Relativistic Interpretation of Fizeau Experiment/ Fresnel Drag

The relativistic interpretation of the Fizeau experiment is given by the velocity addition theorem as explained in https://en.wikipedia.org/wiki/Fizeau_experiment#Derivation_in_special_relativity .
Now here the speed of light in the reference frame of the water is assumed to have a constant value c/n independent of v (the speed of the water relative to the light source). This means however that in the reference frame of the water the light is 100% dragged (as otherwise it would depend on v). Now how is it possible that in one reference frame the light is 100% dragged, but in the other (the lab frame) it is only partially dragged (as indicated by the experiment)?

Thomas

 

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Because there is no such thing as "dragged". You are mixing components of two different theories of space-time.

 

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Well, some mixing of different theories would obviously be required here, as the light speed postulate applies only to vacuum conditions, but the detected experimental effect is on the contrary due to the medium i.e. deviations from the vacuum.

The point is that the velocity addition formula (which follows from the light speed postulate) is here applied to the light signal as

v_lab = (c/n +v)/(1+cv/n/c^2) ~ c/n +v*(1-1/n^2)

which implies that the speed of the light signal in the reference frame of the water is taken as

v_w = c/n

So in the reference frame of the water the speed of the light signal is assumed to be independent of the relative speed of the light source (as it is in the vacuum), but in the lab frame it is dependent on v. This would appear to contradict the relativity principle.

Thomas

 

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Nope. You can't mix ether theories with absolute space and time with relativistic theories.

Irrelvant. The "c" in Relativity doesn't change if you are not in vacuum because it is a physical quantity defined in vacuum, not something that only exists in vacuum. What changes is that the speed of propagation of light in matter is no longer c, and is not invariant under change of standard of rest.

Ignoring the "on the contrary" part , yes it due to the medium, because of the theory of electromagnetism -- materials change the velocity of propagation of EM signals.

\(The point is that the velocity addition formula (which follows from the light speed postulate) is here applied to the light signal as v_lab = (c/n +v)/(1+cv/n/c^2) ~ c/n +v*(1-1/n^2) which implies that the speed of the light signal in the reference frame of the water is taken as v_w = c/n\) That \(v_{\textrm{light in water}} = \frac{c}{n}\) is the definition of n, the refractive index, should be of no surprise. Neither should that is in experimentally determined in the coordinates where the medium is at rest. Thus when the water is moving at speed \(v_{\textrm{water}} = v\) relative to the lab, we compute the propagration of light in the same direction of the moving water at speed:
\(v_{\textrm{lab}} = \frac{ v_{\textrm{water}} + v_{\textrm{light in water}}}{1 + c^{-2} v_{\textrm{water}} v_{\textrm{light in water}}} = \frac{ v + \frac{c}{n} }{1 + c^{-2} v \frac{c}{n}} = \frac{ nvc + c^2 }{nc + v} = \frac{c}{n} + v \frac{n^2 c - c }{n^2c + nv} = \frac{c}{n} + v \left( 1 - \frac{1}{n^2} \right) \frac{1}{1+\frac{v}{nc}} \\ = \frac{c}{n} + v \left( 1 - \frac{1}{n^2} \right) \left( 1 - \frac{v}{nc} + \left(\frac{v}{nc}\right)^2 - \left( \frac{v}{nc} \right)^3 + \dots \right) \\ \approx \frac{c}{n} + v \left( 1 - \frac{1}{n^2} \right) ; \quad \textrm{when} \; \left| \frac{v}{nc} \right| << 1 \)
There is no concept of dragging in this derivation, only the notion that water at rest has a well-defined speed for the propagation of light and that the water and light are moving in the same direction in the lab frame.

Since nowhere the relative speed of the source of the light entered the discussion, it appears you have an unsupported leap of logic. v is the speed of the water relative to the lab. Thus a Lorentz transformation of a velocity which is \(v_{\textrm{light in water}} = \frac{c}{n} \neq c\) gives a result that is frame-dependent, entirely consistently with the theory of relativity.

