Refractive index of a medium in relativistic motion

[Mayan Fung]

TL;DR Summary

What is the speed of light in a medium if the medium is also moving at a relativistic speed?

I once naively think that the speed of light is also a constant in a medium in all inertial frames which is not the case. I tried to derive the result yet there is a discrepancy from the results I read in some articles.

For example, from [Link to unpublished paper redacted by the Mentors], the author derived the speed of light in a medium of refractive index $n$ moving at a relativistic speed v is
$$u = \frac{c}{n}(\frac{1+\frac{nv}{c}}{1+\frac{v}{nc}})$$
He derived it with the full set of Maxwell's equations which I think there is another easier way to the solution.

Here's my approach:

Let's say we have a piece of glass with width $d$ and refractive index $n$ moving at a relativistic speed v.

First, we calculate the time, $t$ the light travels through the entire glass in the glass frame. In this frame, the glass is stationary and the speed of light in the medium is $c/n$. Therefore,
$$t = \frac{d}{c/n} = \frac{n}{c}d$$

The time measured in the lab frame, $t'$, considering the time dilation effect, is:
$$t'=\gamma t, \gamma = \frac{1}{\sqrt{1-v^2/c^2}}$$

Also, the width of the piece of glass, considering length contraction, is:
$$d' = d/\gamma $$

Therefore, in the lab frame, the light in the glass travels a distance of $vt' + d/\gamma$ to reach the other side of the glass. So, the speed of light in the medium, $u$, can be found by:
$$\begin{align*}
ut' &= vt' + d/\gamma\\
u &= v + d/\gamma t'\\
u &= v + d/\gamma^2 t\\
u &= v + \frac{c}{n \gamma^2}\\
u &= \frac{c}{n} (1-v^2/c^2) +v
\end{align*}$$

This is different from the results listed in the beginning. I wonder where I made a mistake.

 

[Ibix]

If ever you use length contraction and time dilation and get the wrong answer, do the problem again using the Lorentz transforms. In fact, it's usually best to just start with the Lorentz transforms.

Here, you're not correct to assume that the light spends $t/\gamma$ in the block in the frame where it moves. You've correctly written the equations for the frame where the glass does not move, so you can write coordinates in that frame for the light's entry and exit events. Use the Lorentz transforms and try again from there.

 

[Sagittarius A-Star]

$$u = \frac{c}{n}(\frac{1+\frac{nv}{c}}{1+\frac{v}{nc}})$$

This is equal to the relativistic velocity addition formula:
$$u = \frac{v+c/n}{1+\frac{v c/n}{c^2}}$$

 

[vanhees71]

If ever you use length contraction and time dilation and get the wrong answer, do the problem again using the Lorentz transforms. In fact, it's usually best to just start with the Lorentz transforms.

Here, you're not correct to assume that the light spends $t/\gamma$ in the block in the frame where it moves. You've correctly written the equations for the frame where the glass does not move, so you can write coordinates in that frame for the light's entry and exit events. Use the Lorentz transforms and try again from there.

I can't agree more with that. For the Doppler effect you only need to know that $k^{\mu}$ in a plane wave is a four-vector. No matter which dispersion relation, $\omega=\omega(\vec{k})$ is valid. If you have the most simple approximation, where $\omega=|\vec{k}|/c_{\text{med}} =\frac{n}{c} |\vec{k}|$ (where $n$ is the refractive index; this simple relation is of course only correct if $n>1$, i.e., in the region of normal dispersion). That holds in the rest-frame of the medium.

You must also be careful that in this case the Doppler effect is not as simple as for light propagation in the vacuum, because here the (local) rest frame of the medium is a distinguished frame. I.e., the $\omega_{\text{obs}}$ and $\vec{k}_{\text{obs}}$ depends on both the velocity of the medium and the velocity of the light-source in the observer's rest frame and not only on the relative velocity of the observer and the light-source as in the vacuum.

 

[Sagittarius A-Star]

I wonder where I made a mistake.

