Annus Mirabilis · Interactive critical edition in preparation
Wave equations under a change of coordinates
Compare coordinate substitutions on the same scalar wave.
Modern SI calibration and illustrative settings. This is not a verified historical dataset or a strict 1904-mode instrument. No FrankenSim WASM execution is claimed.
Change one setting and compare the consequences
Static worked exampleBar range: −0.19 to +0.19. The center line is zero. The scale is recalculated after Apply.
Accepted calculation 0. Every model row shares these inputs.
Inputs used for these results
- Signed frame speed divided by c
- 0.1
- Angular wavenumber k
- 1 rad/m
| Observable | Galilean substitution | Lorentz substitution |
|---|---|---|
| Residual normalized by k² | 0.19 | 0 |
| Maximum absolute residual (1/m²) | 0.19 | 0 |
| Mixed-derivative coefficient (s/m) | 6.671 × 10⁻¹⁰ | 0 |
This checks the same forward-travelling plane wave after each coordinate substitution, not a measured wave or a grid simulation. At zero relative speed both residuals vanish. The Lorentz zero is an analytic identity under the stated map; it does not independently prove a field transformation or select an interpretation of space and time. Reversing the frame velocity while holding the forward wave fixed is a different relative configuration.
A light wave obeys a single wave equation. Describe the same wave from a moving frame with the everyday change of coordinates and the equation no longer holds in its old form; use the change of coordinates Einstein derived, and it does.
Take a wave travelling at c along x, φ = cos(k(x − ct)), which satisfies ∂2φ/∂x2 − (1/c2)∂2φ/∂t2 = 0. Describe it from a frame moving at v. With the Galilean substitution x′ = x − vt, t′ = t, the wave no longer satisfies an equation of the old form in the new coordinates: it leaves a residual of relative size |β(2 − β)|, 0.19 at β = 0.1, and the correct operator there is a different one, (1 − β2)∂2/∂x′2 + (2v/c2)∂2/∂x′∂t′ − (1/c2)∂2/∂t′2. With the transformation of §3 of the relativity paper the equation keeps its form and the residual is 0 at every speed. Section 6 shows the same for the full Maxwell–Hertz equations of empty space, with the fields transforming as well; this instrument checks only the scalar wave operator.
In the moving frame x = x′ + vt′ and t = t′, so x − ct = x′ − (c − v)t′, and the wave is cos(k(x′ − (c − v)t′)): it moves at c − v. Differentiating twice in x′ gives −k2φ, and twice in t′ gives −k2(c − v)2φ. The old-form operator then gives −k2φ + (k2(c − v)2/c2)φ = −k2φ[1 − (1 − β)2] = −k2φβ(2 − β). At β = 0.1, β(2 − β) = 0.1 × 1.9 = 0.19, and since φ is at most 1 the largest residual is 0.19k2. With the relativity transformation, x = γ(x′ + vt′) and t = γ(t′ + vx′/c2), so x − ct = γ(1 − β)(x′ − ct′). The wave is cos(k′(x′ − ct′)) with k′ = γ(1 − β)k: it still moves at c, and the old-form operator gives −k′2φ + (k′2c2/c2)φ = 0. The wavenumber changes, by the Doppler factor γ(1 − β) = √((1 − β)/(1 + β)) = 0.905 at β = 0.1, but the equation does not.
Voigt noticed in 1887 that a transformation of this kind leaves the wave equation unchanged, and Lorentz and Poincaré worked with the full set for electrodynamics by 1904 and 1905. Einstein's §6 applies his transformation to the Maxwell–Hertz equations for empty space and finds them unchanged in form, and §7 draws the Doppler and aberration formulas from that. Before 1905 the invariance was a property of the equations; in the paper it follows from two principles about measurement.
The explanation
Full explanation
Change the frame speed, then change only the wavenumber. Distinguish the dimensional residual from the normalized residual so a scale change is not mistaken for a better transformation.
Show every step of the investigation
Inspect the mixed-derivative term and both residuals for Galilean and Lorentz substitutions. This scalar diagnostic is not the complete Maxwell-Hertz field derivation. Continue to the full reading for the field transformations.
An explanatory model, not an observation of nature. This embed starts from the laboratory’s worked defaults, not a saved run. Presentation options change the surrounding guide, never the numerical inputs.