Relativistic Optics

[yuiop]

Hi,

This is not about the usual gravitational lensing effect but about the speed of light in glass in Special Relativity. I have not analysed it in great detail yet, but on first inspection it seems that at least one of the following list must be true:

1) The refractive index of glass changes with relative velocity.
2) The refractive index of the glass parallel and transverse to the motion may change to different extents and possibly inversely to each other.
3) The light travelling through the the glass is "carried along" by the glass and gets a boost in the forward direction and is retarded in the backward direction.
4) The speed of light through glass probably obeys the relativistic addition of velocities law where the speed of light in the glass medium and the velocity of glass itself are added according to that equation.
5) Extending statements (4) and (5) implies that the speed of light emitted from a source embedded in a moving optical medium is not independent of the velocity of the emitter (and optical medium) relative to the observer.

I was curious if anyone knows exactly which of the above statements are true and if there are already any equations published for the refractive index of a block of glass (or other light medium) that has has velocity relative to the observer?

 

[granpa]

4) The speed of light through glass probably obeys the relativistic addition of velocities law where the speed of light in the glass medium and the velocity of glass itself are added according to that equation.

the speed of light in glass is less than c not greater than c.

 

[yuiop]

4) The speed of light through glass probably obeys the relativistic addition of velocities law where the speed of light in the glass medium and the velocity of glass itself are added according to that equation.

the speed of light in glass is less than c not greater than c.

I never suggested that he speed of light in glass (moving or not moving) is greater than c.

What I am getting at is this. Lets assume there is a block of glass of refractive index RI=2. That implies the speed of light through the block of glass is 0.5c where c is the speed of light in a vacuum. Now if the the block of glass is moving at 0.5c relative to observer Anne will she still perceive light going through the block of glass (in the same direction the block is moving) as going at 0.5c relative to her? I think not because then an observer co-moving with the block of glass would perceive the light to be passing through it at 0.5c. Therefore it seems reasonable that the light passes through the block of glass in the forward direction at (0.5+0.5)/(1+0.5*0.5)= 0.8c and at (0.5-0.5)/(1-0.5*0.5) =0.0c in the reverse direction according to Anne.

It can also be noted that if light moves at 0.5c in the block of glass that when the glass is moving at 0.5c relative to the observer, that the light would have no velocity relative to the glass if the relativistic velocity equation is not used.

 

[Mentz114]

Kev,
I found these publications in my GR library. They might be worth a look but the 2000 one appears to be unrefereed.

http://arxiv.org/abs/physics/0009093

http://arxiv.org/abs/gr-qc/0106045v2

 

[granpa]

its motion relative to the glass is 0.5c. its motion relative to anything else would be calculated the same way anything else would.

 

[Redbelly98]

4) The speed of light through glass probably obeys the relativistic addition of velocities law ...

That was my initial thought too. Just use relativistic velocity addition to figure out the speed of light in moving glass. Then it's just semantics whether you say that the refractive index changes with velocity, or that the glass "carries along" the light wave with it.

 

[Mentz114]

There is a treatment of refraction between media when they are moving relative to one and other, here.

http://www.mathpages.com/rr/s2-08/2-08.htm

RedBelly98;

Then it's just semantics whether you say that the refractive index changes with velocity, or that the glass "carries along" the light wave with it.

This is the impression I got from my scanty exposure to this some time ago.

 

[yuiop]

There is a treatment of refraction between media when they are moving relative to one and other, here.

http://www.mathpages.com/rr/s2-08/2-08.htm

RedBelly98;

This is the impression I got from my scanty exposure to this some time ago.

Hi mentz,

Thanks for the latest link. I have not absorbed all the detail yet but it looks more readable than the first two links which were a bit "heavy" ;)

One problem I am having is the the early experiments conducted by Fizeau on the speed of light in moving water essentially do not agree with Special Relativity. At this stage I have a hunch that the incompatability of Fizeau results with relativity is that he interpreted the interferance seen the fringes as a change in the speed of light relative to the moving media rather than a change in the refractive index and it is not clear if he took doppler effects into account.

 

[Mentz114]

Looking at the situation where light passes ( incident at 90deg) through a block of glass with refractive index (RI) n and thickness d there is a phase change

$$\delta t = \frac{d}{cn}$$ in the frame in which the glass is stationary.

From a frame boosted to beta in the direction of the light beam,

$$\delta t' = \frac{d'}{cn'}$$ from which it seems that if d and n transform the same way, we have invariance of $$\delta t$$. I presume that is required.

This is elementary but brings me to the point. When I tried to calculate the phase change by boosting the velocity of the light in the glass, I get for the transformed speed of light c'

$$c' = c\frac{1 + n\beta}{ n + \beta}$$

and using this value I cannot derive an invariant delta t.

What have I done wrong. I was expecting this to work out.

M

 

[yuiop]

Looking at the situation where light passes ( incident at 90deg) through a block of glass with refractive index (RI) n and thickness d there is a phase change

$$\delta t = \frac{d}{cn}$$ in the frame in which the glass is stationary.

