) will travel at speed s under vacuum conditions. I can then measure the speed of light in a vacuum to determine what velocity s I should use in the theory. If I intended this all along, I may as well replace s with "the speed of light in a vacuum" in the original postulate. This is more or less how relativity is constructed.
Using different versions of SRT for different objects is an inconsistency.

18. ### Physics MonkeySnow Monkey and PhysicistRegistered Senior Member

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This is completely obvious. It would be logically inconsistent to have more than one invariant speed, not to mention a continuum of invariant speeds. Of course, you don't have to take my word for it, just look for yourself.

As I just indicated, having multiple invariant speeds leads to an inconsistent theory. You can have one or none. Furthermore, the postulate of an invariant speed is hardly an ad-hoc assumption, it's one of the most tested ideas in physics.

I've already addressed the logical fate of your multiple invariant speeds idea. As for the Fresnel drag experiment, your comments imply a freedom of interpretation that does not exist. If you accept Maxwell's electrodynamics as providing an accurate description of light waves (and I assume you do), then the phase difference is related to the speed (more properly, one of several speeds that can be defined). You act like we don't know what that phase difference means, but we do know what it means. The only way around is to propose an entirely different descriptive scheme for light waves. Of course, this would be silly since our current scheme already describes all observed phenomenon.

19. ### tsmidRegistered Senior Member

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346
No, you can't. That's exactly the point I am making.

I quote from Einstein's book 'Relativity: The Special and General Theory' (1920). Chpt.11 (Lorentz Transformation) "The relations must be so chosen that the law of the transmission of light <I>in vacuo</I> is satisfied for one and the same ray of light (and of course for every ray) with respect to K and K' "(see http://www.bartleby.com/173/11.html ). The restriction 'in vacuo' is emphasized by Einstein himself by the way, here and numerous times in the previous chapters. However, in the later chapters this restriction is tacitly dropped and for instance not mentioned at all in Einstein's own treatment of Fizeau's experiment (see http://www.bartleby.com/173/13.html ). There is no justification of any kind given by Einstein why the derived Lorentz transformation is still applied here even though there is no vacuum in this case.

Thomas

20. ### tsmidRegistered Senior Member

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346
It would not be logically inconsistent but obviously inconsistent with the assumption that there is only one invariant speed. The point is that Einstein himself originally considered the invariance principle only for the propagation of light in a vacuum (see the post directly above this) and there are no grounds why it should apply in other cases as well (in fact logic would suggest that it doesn't).

Because there are no logical grounds why the invariance principle (as formulated for a vacuum) should also apply to regions of space where there is no vacuum, it could therefore be at best an ad-hoc assumption. However, as should be evident from Einsteins's own publications (see the post above), there is even no explicit assumption of this kind being made. It is merely being introduced tacitly without justification of any sort.

You seem to forget that the Doppler effect for instance will also affect the phase difference: when the molecules in the medium absorb and re-emit the light of a source moving relatively to it, the light propagating in the medium should have a different frequency due to the Doppler effect. This should create a phase shift (compared to a stationary source) even without the observer moving relatively to the medium.

Thomas

21. ### przyksquishyValued Senior Member

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So which refractive index do you think should go in the velocity addition formula? And why doesn't it also appear in the relativistic momentum formula?
The point is that Einstein never explicitly restricted himself to vacuum conditions. He claims that it is the idealized x = ct that remains invariant, regardless of whether light is actually propagating at that speed or being slowed down for some reason locally.
Because this restriction was never assumed in the first place. There's no reason to adopt it, and it is not in the quotation you dug up. Einstein basically said "the vacuum speed of light is invariant." You are adding the "as long as we are in vacuum conditions" yourself. At least part of your misconception is in your following reply to Physics Monkey:
You are taking invariance to be a property of light. I've already told you that relativity takes invariance to be a property of the speed at which light travels under vacuum conditions. There's an important difference between "the speed of light is invariant in a vacuum" and "the speed of light in a vacuum is invariant."

While I'm here, I don't know if it's logically impossible to have multiple invariant speeds, but it is certainly not possible with any linear transformation. Substituting a variable in the Lorentz transformations won't accomplish this. If you make a modification such that c/n is invariant, c will become variant.

Last edited: Aug 2, 2006
22. ### DaleSpamTANSTAAFLRegistered Senior Member

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Hi tsmid,

As you mention, it is inconsistent with experiment (the experiment was set up so that velocity differences would give phase differences). However, I was not talking of experiment, simply of logic. If you don't see any logical problems, then you have not thought it through carefully enough. Just thinking about it last night I came up with 3 logical problems.

1) geometry. A leg of a hyperbola cannot simultaneously asymptotically approach two different lines. So the spacetime geometry would be inconsistent with two invariant speeds.

2) causation. The invariant speed separates future from past and space from time and determines which events can be causally connected. On one side of the line indicated by the invariant speed you have all frames agreeing on the order of events and on the other you have some frames disagreeing on the order. This is only logically acceptable because the events on which they disagree cannot be causally connected. However, if there were two invariant speeds then two events could be causally connected by the higher invariant speed and their order could be disagreed-upon by the lower invariant speed. So you would get some frames where the effect would preceed the cause. This is obviously illogical.

3) algebra. Let's assume that there are two invariant speeds, c and u. Consider the event at s=(ct,ut) and the Lorentz transform s' = ((γ, -γv/c), (-γv/c, γ)).s = (γct-γtuv/c, γtu-γtv). If u is an invariant speed then u = c(γtu-γtv)/(γct-γtuv/c) = c²(u-v)/(c²-uv). Solving for u we get u=c or u=-c. So if there is one invariant speed there cannot be another different invariant speed.

-Dale

23. ### Physics MonkeySnow Monkey and PhysicistRegistered Senior Member

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tsmid,

But that's just it, it would be logically inconsistent. See Dale's post right above this one for just a few reasons why. Einstein said the invariant speed was the speed of light in vacuum, and he applied this postulate consistently. He did not say that it only holds in vacuum, and this is what you appear to be misunderstanding. If light goes slower in some medium, big deal, the speed of light in vacuum is still an invariant. Furthermore, there are plenty of grounds for believing that the invariance postulate holds good in almost every situation.

This is all false. The justification is the countless number of experiments that Einstein's theory correctly describes. The invariance principle is believed to hold good because no experiment has ever been found to contradict it. That's about as logical as one can be in science.

I certainly haven't forgotten about the Doppler effect. The Doppler effect combined with thermal motion broadens the spectrum of the light pulse, but this is known and can be easily accounted for. Why do you keep bringing up these red herrings? We know what the phase means, we know what can effect it, we know how to calculate all these things consistently, and it all agrees with experiment.

I just don't understand, Thomas. Why are you trying to convince us that something makes no sense when it very clearly makes a great deal of sense?