Deflection of light through a relativistic glass rod.

[yuiop]

Let us say we have a very long glass that has a uniform square cross section. Its long axis is aligned with the x axis. A beam of light is directed along the y-axis such that it is normal to the side of the glass rod's side surface. When the rod is stationary wrt to the observer we would expect the beam of light to pass straight through the rod and hit a target on the other side without any deflection. Now it seems to me that if the rod is moving relativistically in the positive x direction that the beam of light would be deflected in the positive x direction and no longer hit the target. The deflection would be a function of the relative speed of the rod to the observer. Agree or disagree?

There is probably an equation for this somewhere. Anyone know of one?

 

[PAllen]

Can't this just be analyzed from the rod's frame? Just Lorentz transform to the Rod's frame getting angle of incidence in that frame. Then, refraction formula for deflection in that frame. Then Lorentz back to get angle of deflected ray in the original frame. Or you could use aberration formula, snell's law, abberration formula, instead of two Lorentz's.

By the way, it seems to me the ray would be bent in the negative x direction. [Edit: Not sure. Would want to calculate it].

 

[yuiop]

Can't this just be analyzed from the rod's frame? Just Lorentz transform to the Rod's frame getting angle of incidence in that frame. Then, refraction formula for deflection in that frame. Then Lorentz back to get angle of deflected ray in the original frame. Or you could use aberration formula, snell's law, abberration formula, instead of two Lorentz's.

By the way, it seems to me the ray would be bent in the negative x direction. [Edit: Not sure. Would want to calculate it].

Your first method is what I had in mind. I was just be lazy and assuming that there was a ready made equation in the literature somewhere and someone here might know where it is.

The curious aspect is that I am almost certain that refraction occurs even though the angle of incidence is perfectly normal to the glass surface in the rest frame of the source and my best guess is still that the deflection is forward.

 

[PAllen]

Your first method is what I had in mind. I was just be lazy and assuming that there was a ready made equation in the literature somewhere and someone here might know where it is.

The curious aspect is that I am almost certain that refraction occurs even though the angle of incidence is perfectly normal to the glass surface in the rest frame of the source and my best guess is still that the deflection is forward.

I did a sample calculation, and agree with forward deflection. Always forward, always by less than the beaming change, for your set up.

 

[PAllen]

For any speed achievable in the lab, I would suspect that deviations from optical imperfections would be larger than this effect.

 

[ardie]

to simplify the discussion, let us take the case of an ideal glass that cannot absorb any photons. on the microscopic level, the E.M. field in the medium experiences a different group velocity due to the presence of charged particles etc... the relative motion of the present particles will not in anyway effect the interaction. the only thing that motion in the y direction can change is the fields due to the charged particles in the glass get stronger along z and x directions. but these fields are not the same as those of the incoming photons. they are in fact independent and do not interact with one another. Now in the rest frame of the glass, the charged particles are also essentially at rest (they look just as they were in the rest frame) and so the net refractive index cannot be a function of translational velocity (in the direction perpendicular to that of the incident light).

 

[PAllen]

to simplify the discussion, let us take the case of an ideal glass that cannot absorb any photons. on the microscopic level, the E.M. field in the medium experiences a different group velocity due to the presence of charged particles etc... the relative motion of the present particles will not in anyway effect the interaction. the only thing that motion in the y direction can change is the fields due to the charged particles in the glass get stronger along z and x directions. but these fields are not the same as those of the incoming photons. they are in fact independent and do not interact with one another. Now in the rest frame of the glass, the charged particles are also essentially at rest (they look just they were in the rest frame) and so the net refractive index cannot be a function of translational velocity.

Simplify? All you need to note is that if the angle of incidence is perpendicular to the moving rod in the lab frame, it is not perpendicular to the rod in frame where the rod is stationary. Thus it suffers refraction. If it suffers refraction in this frame, it also must refract in the lab frame.

 

[pervect]

Your first method is what I had in mind. I was just be lazy and assuming that there was a ready made equation in the literature somewhere and someone here might know where it is.

http://en.wikipedia.org/w/index.php?title=Fizeau_experiment&oldid=479541788

It looks like the approach is the same, I didn't carefully examine the equations.