You are not taking into account the relativity of simultaneity.

If you have mounted 2 clocks to both ends of the glass block, which are synchronous in the glass frame, then in the lab-frame the clock at the rear-end runs ahead of the clock at the front-end. That follows, together with time-dilation and length-contraction, from the Lorentz-transformation.

 

[Vanadium 50]

I wonder where I made a mistake.

At the start. There is no single speed of light in a medium. It depends on direction. For most media we are familiar with, the difference is small or zero. For an object traveling with respect to you near the speed of light, it is large. There is no "n".

For example, from [Link to unpublished paper redacted by the Mentors],

Please don't link straight to the PDF.

This is an unpublished paper that in 17 years never been able to get published. It has exactly two citations, one saying it is wrong and the other saying it is hardly ever right. It is among the crappiest of crap papers.

 

[Sagittarius A-Star]

At the start. There is no single speed of light in a medium. It depends on direction.

I think, that is not an error in the calculation. This is nothing else than the Fizeau-experiment (it was with water). Different formulas, containing $n$, came from Fresnel, Lorentz and Einstein.

Source:
https://en.wikisource.org/wiki/Rela...ition_of_Velocities._The_Experiment_of_Fizeau

 

[vanhees71]

At the start. There is no single speed of light in a medium. It depends on direction. For most media we are familiar with, the difference is small or zero. For an object traveling with respect to you near the speed of light, it is large. There is no "n".
Please don't link straight to the PDF.

This is an unpublished paper that in 17 years never been able to get published. It has exactly two citations, one saying it is wrong and the other saying it is hardly ever right. It is among the crappiest of crap papers.

As a constitutive property of the medium the refractive index is defined in the local rest frame of the medium by convention and thus a scalar. Of course you are right that generally the constitutive relations (permittivity and permeability) of dielectrics are of tensorial form (anisotropic, crystal optics).

The quoted paper is indeed pretty confusing (to put it mildly) ;-(. At the level of sophistication we discuss here the standard constitutive relations like $D=\epsilon E$ etc. have been derived in relativistically covariant form already by Minkowski around 1908.

 

[Vanadium 50]

As a constitutive property of the medium the refractive index is defined in the local rest frame of the medium by convention and thus a scalar.

Sure, but if you define it that way, it's not equal to c/v any more.

The microscopic way to look at this is that if the dielectric is at rest, the atoms that make up the dielectric are little round balls. Boost and they become ellipsoidal. Of course the dielectric constant - or if you prefer, the effective dielectric constant, c/v - becomes anisotropic. The medium itself does.

 

[PAllen]

This may be useful:

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

It is not peer reviewed, but I once worked through it all, and it seemed right to me. Take it as a starting point, but not Gospel. Note that the more general case considered here includes the special case vacuum as one of the two materials (n=1).

 

[vanhees71]

Sure, but if you define it that way, it's not equal to c/v any more.

The microscopic way to look at this is that if the dielectric is at rest, the atoms that make up the dielectric are little round balls. Boost and they become ellipsoidal. Of course the dielectric constant - or if you prefer, the effective dielectric constant, c/v - becomes anisotropic. The medium itself does.

Again, the dielectric constant is a scalar, because it's defined in the rest frame of the medium. You solve the Maxwell equations in the medium, and then you Lorentz boost the plane-wave solutions to any other frame, where the medium moves with some velocity $\vec{v}$ (or four-velocity $u^{\mu}$). The electromagnetic field builds an antisymmetric four-tensor field in the medium as well as in the vacuum.

What you refer to is the fact that in manifestly covariant form the solution depends on $u^{\mu}$. Sometimes you find the pretty misleading statement that in in-medium (quantum) field theory Lorentz invariance is broken, but this is only true if you formulate everything in the preferred frame, i.e., the rest frame of the medium and don't write it in manifestly covariant form using the four-velocity of the medium explicitly.

