Lorentz transformation of electric and magnetic fields

Continue with signature $(+,-,-,-)$, $x^0=ct$, and $F^{0i}=-E_i/c$. For a primed frame moving at $V\hat{\mathbf x}$ relative to the original frame, each upper tensor index receives a boost factor. Unlike a four-vector transformation, the field transformation therefore contains two copies of the Lorentz matrix. Applying the rule to the time-x entry produces two contributions whose coefficient is $\gamma^2(1-\beta^2)$. They combine to leave the longitudinal electric component unchanged; a similar calculation preserves the longitudinal magnetic component.

A transverse entry mixes a time-space component with a space-space component. For example, $F'^{02}=\gamma F^{02}-\gamma\beta F^{12}$ becomes $-\gamma(E_y-VB_z)/c$. Reading the electric field from the same table gives the correct transverse sign. This small derivation is safer than remembering a formula while forgetting whether V denotes observer motion or source motion. The inverse transformation uses negative V, and restores the original field when applied to both transformed fields together.

The magnetic transformation has an additional factor of $c^{-2}$ in the electric contribution. This follows from the two factors of $c^{-1}$ already present: the boost uses $\beta=V/c$, and the tensor's electric entry uses E divided by c. Dropping either factor can make a numerically enormous fictitious magnetic field. Dimensional analysis is decisive: a velocity multiplied by an electric field is not measured in tesla until divided by the square of a speed. The formulas have a smooth low-relative-speed limit but that limit still contains the mixing terms.

The compact transverse relations are $\mathbf E'_\perp=\gamma(\mathbf E+\mathbf V\times\mathbf B)_\perp$ and $\mathbf B'_\perp=\gamma(\mathbf B-\mathbf V\times\mathbf E/c^2)_\perp$. Components parallel to V retain their original values. These are vacuum field transformations between inertial Cartesian frames, regardless of whether local sources are present. They do not by themselves provide constitutive relations for a moving material. In matter one must transform charge and current, and account for polarization and magnetization consistently.

If a frame has only a transverse electric field, a moving observer generally measures a magnetic field as well. That does not mean the observer has created a new physical source. The decomposition of the same electromagnetic tensor has changed. For crossed fields with $\mathbf E\perp\mathbf B$ and $E<cB$, an observer moving at $\mathbf V=\mathbf E\times\mathbf B/B^2$ measures zero electric field. Its speed is subluminal precisely because the original field is magnetically dominated. The remaining magnetic field is smaller than the original one.

A frame change is also different from varying an experimental knob over time. If a source is switched or accelerated, retarded fields and radiation may enter. A Lorentz transformation simply compares two coordinate descriptions of an already specified field at the same event. For a nonuniform field, transform both its component values and the coordinates on which those values depend. Transforming a static spatial profile's components while retaining its old arguments generally fails to describe the field measured by the moving observer.