TL;DR

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/

robphy said:

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

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.

Last edited: Sep 20, 2022

vanhees71 said:

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).

Last edited: Sep 20, 2022

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 said:

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).

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/

robphy said:

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.

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/

A few updates:

• Dark mode.
• Sliders (originally suggested by @robphy), though the input-boxes are still there for "custom" values.
• On desktop, the options and controls now go side-by-side with the visualization, so that it's easier to adjust the parameters while viewing the results.

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

Sagittarius A-Star said:

In the following example, for E' the graphics shows y'=2 and the the number field "Boosted electric field" y'=3.96.

https://em-transforms.vercel.app/?e...false&showA=false&hideV=false&hideEandB=false

I also wanted to thank you for leaving this comment. I'm guessing that you simply weren't aware that the camera could be controlled.

I've just made some changes to make it more apparent to the user that the visualization is 3D and comes with camera controls. Previously the instructions for controlling the camera were "buried" in the expandable "Instructions" panel. Now they're in their own top-level panel.

SiennaTheGr8 said:

I also wanted to thank you for leaving this comment. I'm guessing that you simply weren't aware that the camera could be controlled.

I've just made some changes to make it more apparent to the user that the visualization is 3D and comes with camera controls. Previously the instructions for controlling the camera were "buried" in the expandable "Instructions" panel. Now they're in their own top-level panel.

Yes, I am now aware. The following screenshot shows a rotation by 90° around the y-axis (the x-axis is perpendicular to the paper-plane). Only the numbers at the y-axis cannot be read from this viewpoint.

I've made some more improvements to this app, including:

• Made the two-column desktop layout more convenient: the visualization remains "fixed" on the left (at full-screen height) as you scroll the rest of the page-contents on the right. This way you can read all the content and interact with all the controls without losing sight of the visualization.
• Switched to the "physics" convention for spherical coordinates (previously was just using the software's default, which was idiosyncratic—neither the "physics" convention nor the standard "math" convention).
• All labels in the visualization now always "face" the camera (resolves an annoyance that @Sagittarius A-Star pointed to in the previous comment).
• Added a few new options, including displaying the field invariants in a corner of the visualization (partly a sanity check for the transformations, since the unprimed and primed values are calculated independently).

Speaking of the invariants, I've been trying to think of a way to visualize them for the unprimed and primed vector-pairs, so that the user could "see" them staying the same for each pair (in the existing visualization, I mean). Can't think of anything practical so far. Any ideas? Maybe there just isn't a good way.

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

pervect said:

Could you add a brief discussion of the units you use to the instructions/descriptions? Or which unit systems are compatible with it? I assume it's not SI because there are no annoying factors of c.

Heaviside–Lorentz units with c=1. They're mentioned there somewhere I think, but I should make it more prominent.