 

Since the light source is at rest in the lab, v is also the speed of the light source relative to the water. And if the speed of light in the water's frame is c/n independent of the speed of the source (the speed of the water in this case), this is essentially the light speed postulate, only with c replaced by c/n. But in the lab, the light speed is found to be not invariant if the speed of the water is changed.

Thomas

 

Irrelevant. Because the speed of the light source in the coordinates where the water is at rest is -v and that never enters the calculation.

Logically, it can't enter the calculations because to reach the water, the light already must propagate through empty space, perhaps millions of miles of it, so it would be very weird physics for the light to remember anything about the motion of its source only after contact with water. We don't have any such physical theory and therefore the motion of the source doesn't enter the calculations. Indeed, refracting optical systems work because the index of refraction has the same meaning regardless of the motion of the source of the light.

So if the speed of the source of the light does not enter the calculations of the velocity of light in a transparent medium when the medium is at rest, it must be irrelevant to the discussion when the medium is in motion because of the principle of relativity.

No, it isn't. You can't create additional postulates by analogy.

The independence of the speed of propagation in water of light from the state of motion of the source of the light is a statement that water at rest has a well-defined propagation speed of electromagnetic phenomena, which is empirical confirmation of Maxwell's theory of electromagnetism.

The speed of light postulate is “light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body” . The independence from the motion of the source is not the important part of the postulate except in combination with the principle of relativity, which establishes the velocity of light is always c in any choice of inertial coordinates, thus c is a constant of space-time, not just electromagnetism. \(\frac{c}{n}\), however, is a constant of water and thus has meaning tied to coordinates where water is at rest.

The speed of light postulate combines with the principle of relativity to establish that the law of velocity composition for c and v is c, not c+v as Newton and Galileo would have it. \(\frac{c}{n}\), being a statement about a ponderable medium with definite state of motion cannot be substituted into the speed of light postulate by analogy. Spacetime supports only one such velocity for massless phenomena.

Correct, the speed of light in water changes co-variantly in transparent ponderable medium to preserve the laws of physics, in this case, the index of refraction of water which is defined, along with a myriad of physical properties, in water's rest frame.

 

But assuming an observer moving with the water measures the speed of light, the velocity addition formula as employed here would imply in this case

“light is always propagated in a medium with a definite velocity c/n which is independent of the state of motion of the emitting body"

At least, that is what this observer would have to conclude. Is that confirmed by experiments?

Thomas

 

To gain an important physical insight, try setting up and solving for a toy model reproducing the essentials of slowed propagation in a dielectric medium. A regularly spaced linear array of RT's (receiver/transmitters). Assume in between each RT is vacuum, with a precise finite delay between reception and re-transmission for each RT. In the rest frame of such array, equate an effective refractive index n to the RT array spacing and delay time.
Now, move to lab frame where the array has some relative speed v. Then, just apply the known rules of SR to work out the transformed mean light speed c'. Make sure to carefully apply non-simultaneity correctly! Please report back your own results. Since the Fizeau result and that of later variants has been amply demonstrated to be true, double and triple check before posting - if you initially obtain anything that doesn't reduce to the standard formula(s) - exact and approximate.

 

Your model seems to be implying that the observer co-moving with the water will indeed measure an invariant speed of light regardless of the relative motion of the light source. Again, is there any actual experimental confirmation for this ? (the Fizeau result only makes a statement about the observed speed of light in the lab frame not the water's reference frame)

Thomas

 

In the coordinates of the frame of the water, we have a regular 1-d spatial lattice of delay-objects, closely spaced \(\Delta L_0\) apart and each delaying propagation of light for a short while, \(\Delta T_0\). Then in these coordinates, the space-time picture of light propagation is an alternating pattern of moving for time \(\frac{\Delta L_0}{c}\) at speed c and then pausing for time \(\Delta T_0\). This is an average speed of \(\frac{\Delta L_0}{ \frac{\Delta L_0}{c} + \Delta T_0 } = \frac{c}{\frac{1} + \frac{c \Delta T_0}{\Delta L_0}}\). Setting this equal to the empirically-modeled value of \(\frac{c}{n}\) we have \(\Delta L_0 = \frac{c \Delta T_0}{n - 1}\).



