Special relativity · Electrodynamics §6
The field equations
keep their form.
How do the Maxwell-Hertz equations keep their form under the transformation, and what must the electric and magnetic fields do?
Static worked example (algebra) and ideal model, host calculation (residuals)
The field equations keep their form
Carry Maxwell's equations into the moving frame. They keep the same form, provided the electric and magnetic fields transform together.
Section 6 substitutes the chain-rule operators from the §3 map, groups terms, and identifies the combinations that play the role of the transformed fields. In paper 3, X, Y, Z are electric components and L, M, N are magnetic components. Einstein's β is the modern γ. The algebra is a static worked chain; the residuals are a live host calculation on an admitted plane wave.
Select one of the six Maxwell-Hertz equations and step it. The printed layer is Gaussian, as in the 1905 paper. The modern layer is SI; the conversion is labeled, never silent. Validation uses closed-form derivatives, not finite differences. |v| ≥ c is refused.
Form invariance of the homogeneous equations does not by itself prove every physical identification of the fields. The later layer says so and stops. L and N here are magnetic components, not the speed of light and not Avogadro's number.
Unit layer: Printed Gaussian (Annalen 1905). X, Y, Z electric; L, M, N magnetic.
| Printed | Role in this paper | Modern SI (labeled conversion) |
|---|---|---|
| X | electric x in K | E_x |
| Y | electric y in K | E_y |
| Z | electric z in K | E_z |
| L | magnetic x in K (not the speed of light) | B_x |
| M | magnetic y in K | B_y |
| N | magnetic z in K (not Avogadro's number) | B_z |
K is the stationary system. τ is the moving-frame time, not the modern proper time. Einstein's β is the modern γ.
| Form invariant? | yes |
|---|---|
| Max residual | 0.000000e+0 |
| Faraday x, y, z | 0.000000e+0; 0.000000e+0; 0.000000e+0 |
| Ampere-Maxwell x, y, z | 0.000000e+0; 0.000000e+0; 0.000000e+0 |
| Amplitude factor γ(1 − β) for a +x wave | 5.000000e-1 |
| Frequency factor | 5.000000e-1 |
Residuals are an oracle on an admitted analytic wave. They are not the section 6 argument. The algebra above is a static worked example.
Show the validation code
Residuals come from maxwellResidualsPlaneWave and transformSI in the host evaluator src/physics/reference/fields.ts. The derivation steps are an authored static chain, not that function.
Same scientific action without the display equation
Choose an equation from the list, advance or go back a named step, and read the residual table. Ask whether the transformed equations still have the Maxwell-Hertz form.
Not modeled: sources and currents (SR-12); media; boundary conditions; radiation reaction; field configurations other than the admitted analytic validation waves.
Worked case (readable without JavaScript)
The printed layer is Gaussian, as in Annalen 1905. X, Y, Z are electric-field components. L, M, N are magnetic-field components. Einstein's β is the modern γ. V is the speed of light in this paper, not L.
After the section 3 chain rule and grouping, the same form holds in the moving frame for the combinations X, β(Y − (v/V)N), β(Z + (v/V)M) and L, β(M + (v/V)Z), β(N − (v/V)Y). A plane wave along +x at v = 0.6c has transformed amplitude and frequency factors γ(1 − β) = 1/2.