From a frame boosted to beta in the direction of the light beam,

$$\delta t' = \frac{d'}{cn'}$$ from which it seems that if d and n transform the same way, we have invariance of $$\delta t$$. I presume that is required.

This is elementary but brings me to the point. When I tried to calculate the phase change by boosting the velocity of the light in the glass, I get for the transformed speed of light c'

$$c' = c\frac{1 + n\beta}{ n + \beta}$$

and using this value I cannot derive an invariant delta t.

What have I done wrong. I was expecting this to work out.

M

Hi Mentz,

here are some other things about relativistic optics that do not work out. In the 1851 experiment by Fizeau, the empirical results were consistent with:

$$c' = \frac{c}{n}+v\left(1-\frac{1}{n^2}\right)$$

where c' is the speed of light relative to the block of glass, v is the velocity of the block of glass, c is the speed of light in a vacuum and n is the refractive index of the block of glass.
Of course Fizeau used moving water rather a moving block of glass but the principle should be the same.

By casual inspection it can be seen that Fizeau's conclusion violates the theory of Special relativity because if the speed of light is relative to the block of glass is anything different from what it was before the block of glass was accelerated then absolute velocity can be determined by measuring the change in the proper velocity of light relative to the block of glass by an observer co-moving with the block of glass. One can only conclude that that Fizeau's interpretation of the results was incorrect.

Now it is known that the speed of light in a block of stationary glass (u) is proportional to c/n. By substitution the above equation becomes:

$$c' = u+v\left(1-\frac{u^2}{c^2}\right)$$

Inserting values of u=0.5 (which corresponds to a not unrealistic refractive index of 2.0) and v=0.8c for the velocity of the block of glass we get a result of c'=1.1c. Clearly Fizeau's equation is not correct for high velocities of the optical medium relative to the observer. For some reason I can not find any authoritive texts that cleanly resolve this issue despite the experiment being over a hundred years old.

I think reading between the lines that some of the replies to this thread already indicate that other people also realise that the speed of light relative to observer when propogating through a medium moving relative to the same observer can not be anything other than that obtained by the relativistic velocity addition equation if Special Relativity is a correct theory, and yet Fizeau's experiment does not support that. As I mentioned before, I am not saying Special Relativity is wrong, but simply that Fizeau's equation is based on some false assumption and that the issue needs resolving.

There are reasons I believe that the refractive index of a material changes with relative velocity and that when phase changes at vacuum/glass interfaces and doppler shifts are fully taken into account, that the apparent anomally will be resolved.

 

[Mentz114]

Hi Kev,
I can see the problem. How did Fizeau and co. get that formula ? Is it a best fit to the data, or did they use Galilean relativity to calculate it ?

If we have 2 events, E1 being the light entering and E2 being the light exiting, then $$ds^2 = c^2(t_2-t_1)^2 - (x_2 - x_1)^2$$ should be preserved by Lorentz transformation. I'll work this out later and make another post.

M

 

[yuiop]

Hi Kev,
I can see the problem. How did Fizeau and co. get that formula ? Is it a best fit to the data, or did they use Galilean relativity to calculate it ?

If we have 2 events, E1 being the light entering and E2 being the light exiting, then $$ds^2 = c^2(t_2-t_1)^2 - (x_2 - x_1)^2$$ should be preserved by Lorentz transformation. I'll work this out later and make another post.

M

This snippet http://www.jstor.org/pss/228961 indicates a Galilean transform was used in some part of the calculation, which was perhaps reasonable since the velocity of the water relative to the apparatus in the actual experiment was only about 7 meters per second. I read somewhere that the experiment was repeated on a number of occasions using different fluids with different refractive indexes to confirm the relationship indicated in the Fizeau equation.

 

[Mentz114]

Useful snippet. I can see how the expression derived from aether theory, and is clearly not relativistic. It must be a good low speed approximation if the data fits. Some relevant papers refer to light in 'slow moving media' so I guess this is widely understood.
The exact solution is going to need some powerful analysis to find.

Things go pear-shaped when we have light moving at less than c. For instance, referring to my event/proper length analysis, in the rest frame of the glass the proper interval between the entry and exit events is ( in length units)
$$ds^2 = \frac{d^2}{n^2} - d^2$$ which is not zero although we are talking about light. Hmmm.

Now if I boost the events I get
$$\left( \frac{d}{n}, d \right) \rightarrow \left( \gamma d\left( 1 - \frac{\beta}{n}\right), \gamma d\left( \frac{1}{n}-\beta\right)\right)$$

and the transformed proper interval is

$$\gamma\beta d\left(\frac{1}{n^2}-1\right)$$

compare with

$$d\left(\frac{1}{n^2}-1\right)$$

I could have made an error but I can't find it. I would expect to get the same value.

One of thae papers I cited suggests that the Minkowski metric should be replaced by one where the relevant space-space component be replaced with $$(n^2 - 1)\beta^2$$

Proper lengths in this space would be

$$dt^2 - dx^2(n^2 - 1)\beta^2$$

which doesn't resolve the issue. But it's a start. This is interesting so I'll do some more tomorrow.