Now if we do a Lorentz transformation in the direction of light-propagation, v, our alternating pattern consists of propagation at c for some time and propagation at v for some time, and some overall average speed.
What was propagation of light across a time-space interval of \((\Delta t_1, \Delta x_1) = ( \frac{\Delta L_0}{c}, \, \Delta L_0 ) \) becomes \((\Delta t'_1, \Delta x'_1) = ( \frac{\sqrt{1 + \frac{v}{c}}}{\sqrt{1 - \frac{v}{c}}} \frac{\Delta L_0}{c}, \, \frac{\sqrt{1 + \frac{v}{c}}}{\sqrt{1 - \frac{v}{c}}} \Delta L_0 ) \).
What was a delay of light across a time-space interval of \((\Delta t_2, \Delta x_2) = ( \Delta T_0, \, 0 ) \) becomes \((\Delta t'_2 \Delta x'_2) = ( \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \Delta T_0, \, \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} v \Delta T_0 ) \).
Thus what was an average propagation speed of \(u = \frac{\Delta x_1 + \Delta x_2}{\Delta t_1 + \Delta t_2} = \frac{c}{1 + \frac{c \Delta T_0}{\Delta L_0}}\) becomes \(u' = \frac{\Delta x'_1 + \Delta x'_2}{\Delta t'_1 + \Delta t'_2} = \frac{ \Delta L_0 + \frac{v}{c} \Delta L_0 + v \Delta T_0 }{ \Delta L_0 + \frac{v}{c} \Delta L_0 + c \Delta T_0} c\).

Or with \(\Delta L_0 = \frac{c \Delta T_0}{n - 1}\), \(u = \frac{c}{n}\) , \(u' = \frac{ \frac{1 + \frac{v}{c}}{n -1 } c + v }{ \frac{1 + \frac{v}{c}}{n -1 } c + c } c = \frac{c^2 + c n v}{ c^2 n + c v} \approx \frac{c}{n} + \left( 1 - \frac{1}{n^2} \right) v\).

 

Not a correct statement, since invariance relates to the constancy of c whatever the inertial frame of *observer*. But the model does quite naturally imply effective c in rest frame of iterated medium is independent of relative motion of source, which is what you meant.

Huh? The raw Fizeau experiment results makes a statement about both since the water speed in lab frame can readily be made zero i.e. no flow.
Thanks to rpenner's derivation in #10 you now have it all on a platter - unless you fancy there is a mathematical flaw somewhere in it (good luck with that route!).
Handily supplying URL of your own, what most would brand a crank site, this should now become an interesting test of character. Now having a fully worked example explicitly showing full consistency between Fizeau experiment and transformation rules of SR, will you now publish an addition to your site specifically acknowledging that fact?
I and maybe others will be checking in every so often to see what happens. Have a nice day - Thomas.

 

If there is no flow then the light source is at rest with regard to the water and the speed of light in the water's reference frame is naturally c/n. Your toy model tries to justify the use of the same light speed c/n in the velocity addition formula also in case the light source is moving with regard to the water. My question was whether this toy model is based on experimental fact. Has the speed of light in the water actually been experimentally measured to not depend on the state of motion of the light source? (and with this I do not mean measuring the speed in the lab's frame and then making conclusions from this regarding the speed in the water's reference frame by means of certain assumptions).

But let's for the sake of the argument assume an explicit speed measurement in the water's reference frame would confirm your toy model (and with it the use of the constant value c/n in the velocity addition formula).
Now do you agree that in general the velocity addition formula

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

holds not only for light signals, but u could indeed be the speed of anything? So you could change your toy model to work with ping pong balls rather than RF waves. You would measure the same dependence as in the Fizeau experiment. The result of the latter would have nothing specifically to do with light.

Thomas

 

The fact that water travels slower in water than a vacuum is the result of the photons be absorbed and re-emitted by the water molecules. To the extent that your ping pong balls behaved similarly the calculations would be the same, otherwise no.

 

How about Fizeau experiment, and variants, as experimental fact. The only assumptions being that basic logic applies.

