Electromagnetism for Media in Arbitrary Motion

[Cryo]

Hello

To develop one interesting idea I need to be able to do calculations on (1) scattering of light from bodies in arbitrary motion, possibly at relativistic speeds; (2) Propagation of light in electromagnetic media that are in arbitrary motion (possibly relativistic). For example, I would like to know how to tackle the problem of light scattering on an accelerating dielectric sphere of radius larger than the wavelength of light (assuming that acceleration is such that sphere reaches relativistic speed (say $0.9c$) within one period of the incident wave).

Can anyone please suggest some literature on this topic? I did not find it in Jackson, Lifshitz&Landau, Stratton...

My background: fairly confident with special relativity, somewhat familiar with general relativity, comfortable with classical electromagnetism, including scattering and radiation problems.

Thank you!

 

[Paul Colby]

In scattering problems one usually transforms to the rest frame of the scattering body, solves the problem then transforms back. In the rest frame of the sphere it's just the Mie series solution for a dielectric sphere. The transforming the fields, just use the Lorentz transformation.

 

[Cryo]

Thank you. But is Lorentz transformation the correct thing to use here? Let's say I want to look at object under uniform acceleration. The object has finite size.

My first problem is how to describe such object sensibly. Will all of its constituents simply follow the hyperbolas of uniform acceleration? Let's say the lab-frame coordinates are conventional Cartesian coords: $ct, x, y, z$, and the square of the interval length is $ds^2=c^2dt^2-dx^2-dy^2-dz^2$. If I intoduce another coordinate system (e.g.) with coordinates $\mu, x, y, \alpha$, related to normal coordinates through:

$ct=\alpha \sinh\left(\frac{\mu}{\alpha}\right)$
$z=\alpha \cosh\left(\frac{\mu}{\alpha}\right)$

Where lines of constant $\alpha$ are hyperbolas. Can I then obtain the world-line of the small 'bit' of the object by the following procedure? If that 'bit' is at rest in the lab frame at time $t_1$ and, at that time, is at the position $\vec{r}=(x_1\,y_1\,z_1)$, which corresponds to $\alpha_1=\frac{z_1}{\cosh\left(\mbox{atanh}(c t_1/z_1)\right)}$ and $\mu_1=\alpha_1 \mbox{atanh}\left(c t_1/z_1\right)$. Can I assume that the world-line of that bit is simply

$(\mu, x, y, \alpha)^\mu=(c\tau_1+\mu_1, x_1, y_1, \alpha_1)^\mu$, where $\tau_1$ is the proper time of that bit, and $\tau_1=0$ corresponds to $t=t_1$ ? My hunch is that it may not be so simple, and that I may need to know something about the elastic properties of the material from which the object is made.

Lets assume that this description is somewhat correct, let's also assume that the object is small enough (and acceleration is low enough) for it be desribed by single proper time. We then recover something resembling the 'rest frame' in $\mu, x, y, \alpha$-coordinates. You then suggest to represent the incident field in this coordinates and solve the problem in these coorinates. Right?

I then have problems with boundary conditions and constituitive relations. I can assume that the object's response to electromagnetic field is well described by polarization $\vec{P}$ and magnetization $\vec{M}$ (when it is not accelerating). I could then bundle both of them into magnetization-polarization tensor $\mathcal{M}^{\mu\nu}$, I could then postulate a linear relationship between $\mathcal{M}$ and the electromagnetic tensor $F_{\mu\nu}$, and introduce a polarization tensor $\mathcal{P}^{\mu\nu\kappa\sigma}$ such that $\mathcal{M}^{\mu\nu}=\mathcal{P}^{\mu\nu\kappa\sigma}F_{\kappa\sigma}$

Is this the correct way to handle the constituitive equations? After that come the boundary and continuity conditions. I could try to redrive them based on the same approach as always: start with Maxwell Equations, consider the boundaries, assume that fields remain finite at all times (and squeeze the contours around the material boundaries). The Maxwell equations would be:

