# Maxwell's Equations in a Rotating Frame of Reference: How to transform E and B Fields?

I know there is allready a question on the topic, here, however the answer to this question deals with GRT, and I want to keep the level of the question basic (however enough to deal with thing like "Maxwell's Equations".

Of what I know, Maxwell's Equations are not invariant when changing to a rotating frame of reference. I tried to work out how they will turn out: My idea was to simply calculate things like \begin{align} \nabla \times E \end{align} in the rotating frame. Before I can do that, I have to define how the Fields $$\vec{E}$$ and $$\vec{B}$$ will look like, so I need the transformation laws. My naive approach here was to define a time dependent rotation-matrix $$R_t$$ which turns points around the z-axxis, and then define: \begin{align} \tilde{\vec{E}}(\vec{x}) = R_t \vec{E}(R_{-t}\vec{x}) \end{align} Here $$R_{-t}$$ indicates the reverse matrix. At first this seemed like a reasonable approach to me, but now I thought about that the same method would yield wrong results for the case of changing to a uniformly moving frame of reference.

So - What is the general way of transforming electric and magnetic Fields to a rotating frame? Is this even a well defined operation? Is it possible to give an answer to this question that doesn't rely on the mathematics of General Relativity Theory?

• Map $\vec{x}\mapsto R\vec{x}$ and $\vec{\nabla}\mapsto R\,\vec{\nabla}$ into the Maxwell equations and see which "new" equations you get? – Eli Jan 24 '19 at 9:33
• @Eli This is what I wanted to do. But this alone is not sufficient, since not only $\vec{x}$ and $\vec{\nabla}$ do transform. $\vec{E}$ and $\vec{B}$ do transform as well. Proceding the way you suggested would ignore this. – Quantumwhisp Jan 24 '19 at 14:59

This has occasionally come up in e.g. modeling pulsar magnetic fields. One reference is Paul Arendt’s thesis. In that form, one chooses to define time by the observer on the axis of rotation; other choices are possible.

it is difficult to define the electric and magnetic fields in this frame. This situation worsens with distance from the axis of rotation, and becomes critical at the ‘light cylinder’ distance, $$r_L$$ = 1/Ω, where Ω is the angular frequency of rotation (here and throughout, we use units where c = 1). This situation will be shown to be an essential feature of all such frames whose metric tensor gμν has off-diagonal elements. For this reason, all electromagnetic quantities used here will be given careful definitions, with reference to their values in an inertial non-rotating frame (where the ambiguities disappear).

It is pointed out in Misner Thorne Wheeler Gravitation (Part II.6) that it is not possible to define a global reference frame for accelerating observers. For example, the temporal axis of a rest-frame for an observer is always tangent to the world-line of this observer. What happens if the observer is accelerating? The world-line is no longer straight and its tangent changes at different points in observer's history. So it becomes impossible to define a single direction for the time-axis. Why does this matter? Because this affects in which 'direction' you differentiate when you take a derivative with respect to time.

Having said this, it is still possible to define a reference frame locally, also you can use the concept of instantaneous reference frame, but do not expect your Maxwell's equations to take a simple form.

If I wanted to work it out, I would go back to the EM-lagrangian density, assume a suitable coordinate system (is there such a thing as rotating coordinates), write the Lagrangian density in this coordinate system, then extract the new equations of motion.

If you use the formulation in terms of differential forms, it becomes immediately clear how everything should transform under coordinate transformations (if you know how differential forms transform).

The point is that if you start with and stick to Cartesian coordinates on $$\mathbb R^{3}$$, then you don't really get into trouble if you treat everything with 3 coordinates as a vector field. If you are more precise, neither the electric field nor the magnetic field are vector fields, the first is a 1-form

$$E = E_xdx + E_ydy + E_zdz,$$

the last is a 2-form

$$B = B_xdydz + B_ydxdz + B_zdxdy$$

and in the Maxwell equations both they and their Hodge duals appear. Even better, they can be combined into a 3-form on Minkowski space $$\mathbb R^{1,3}$$, called the Faraday tensor

$$F = B + E\wedge dt$$

in which formulation the special covariance becomes manifest.

In any case, of these quantities the transformations under general diffeomorphisms are unambiguously defined. All this can be found in beautiful detail in chapter 5 of Baez and Muniain's Gauge Fields, Knots and Gravity.