 

[PAllen]

Here is an extensive discussion:

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

Of course the OP setup, which I assumed was vacuum to glass, and for which the lab angle is perpendicular, is an extremely simplified case compared to above.

 

[yuiop]

Thanks all for the contributions

... Now in the rest frame of the glass, the charged particles are also essentially at rest (they look just they were in the rest frame) and so the net refractive index cannot be a function of translational velocity.

I do not think it is about refractive index. The simple experiment in the OP uses a incident beam that is perpendicular to the glass and any change in refractive index cannot possibly have any effect if Snell's law holds. By very straight forward analysis as PAllen explained, the light must deflect due to aberration of the light paths in the different reference frames.

I did a sample calculation, and agree with forward deflection. Always forward, always by less than the beaming change, for your set up.

I did too and got the same result. I used your second method, relativistic aberration to the glass rod rest frame, Snell's law and then relativistic aberration back to the lab frame.

This is the method in more detail.

$$\theta_1 ' = 2*atan \left( tan (\theta_1 /2)*\sqrt{\frac{(1-v)}{(1+v)}} \right)$$

$\theta_1$ is the beam angle in the rest frame of the source (lab frame). $\theta_1 '$ is the beam angle in the rest frame of the glass rod.

Both are measured clockwise from the negative x axis.
It is assumed the glass rod is moving to the right in the positive x direction in the lab frame.

Next apply Snell's law to obtain the refracted angle $\theta_2 '$ as measured in the rest frame of the glass rod:

$$\theta_2 ' = \pi/2 - asin(sin(\pi/2 - \theta_1 ') /n )$$

where n is the refractive index of the glass rod.

I have adapted the equation so that the angles are consistently measured clockwise from the negative x axis.

Now the aberration formula is used again to transform the refracted ray angle back to the lab frame to obtain the total deflection:

$$\theta_2 = 2*atan \left( tan(\theta_2 ' /2)*\sqrt{\frac{(1+v)}{(1-v)} }\right)$$

A numerical example is that for a relative velocity of 0.8c, a refractive index of 1.5 for the glass and the incident beam at right angles in the lab frame, the ray is deflected to an angle of 2.0545 radians clockwise from the negative x axis, which is in the forward direction.

For any speed achievable in the lab, I would suspect that deviations from optical imperfections would be larger than this effect.

Perhaps if rotating glass discs were used it would make the experiment doable. There might be ways to multiply the effect optically. I think similar experiments have been done with microwaves to study relativistic Doppler effect. One advantage with a rotating disc is that the imperfections can be filtered out by noting that they will show up as a beat with the same frequency as the rotation of the disc.

Lastly, although I have not calculated it, I suspect that when the beam exits the glass rod it will be parallel to the original incident beam but offset.<EDIT> For those that like relativistic curiosities, there is a point on the leading edge of a glass sphere that is moving in the x direction of the lab frame, that will allow a pulse of light traveling in the y direction to pass straight through the glass sphere with no deflection at all. Agree or disagree?

 

[yuiop]

Here is an extensive discussion:

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

Of course the OP setup, which I assumed was vacuum to glass, and for which the lab angle is perpendicular, is an extremely simplified case compared to above.

Thanks PAllen (Paul, Pete?). The mathpages article hints that with certain combinations of refractive index media and non perpendicular incident angles, the effects are exaggerated and possibly measurable in a realistic lab.

 

[PAllen]

By the way, I now think that (aberration, snell, aberration) is not right. The issue is the last aberration: light does not travel at c in glass, which is assumed in aberration. The (Lorentz, snell, Lorentz) works, as long as you set up an equation for the path of light in the glass that accounts for speed of c/n in the rod, in the rod's frame. It doesn't change the qualitative result, but does change the details.

The mathpages analysis shows that if you are going medium to medium, you need a more robust approach, since you have two refractive media, for each of which you know its rest refractive index (and light speed), but not (a priori) the characteristics of the moving medium. That is, no matter what frame you use, you have a moving refractive medium. With vacuum to medium, you can fundamentally simplify.

