Lorentz Electromagnetic Field Invariants

Worked Example

Let's calculate the transformed fields and invariant for a specific case.

1. Assume normalized units c = 1.
2. Let E_y = 1 and B_z = 2. The frame moves at β = 0.5 along the x-axis.
3. Calculate the Lorentz factor: γ = 1 / √(1 - 0.5²) = 1 / √(0.75) ≈ 1.154700538.
4. Calculate the transformed electric field: E'_y = γ(E_y - β B_z) = 1.154700538(1 - 0.5 × 2) = 0.0.
5. Calculate the transformed magnetic field: B'_z = γ(B_z - β E_y) = 1.154700538(2 - 0.5 × 1) ≈ 1.732050807.
6. Calculate the invariant in the original frame: B² - E² = 2² - 1² = 4 - 1 = 3.
7. Calculate the invariant in the transformed frame: (B'_z)² - (E'_y)² = 1.732050807² - 0.0² = 3.

Since |B| > |E|, the field is magnetic-dominated, and there exists a frame (β = E_y / B_z = 1 / 2 = 0.5) where the electric field is completely eliminated (E'_y = 0).

Experiment

You can use the interactive module to explore these transformations and discover the cancellation frames. Follow these steps:

1. Set Electric y to 1 and Magnetic z to 2.
2. Set Boost beta to 0.5.
3. Observe that the Transformed electric y becomes exactly 0, showing that a frame moving at β = E/B cancels the electric field.
4. Change Electric y to 2 and Magnetic z to 1. The field is now electric-dominated.
5. Change Boost beta to 0.5. Now, the Transformed magnetic z becomes 0, because β = B/E cancels the magnetic field.
6. Set both to 1 to observe a null field where no subluminal frame can cancel either field.

Link to the interactive module: Lorentz Electromagnetic Field Invariants

Limitations

This module uses normalized units where the speed of light c = 1. When reintroducing standard SI units, the electric field E and magnetic field B must be scaled by c appropriately (e.g., E and cB have the same units). Additionally, this visualization assumes the fields are uniform in space and constant in time, ignoring wave propagation or localized sources. Null fields (|E| = |B| ≠ 0) have no valid subluminal frame where either field vanishes; such fields characterize electromagnetic plane waves. We explicitly restrict |β| < 1 because transformations are singular at the speed of light.

References

The transformation of electromagnetic fields and the construction of Lorentz invariants are foundational concepts in classical electrodynamics and special relativity. For a detailed derivation and physical discussion, refer to The Feynman Lectures on Physics, Volume II, Chapter 26, which covers Lorentz transformations of the fields.
https://www.feynmanlectures.caltech.edu/II_26.html