# Covariant Description of Light Scattering at a fastly rotating Cylinder

Let us consider the following Gedankenexperiment:

A cylinder rotates symmetric around the $z$ axis with angular velocity $\Omega$ and a plane wave with $\mathbf{E}\text{, }\mathbf{B} \propto e^{\mathrm{i}\left(kx - \omega t \right)}$ gets scattered by it.

We assume to know the isotropic permittivity $\epsilon(\omega)$ and permeability $\mu(\omega)$ of the cylinder's material at rest. Furthermore, the cylinder is infinitely long in $z$-direction.

The static problem ($\Omega = 0$) can be treated in terms of Mie Theory - here, however, one will need a covariant description of the system for very fast rotations (which are assumed to be possible) causing nontrivial transformations of $\epsilon$ and $\mu$.

Hence my question:

### What is the scattering response to a plane wave on a fastly rotating cylinder?

RotatingDisc http://www.personal.uni-jena.de/~p3firo/PhysicsSE/RotatingDisc.png

• What is "infinite" on that cylinder? Commented Jan 29, 2011 at 13:35
• Thank you @Georg for pointing out to the misleading formulation. I mean infinitely in $z$-direction. I will change it in a second :) Greets Commented Jan 29, 2011 at 13:38
• @Carl: You might consider that there are still some things in classical electrodynamics which are somehow basic but not standard homework problems. To my mind, the covariant description of electrodynamics in media belongs to this class. Greets Commented Jan 30, 2011 at 12:24
• Robert, it might be useful to begin with the case for light impinging on a moving half-infinite media (i.e. infinite plane dividing space into two different materials). That case solves trivially (just boost the case for non moving material), and can be summed up (I think) to give a limiting case for the rotating cylinder (in the limit of small wave length). But it's been 30 years since I took E&M and it was never my "best" subject. Commented Jan 30, 2011 at 23:51
• @Carl: Thank you for the hint. The difference to the reflection problem at a half-space is the rotational character of the system. One attempt to solve the problem is to go into a co-rotating coordinate system and transform the plane wave accordingly - In this case I am not sure if such a framework is physically correct. The other way would be to just covariantly transform the medium - this is much more general since we would learn about the special relativistic relation of $\epsilon$ and $\mu$. Greets Commented Jan 31, 2011 at 9:13

This is a very interesting problem. First of all, it can be shown that every component of the electromagnetic tensor can be written in terms of $F^{03}$. From axial symmetry, the solution will be $$\frac{f(r)}{\sqrt{r}}e^{i(m\phi - \omega t)}$$ It turns out that, when $$0<\omega<m\Omega,$$ where $\Omega$ is the angular velocity of the cylinder in the laboratory's frame, the reflection coefficient is greater than 1! This phenomenon is called Superradiance.
• Can you at least solve the problem analytically for some special cases where still $\Omega\neq 0$? Commented Mar 18, 2012 at 15:10
• Thank you @akmeteli for your input. I however think that the determination of the properties of the rotating cylinder is at the core of this problem - how do $\epsilon$ and $\mu$ transform? Greets Commented Apr 22, 2012 at 10:06