 

[PAllen]

Thanks PAllen (Paul, Pete?).

Peter, Paul? What about Mary.

 

[PAllen]

<EDIT> For those that like relativistic curiosities, there is a point on the leading edge of a glass sphere that is moving in the x direction of the lab frame, that will allow a pulse of light traveling in the y direction to pass straight through the glass sphere with no deflection at all. Agree or disagree?

Agree to no angular deflection. I think there will be a displacement.

 

[yuiop]

Peter, Paul? What about Mary.

In best Father Ted voice... "Well have it as you will, Mary it is".

By the way, I now think that (aberration, snell, aberration) is not right. The issue is the last aberration: light does not travel at c in glass, which is assumed in aberration. The (Lorentz, snell, Lorentz) works, as long as you set up an equation for the path of light in the glass that accounts for speed of c/n in the rod, in the rod's frame. It doesn't change the qualitative result, but does change the details.

The mathpages analysis shows that if you are going medium to medium, you need a more robust approach, since you have two refractive media, for each of which you know its rest refractive index (and light speed), but not (a priori) the characteristics of the moving medium. That is, no matter what frame you use, you have a moving refractive medium. With vacuum to medium, you can fundamentally simplify.

Agree with all the you say here. Medium to medium is much more complex. I am not convinced that even the mathpages article get the universal equation which contains n1 and n2 is correct in they are both greater than unity. I say this because when n1=1 it should agree with second version that is taylored to the vacuum to medium situation and it does not always do that. Secondly when n1=n2 >1 there should be no deflection at all, (but the mathpages equation suggests there is). Certainly there should be no refraction, but I cannot rule that odd things happen when two mediums are moving relative to each other, even they have the same refractive index. Need to look into that some more. If anyone knows how to contact the author of mathpages (Kevin Brown) tell him he needs to check that equation out, it seems to be flawed. http://mathpages.com/rr/s2-08/2-08.htm

Here is a corrected version of my analysis that takes the reduced speed of light in the medium into account, but for simplicity I will stick with the n=1 for the first medium (vacuum).

I am now using a more conventional notation where all angles are measured anticlockwise from the y-axis in radians. The rod is moving along the x-axis as before and the primed symbols are the measurements in the rest frame of the rod. v is the relative velocity of the reference frames.

First step is the transformation of the incident light path to the rest frame of the glass rod (aberration):

$$tan(\theta_1 ') = \frac{ tan (\theta_1 ) + n1*v/cos(\theta_1)}{\sqrt{1-v^2/c^2}}$$

Again this is only valid for n1=1.

As in the previous post, next apply Snell's law to obtain the refracted angle $\theta_2 '$ as measured in the rest frame of the glass rod:

$$sin(\theta_2 ') = sin(\theta_1 ' )\frac{n1}{n2}$$
where n2 is the refractive index of the glass rod.

Now the refracted ray angle is transformed back to the source rest frame

$$tan(\theta_2) = \frac{ tan (\theta_2 ' ) - n2*v/cos(\theta_2 ')}{\sqrt{1-v^2/c^2}}$$

Note that this is the same as the first equation except for the change of sign of v and use of n2>1 instead of n1=1.

Earlier I gave this example:

... using a relative velocity of 0.8c, a refractive index of n2 = 1.5 for the glass and the incident beam at right angles in the lab frame, the ray is deflected to an angle of 2.0545 radians clockwise from the negative x axis, which is in the forward direction.

Now I get -0.92 radians anticlockwise from the y axis, which is equivalent to 2.49 radians clockwise from the x-axis for comparison to the earlier example.

The deflection is quite large and increases with the refraction index of the glass and increasing the thickness of the glass increases the displacement, raising the possibility that the effect would be detectable in a lab with a spinning disk of high strength, high index material at realistic speeds.

My equation was derived independently from the mathpages equations and looks very different, but is numerically equivalent to the second aberration equation given near the end of the page which is a good thing.

 

[PAllen]

Secondly then n1=n2 >1 there should be no deflection at all. Certainly there should be no refraction, but I cannot rule that odd things happen when two mediums are moving relative to each other, even they have the same refractive index. Need to look into that some more.