The simple co-linear velocity formula above is correct but your conclusion is not. Instead of asking rather pointless questions of me, answer them yourself. For instance, based on the derivation in #10 (which I notice you have not challenged), make the appropriate substitution and it will be perfectly evident if Fizeau formula is recovered in ping pong balls case. It won't. Something broadly similar within a quite limited range of relative speeds will. But prove that for yourself.
Suggested reading: http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html

If you still want to argue after that, lay all your cards on the table. Set out for everyone to see just what is your philosophical position.
For instance, do you believe in an 19th century style aether? Newtonian physics/Galilean relativity as fundamental? Which conservation laws/symmetry principles hold?

 

Shorter: If Fizeau experiment is empirically right and the Einstein velocity composition law is correct, then the velocity of light in water in coordinates where the water is at rest calculated from measurement of the velocity of light in moving water agrees with the velocity of light in water which is not moving. The idea that the motion of the source has anything to do with the velocity of light propagation (ballistic theory) is rejected by the success of Maxwell's equations. Therefore the only theoretical framework that unites all these motion and electromagnetism phenomena with classical motion studies at low velocity as documented by Newton and Galileo is Special Relativity.

Therefore, the progress of science demands that, without good empirical cause, one must reject Newtonian mechanics in favor of Special Relativistic mechanics whenever the difference of the predictions of the theories is within experimental limits.

Quibbles about the meaning of why Einstein focused on the movement of the sources rather than adopt the broader principle immediately that "the speed of light in empty space in c to any inertial observer" ignores 1) the history of light theories 2) the state of knowledge of his audience 3) the principle that you advocate minimal postulates in math and geometry to prove grand propositions on simple principles that people can easily convince themselves of and 4) the concepts don't yet exist at that stage of the writing in 1905. Between the experimental picture from telescopic resolution of binary stars and success of Maxwell's theory, the claim that “light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body” was on firm empirical and theoretical ground.

 

This is a little loose on the language, and not quite correct. If a photon source is moving relative to an object that has an index of refraction (n), the frequency of the photon will change based on the relative motion relative to the medium. Because the index of refraction is different for different frequencies, you will have different change in the angle (upon entry into the medium) based on the relative motion of the photon-emitter to the medium with the index of refraction. You know this, of course, just commenting on the language. Thus, we see the angle change for visible light photons varies based on the frequency, as for instance the beautiful display of a rainbow.

"The refractive index varies with the wavelength of light. This is called dispersion and causes the splitting of white light into its constituent colors in prisms and rainbows, and chromatic aberration in lenses." https://en.wikipedia.org/wiki/Refractive_index

 

Not quite correct, but because the refractive index is near a local minimum at optical frequencies, especially the sodium spectral doublet, the dispersion interaction results in an effect about 44 times smaller than that measured by Fizeau.

Calculated from: https://en.wikipedia.org/wiki/Optical_properties_of_water_and_ice

 

It may be more instructive if we try to consider the problem in terms of phase differences (rather than speeds) because that is what is in fact observed in the Fizeau experiment. And in this sense, do we agree that all of the phase difference for the two propagation directions arises only in the medium? So for the sake of determining the phase difference, we can ignore anything that happens before entering and after exiting the lattice.

Thomas

 

I don't see why since \(\frac{c}{n}\) is the phase velocity of light in still water. The elementary math of finding average velocity in a stop-go lattice is the same. The conclusion about the velocity composition law is the same.

We have ignored it so far and wound up with the correct model, eh?

 

Well, then let' s see what we obtain for the phase difference with this model:

if the medium (our stop-go lattice) has the length L, the number of cycles within the medium is obviously

N = L/λ

where λ is the wavelength of the light.

Now the wavelength in the medium is

λ = c/n/ν

with n the refractive index and ν the driving frequency of the first array element of our model. So we have

N = L*n*ν/c

Now if the array moves with regard to the external light source with speed v, the driving frequency ν is shifted from the original frequency ν0 due the Doppler effect i.e.

ν = ν0 * ( 1±v/c)

so for the difference between the two opposite velocities we get

ΔN = 2*L*v*n*ν0 /c^2

So the Doppler shift due to the motion would be the only cause for the phase difference here.

Thomas