M

 

[Mentz114]

I plotted the transformed velocities using the SR and Fizeau formulae. Slightly interesting.

 

[Mentz114]

I think I've got some of it...

I made a mistake my event/proper length analysis above, it actually works out correctly.

In the rest frame of the glass the proper interval between the entry and exit events is ( in length units)
$$ds^2 = \frac{d^2}{n^2} - d^2$$
which is not zero although we are talking about light.
Now if I boost the events I get
$$\left( \frac{d}{n}, d \right) \rightarrow \left( \gamma d\left( 1 - \frac{\beta}{n}\right), \gamma d\left( \frac{1}{n}-\beta\right)\right)$$

and the transformed proper interval is

$$d^2\left(\frac{1}{n^2}-1\right)$$.

just as it should be. But best of all, note that

$$\frac{x'}{t'} = \frac{\frac{1}{n}-\beta}{1 - \frac{\beta}{n}} = \frac{1 - n\beta}{ n - \beta}$$

which is just the result of relativistic velocity addition as worked out in post #9.

I think this shows that transforming n gives the same result as relativistic velocity addition, which was what I wanted to do.

 

[Mentz114]

Lorentz transformation of refractive index

I thought I'd complete my bit by working out the transformation law for refractive index (n).

If d is the thickness of the glass then the proper interval between the entering and exiting of the light in the rest frame of the glass is

$$d^2\left(\frac{1}{n^2}-1\right)$$

In a frame boosted by beta in the direction of the light, the same interval is

$$\frac{d^2}{\gamma^2}\left(\frac{1}{m^2}-1\right)$$

where m is the transformed refractive index. Equating these gives

$$\frac{1}{\gamma^2m^2} - \frac{1}{\gamma^2} = \frac{1}{n^2} -1$$

and after some algebra

$$m^2 = \left( 1 + \gamma^2\left(\frac{1}{n^2}-1\right)\right)^{-1}$$

It seems to work, if gamma=1 then m=n, if n=1 then m=1.

And it preserves the proper interval.

 

[Mentz114]

Something strange emerges from this, which I don't fully understand.

The transformation I've written above undoubtedly preserves the invariant proper length between the entry and exit events, but it breaks down if the boost velocity v is greater than the speed of light in the glass

$$v \ge \frac{c}{n}$$.

What can it mean ?

 

[yuiop]

Something strange emerges from this, which I don't fully understand.

The transformation I've written above undoubtedly preserves the invariant proper length between the entry and exit events, but it breaks down if the boost velocity v is greater than the speed of light in the glass

$$v \ge \frac{c}{n}$$.

What can it mean ?

I am not sure yet. As far I can tell doppler shift due to a stationary light source and moving glass is self cancelling on entry and exit from the glass back to a stationary receiver is self cancelling. I think the phase shift of a light wave as it reflects off a mirror at 45 degrees to the path of the light (half a wavelength?) is also self cancelling as there are two mirrors in each minimal path of Fizeau's experiment.

I am going to try and analyse quantitatively what happens when a light from a source in a moving frame passes through a moving lens and is focused on a moving target as the result of that analysis will have to be consistent with everything else discussed above.

A brief qualatative analysis.

Let the source, lens and real image all be moving relative to an observer in frame S' but stationary with respect to each other in frame S.

The lens is thinner due to length contraction so it has a longer focal length according to S' The light from the source hits the glass of the lens at a shallower angle according to S' because the lens is moving away from the point the the light was emitted from and becasue the lens is thinner. The target where real image is formed is also moving away the point where the light is emitted and the longer focal length of the lens seems to work out Ok in this instance.

Now when we reverse the order of the light source and the target image, the lens and target image are moving towards the source according to S' and the longer focal length of the thinner lens seems to work against the source forming a focused image. This suggest the refractive index of the glass must change (possibly anisotropically) to compensate for this apparent anomally.

 

[yuiop]

Something strange emerges from this, which I don't fully understand.

The transformation I've written above undoubtedly preserves the invariant proper length between the entry and exit events, but it breaks down if the boost velocity v is greater than the speed of light in the glass

$$v \ge \frac{c}{n}$$.

What can it mean ?

Well I have just checked your equation

$$m^2 = \left( 1 + \gamma^2\left(\frac{1}{n^2}-1\right)\right)^{-1}$$

and you are right that there is a problem when v>=c/n because the transformed refractive index becomes imaginary. hmmmm

..maybe it just coincidental that the velocity of light u relative to a medium is proportional to c/n in the rest frame of the medium?

I'll just quickly brush up on the equations for the relationship of refractive index and light paths in classical optics and then try tackling this problem from a different angle.

 

[Mentz114]

Strange case of moving glass...

I've plotted space-time diagrams to show the rest-frame of the glass, and the glass boosted to 0.522c. This shows the 'light' stopping and changing direction in the rest frame of the onserver.

The only way this cannot violate SR is if there is no light in the glass, or the light follows a complicated path in the glass. Maybe the latter is not feasible.