This takes the effect you referring to into account. In the most simple form the constitutive equations for a dielectricum read in covariant form
$$H^{\mu \nu} u_{\nu}=\epsilon F^{\mu \nu} u_{\nu}, \quad ^{\dagger}F^{\mu \nu} u_{\nu} =\mu ^{\dagger} H^{\mu \nu} u_{\nu}.$$
In 3D formulation this reads
$$\vec{D}+\frac{\vec{v}}{c} \times \vec{H}=\epsilon \left (\vec{E}+ \frac{\vec{v}}{c} \times \vec{B} \right), \quad \vec{B}-\frac{\vec{v}}{c} \times \vec{E} = \mu \left (\vec{H}-\frac{\vec{v}}{c} \times \vec{D} \right).$$
This includes all the "anisotropy" of the constitutive relations in terms of the four-velocity $u^{\mu}$ of the medium (or $\vec{v}=\vec{u}/u^0$ in the (1+3)-formulation).

 

[Ibix]

So, to summarise for @Chan Pok Fung, the calculation in the OP is incorrect because it uses length contraction and time dilation when it should have used either the Lorentz transforms or the relativistic velocity addition equation. Also, although you can calculate the velocity of light in an arbitrary frame by this method, the concept of "the" refractive index of a moving medium is mistaken. You must either define refractive index in the medium's rest frame, or deal with the fact that the medium is anisotropic and the speed of light varies with direction.

Also, it is more helpful to link to the arxiv landing page (https://arxiv.org/abs/physics/0408005) rather than the PDF, since that page includes metadata such as publication history and citations, and makes looking up the authors easier.

 

[Vanadium 50]

Again, the dielectric constant is a scalar, because it's defined in the rest frame of the medium.

I'm going to push back on this. For air or water, sure. For calcite or even ice, it is not. Calcite (ice too) has different indices of refraction in different directions even at rest. I don't think it's helpful to adopt a convention that doesn't even work pre-relativity, at least not for all materials. It's not a good starting point.

To go a bit more beyond B-level for a moment, you call ε a scalar, but it's really not. It links components of D with components of E, so (in 3-space) it's a tensor: D_j =ε^ij E_i. Requiring ε to be a scalar invokes a symmetry that isn't there - at least not for many materials.

So I think the best answer is "the speed of light is not isotropic (the OP says "constant" but means "isotropic") for many materials even at rest". Why should this be any different when they are in motion?

 

[Mayan Fung]

Thank you all for the comments. Some of them are out of the scope of my original problem, yet they are very educational. I didn't intend to put much attention to the isotropy of a dielectric under relativistic conditions. My original attempt is just to study a light beam with a normal incidence in a medium moving at a relativistic velocity parallel to the light, which simplifies the situation into a 1D case. I shall adopt the suggestions by @Ibix and try to spot my mistakes by Lorentz transform. Thanks all!

 

[vanhees71]

I'm going to push back on this. For air or water, sure. For calcite or even ice, it is not. Calcite (ice too) has different indices of refraction in different directions even at rest. I don't think it's helpful to adopt a convention that doesn't even work pre-relativity, at least not for all materials. It's not a good starting point.

To go a bit more beyond B-level for a moment, you call ε a scalar, but it's really not. It links components of D with components of E, so (in 3-space) it's a tensor: D_j =ε^ij E_i. Requiring ε to be a scalar invokes a symmetry that isn't there - at least not for many materials.

So I think the best answer is "the speed of light is not isotropic (the OP says "constant" but means "isotropic") for many materials even at rest". Why should this be any different when they are in motion?

Sure, I was talking about homogeneous media. For "crystal optics" you need a dielectric tensor, which again is defined in the restframe of the medium (crystal) and as such doesn't change under Lorentz transformations. You can solve the wave equation in the medium rest frame and then Lorentz boost the solution to any frame, where the medium is moving (with constant velocity). The solution of course depends on the velocity $\vec{v}=c \vec{\beta}$ of the medium (or rather the normalized four-velocity $u_m^{\mu}$, $u_{\mu} u^{\mu}=\gamma(1,\vec{\beta})$).