$\nabla_\mu\left[F^{\mu\nu}-\mu_0\mathcal{P}^{\mu\nu\kappa\sigma}F_{\kappa\sigma}\right]=0$
$\nabla_\mu\left[\frac{1}{2\sqrt{g}}\epsilon^{\mu\nu\kappa\sigma}F_{\kappa\sigma}\right]=0$

where $\epsilon^{\mu\nu\kappa\sigma}$ is the Levi-Civita (relative) tensor and $g$ is the determinant of the metric. Clearly since $\mu, x, y, \alpha$-coordinates are not trivial one will get additional terms due to Levi-Civita connection ( $\Gamma^{\dots}_{\dots , \dots}$ ). So I will have to be careful

----------------------------

Now all of this seems to be a lot of work, with a lot of opportunities to get it wrong. So I was wondering if there is standard literature I can have a look at.

 

[DrGreg]

The simplest model is a Born rigid object, in which case you can use Rindler coordinates
$$\begin{align*}
ct &= Z \sinh \frac{gT}{c} \\
x &= X \\
y &= Y \\
z &= Z \cosh \frac{gT}{c}
\end{align*}$$

 

[Paul Colby]

Thank you. But is Lorentz transformation the correct thing to use here? Let's say I want to look at object under uniform acceleration. The object has finite size.

Well, it's likely to be 90% of the answer sought if not much more. Uniform acceleration doesn't happen without a motive force which itself is (must be?) electromagnetic. Can't think of any others off hand.

Also, I can't see anything wrong with the usual instantaneous rest frame approach beyond body deformations and that would only be a shape change to some order.

 

[Cryo]

Well, it's likely to be 90% of the answer sought if not much more. Uniform acceleration doesn't happen without a motive force which itself is (must be?) electromagnetic. Can't think of any others off hand.

Also, I can't see anything wrong with the usual instantaneous rest frame approach beyond body deformations and that would only be a shape change to some order.

The problem I see with instantaneous rest-frame approach is when it comes solving the Maxwell's equations. If you express these equations via four-potential, you end up with two derivatives, which can lead to terms in your equations that depend on the curvature of the worldline. For hyperbolic motion the curvature is constant and will not vanish no matter how small is the scale of your problem. For example, an accelerated charge will radiate. This is not something you can establish with applications of Lorentz transforms (only) - at some point you will need to solve the wave-equation. Now the radiation may be small, if the acceleration is small, but if you go far enough from the charge you can always get it to dominate over all other effects.

I think, I agree with the gist of your suggestion - to solve the problem in the well-suited coordinates, but I think the right coordinates must be

The simplest model is a Born rigid object, in which case you can use Rindler coordinates

or something similar.

Of course these are just words - work is still ahead :-)

DrGreg and Paul Colby, thank you!

 

[Dale]

Thank you. But is Lorentz transformation the correct thing to use here? Let's say I want to ...

From what you describe I think that you need a proper covariant formulation of Maxwell’s macroscopic equations. Here is a good starting point, but for a project like this I would recommend going beyond Wiki and at least looking at the references.

https://en.m.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism

 

[Paul Colby]

Well, you could solve for the radiation of every charge making up the object using

https://en.wikipedia.org/wiki/Liénard–Wiechert_potential

and simply add the Compton Scattering of the ambient radiation form each charge

https://en.wikipedia.org/wiki/Compton_scattering

to that solution. Or is the momentum transfer of the illuminating radiation enough to ionize the object? Of course you'll need to sum the multiple scattering series. Good luck.

 

[Cryo]

Well, you could solve for the radiation of every charge making up the object using

My example of radiation from an ionized charge was simply to illustrate what I think could be a problem with relying to much on Lorentz transforms straight away. Lorentz transforms will eventually creep in, I am sure, but I think it is better for them to appear naturally, as a result of taking a limit at some point (so that it is clear what is the assumption behind this limit).

In any case, thank you very much for your help. I will now need to find some time to tackle this problem properly.