Lorentz Transformation of Electric & Magnetic Fields Visualized

[SiennaTheGr8]

TL;DR Summary

I made a tool for visualizing how electric and magnetic fields transform under a Lorentz boost.

I made a tool for calculating and visualizing how the electric and magnetic fields transform under a Lorentz boost. Thought I'd share it here, in case anyone finds it interesting.

https://em-transforms.vercel.app/

 

[Sagittarius A-Star]

I tried some examples and they were correctly transformed.

 

[robphy]

Nice.

It might be interesting to optionally show the components of E and B,
that are parallel and perpendicular to the boost velocity.

 

[SiennaTheGr8]

Nice.

It might be interesting to optionally show the components of E and B,
that are parallel and perpendicular to the boost velocity.

Good idea. Couple of other possibilities I'm considering:

- Poynting vector
- Lorentz force for a particle whose charge and unprimed velocity can be specified by the user

 

[robphy]

In the component form,
https://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity#The_E_and_B_fields
the field-components along axes perpendicular to the boost direction have a familiar-looking transformation.

In addition, it might be helpful to see both as vector fields (not just a single pair of vectors) being transformed,
possibly controlled by a slider... so that one could visualize a smooth transition,
rather than just the start and end configurations.

 

[vanhees71]

What's also a great insight is that the electromagnetic field components establish the representation of the proper orthochronous Lorentz group as $\mathrm{SO}(3,\mathbb{C})$, i.e., the complex $\mathbb{C}^{2\times 2}$ matrices that keep the bilinear form $\vec{x} \cdot \vec{y}=x_1 y_1 +x_2 y_2 +x_3 y_3$ invariant, where $\vec{x},\vec{y} \in \mathbb{C}^3$. Note that this of course does NOT define a scalar product on $\mathbb{C}^3$, which must be defined as a sesquilinear form!

That can be seen by introducing the Riemann-Silberstein field,
$$\vec{F}=\vec{E}+\mathrm{i} \vec{B}.$$
The rotations are of course represented by the usual real rotation group $\mathrm{SO}(3)$ (indeed the elctric and magnetic field components transform as vectors under rotations), while the boosts are represented by rotation matrices with purely imaginary angles $\mathrm{i} \eta$, where $\eta$ is the rapidity of the boost.

 

[Dale]

I wonder if there is a good graphical way to represent a tensor. I mean, a vector is graphically represented as an arrow, but a tensor is a different thing and it would be nice to represent the EM tensor directly as a tensor.

 

[vanhees71]

For one-forms you find a nice picture in MTW (Fig. 2.4 on p. 55).

 

[robphy]

Along the lines of MTW's visualizations of "directed quantities" (based on Schouten and further developed by Burke),
I've been trying to visualize the EM-tensor in spacetime as a pair of bivectors (ideally, a pair of one-forms).
Somehow, computational-graphically, the next step is to extract the spatial quantities (a one-form and two-form in space) based on a chosen 4-velocity. Then [using the Euclidean spatial metric and 3-volume-form] obtain the Electric field vector and the Magnetic field pseudovector.

If done correctly, this should agree with the transformation formulae.

 

[PeterDonis]

For one-forms you find a nice picture in MTW (Fig. 2.4 on p. 55).

There are also pictures in MTW of the Faraday tensor and its dual (which they call the Maxwell tensor) in Chapter 4 (Figures 4.4 and 4.5).

 

[robphy]

Here's are 3 of Maxwell's Equations as spatial [twisted-] differential forms from Burke's Applied Differential Geometry

 

[vanhees71]

Along the lines of MTW's visualizations of "directed quantities" (based on Schouten and further developed by Burke),
I've been trying to visualize the EM-tensor in spacetime as a pair of bivectors (ideally, a pair of one-forms).
Somehow, computational-graphically, the next step is to extract the spatial quantities (a one-form and two-form in space) based on a chosen 4-velocity. Then [using the Euclidean spatial metric and 3-volume-form] obtain the Electric field vector and the Magnetic field pseudovector.

If done correctly, this should agree with the transformation formulae.

I guess this is somehow equivalent to the calculation starting with (4.2.34) in

https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf

 

[robphy]

I guess this is somehow equivalent to the calculation starting with (4.2.34) in

https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf

Yes.
But the calculation I seek must be via geometrical construction first,
and, for that approach to be correct, it must agree with the algebraic calculation.In other words,
I am going to (for example)

• use a geometrical construction with a "circle" to map a 1-form to a vector, and this mapping should be reversible [assuming a nondegenerate metric].
• use a two-form and a vector and construct a one-form from it... and that one-form should yield zero contraction with that vector.
• of course, everything above should be multilinear

I don't want to just do the algebraic calculation, then make a picture of it.

 

[SiennaTheGr8]

I've added the option to show component-vectors parallel and perpendicular to the boost-velocity, and also the option to show Poynting vectors.

Oh, and fixed a bug that nobody seems to have noticed (or at least mentioned): I had a sign-error that was causing boosts in the negative x-direction to give exactly the same results as boosts in the positive x-direction.

 

[SiennaTheGr8]

Oh, and fixed a bug that nobody seems to have noticed (or at least mentioned): I had a sign-error that was causing boosts in the negative x-direction to give exactly the same results as boosts in the positive x-direction.

It was actually a little tricky to figure out whether this was indeed a bug. Since I've never seen a similar visualization, my intuition wasn't quite there. What convinced me was configuring E = [0, 1, 0] and B = [0, 0, 1], which is like a light-wave moving in the positive x-direction, and reasoning that the fields should shrink when boosting "with" the wave and grow when boosting "against" it (they were shrinking both ways before).

 

[SiennaTheGr8]

I made some updates:

• You can now specify the mass, charge, and velocity of a particle co-located with the field vectors.
• You can optionally have the visualization display the particle's velocity-vector, as well as the Lorentz force acting on it and the resulting acceleration vector.
• You can hide the boost-velocity and the boosted quantities (helpful when you're interested in focusing on the particle's behavior in one frame).
• The velocities (boost and particle) are now entered in spherical components and can point in arbitrary directions (before, the boost-velocity was limited to the x-axis).
• I've added a few preset field-configurations. Would be interested in ideas for others to include.

Haven't really confirmed that the particle's dynamics are getting calculated correctly. Passes the eye-test, but I can't rule out bugs.

Link: https://em-transforms.vercel.app/

 

[SiennaTheGr8]

possibly controlled by a slider... so that one could visualize a smooth transition,
rather than just the start and end configurations.

No slider, but I've added hotkeys, so now you can control the vectors without having to scroll down to the input-boxes. Also added some text areas at the top, including one that explains the hotkeys.

 

[SiennaTheGr8]

I made a couple of updates on this, in case anyone's interested:

• You can now hide the field-vectors (and all quantities derived from them). If you do that, then the only displayable quantities are the boost velocity, the "unprimed" particle-velocity, and the "primed" particle-velocity, effectively turning the app into a "velocity-addition visualizer": https://em-transforms.vercel.app/?e...=false&showA=false&hideV=false&hideEandB=true
• The app's state now saves to the URL as you fiddle with the options and controls, so that you can share or bookmark configurations of interest (like I did with the link above). The camera-state doesn't save, but everything else does.

https://em-transforms.vercel.app/

 

[SiennaTheGr8]

The camera-state is now encoded in the URL, too (this was a little tricky!): https://em-transforms.vercel.app/?a...false&showA=false&hideV=false&hideEandB=false