Laboratory · Historical model comparison · Explanatory preview
Does the wave equation keep its form?
Apply two coordinate substitutions to the same forward-travelling plane wave and compare the analytic residuals.
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.
These are host reference calculations with modern SI calibration and illustrative settings. They are not historical measurements, a reviewed historical dataset, or publication of the strict 1904 shelf. No FrankenSim WASM calculation is claimed.
What is being calculated?
The diagnostic is the scalar forward-travelling wave cos(k(x − ct)) and the vacuum operator ∂²/∂x² − c⁻²∂²/∂t². The six-component Maxwell–Hertz field equations are not modeled: this residual belongs to the scalar wave alone. The largest absolute residual is the analytic amplitude over phase, not a sampled numerical maximum.
Hold the frame speed fixed and double k. The absolute residual changes while the normalized one does not. Then set the frame speed to zero. A preserved wave operator alone does not fix the physical meaning or normalization of all transformed fields.
Continue to the Maxwell–Hertz field-equation laboratory · Construct the coordinate map from its constraints
Prediction is not a measurement
No observed points, digitized fringe shifts or experimental confidence bounds have been added to these plots. A reviewed source-specific dataset, with its geometry, wavelength, protocol and uncertainty, is still needed before making a numerical comparison with a historical experiment.
The reference implementation is src/physics/reference/shelfOptics.ts. Its functions own the arm times, fringe shifts, drag speeds and wave residuals; the interface only projects their results. Numerical precision in a table does not imply measurement accuracy.
This comparison belongs to the special-relativity discovery route.