I haven't done a calculation for n1=n2>1, but, in general, I don't trust Snell's law for the moving medium case unless one is a vacuum, and you apply it in the rest frame of the medium. Snell's law is a special consequence of Least Time for the case of stationary media. For two relatively moving media, I would only trust a direct calculation from least time, as done the mathpages reference. A plausibility argument the two relatively moving slabs of identical index are fundamentally different from stationary is that, in the frame of either, the speed of light is anisotropic and different from c/n in the other (moving) slab. This could lead to surprising least time consequences.

 

[yuiop]

I haven't done a calculation for n1=n2>1, but, in general, I don't trust Snell's law for the moving medium case unless one is a vacuum, and you apply it in the rest frame of the medium. Snell's law is a special consequence of Least Time for the case of stationary media. For two relatively moving media, I would only trust a direct calculation from least time, as done the mathpages reference. A plausibility argument the two relatively moving slabs of identical index are fundamentally different from stationary is that, in the frame of either, the speed of light is anisotropic and different from c/n in the other (moving) slab. This could lead to surprising least time consequences.

I am coming around to the idea that maybe when two medium are moving parallel to each other that there might be refraction across the interface even when they have the same refractive index. I guess they might be an effective change in refractive index at relativistic speeds. I think one approach might to do the aberration transformation to the n2 rest frame and work out the new speed of light in n1 as measured in that reference frame. This is not too difficult when we break down the speed into its x and y components. Refractive index is just the inverse of the light speed, so we can use this new speed to work out the effective refractive index of n1 in the n2 reference frame and then use Snell's law with that new value of n1 together with n2.

The more serious criticism of the mathspage is that the first equation does not agree with the second version when they both have n1=1.

 

[Kaelpatch]

Here is a corrected version of my analysis that takes the reduced speed of light in the medium into account... My equation was derived independently [sic] from the mathpages equations... My equation ... looks very different, but is numerically equivalent to the second aberration equation given near the end of the page...

Different? Your original equation looked (and was) very different, but your corrected equation is identical to the mathpages equation, if you clear the cosine factor from the numerator and denominator.

I am coming around [independently, one assumes] to the idea that maybe when two medium are moving parallel to each other that there might be refraction across the interface even when they have the same refractive index... The more serious criticism of the mathspage is that the first equation does not agree with the second version when they both have n1=1.

Not so. If n1 equals 1 the two equations are identical, if you're careful to take the appropriate signs for the square roots. (Also, note that v in the first equation is defined as the speed of the n2 medium relative to the n1 medium, and in the second equation it is defined in the opposite sense, so you have to negate the sign of v.)

 

[yuiop]

Different? Your original equation looked (and was) very different, but your corrected equation is identical to the mathpages equation, if you clear the cosine factor from the numerator and denominator.

All I said was it looked different but was numerically the same. Obviously if I had continued to simplify I would have arrived at the mathpages equation. The derivation using using relativistic velocity equations is pretty straight forward and even I with my limited mathematical ability managed to do it myself it a slightly more straight forward way than the mathpages derivation which was not entirely clear to me the first time I read it.

Not so. If n1 equals 1 the two equations are identical, if you're careful to take the appropriate signs for the square roots. (Also, note that v in the first equation is defined as the speed of the n2 medium relative to the n1 medium, and in the second equation it is defined in the opposite sense, so you have to negate the sign of v.)

I concede that with using the opposite sign of v for the two equations (or the opposite sign for the incident angle) does make the mathpages equations consistent with each other for n=1. Thanks for clearing that up. I must have missed the bit where that was made clear.

I also concede that my idea of using an effective refractive index based on the transformed speed of light in the n2 rest frame does not appear to work having tried it out. Taking the mathpages advanced equation as correct (and I now have no reason to think it is not and I admit the derivation of that particular equation is beyond my simple abilities), then it does appear that when two mediums with the same refractive index are moving relative to each other, that refraction does occur.

It is also interesting to note that total internal refraction can occur when n1=n2>1 and the incident ray is exactly perpendicular to the face of medium n2.