The Speed of Light
Wave speed in any medium is set by its stiffness and its density. Light is no exception, and the derivation is Maxwell's.
Draft. This memo is a working draft, published for scrutiny rather than as a settled result. Figures, derivations and conclusions are open to revision, and the ledger at the foot records what is derived, what is measured, what is assumed and what remains unanswered. Where a number is carried from a secondary source and not verified against the original, the sources section says so.
The TOE series supersedes the earlier A Classical Aether Model and Governor Atom Model white papers one memo at a time. Where this memo and those papers disagree, this memo is the later position.
Proven in this memo
- The ratio of shear modulus to density is fixed exactly by the measured speed of light, with no freedom remaining.
What is not fixed is either quantity on its own. Splitting the ratio into two numbers requires the void density assumed in memo 1.1, and every absolute value in Part 1 inherits that assumption.
In plain terms
How fast a wave travels through a material is not a property of the wave. It is a property of the material — how stiff it is and how heavy it is. Sound crosses steel at five kilometres a second and air at a third of a kilometre, and the difference is entirely in the steel and the air.
On this account the speed of light works the same way. It is not a rule imposed on the universe from outside. It is a reading of how stiff the Aether is and how much it weighs, in the same sense that the speed of sound in steel is a reading about steel.
Maxwell got there first, in 1862. He built a mechanical model of the medium, worked out its wave speed from its stiffness and density, and found the answer sat within about one per cent of the measured speed of light. That was what convinced him light is an electromagnetic disturbance. The derivation in this memo is his; what is new is only the identification of the stiffness with a specific lattice.
1. The numbers
- Speed of light: 299,792,458 m/s, a defined constant since 1983, not a measured one.
- Shear modulus of the Aether: 8.089 × 1016 Pa.
- Maxwell's 1862 comparison: 3.107 × 108 from electrical units, against 3.149 × 108 measured for light — about 1%.
- Light in air is slower than in vacuum by 0.029%, which this framework reads as air holding slightly more Aether.
2. Wave speed is a property of the medium
For any elastic medium the speed of a transverse wave is set by two quantities and nothing else:
G is the shear modulus — how hard the material resists being deformed sideways. ρ is its density. Stiffer means faster; heavier means slower. The same relation gives the speed of sound in steel, the speed of seismic S-waves through the mantle, and the speed of a wave on a guitar string.
| Medium | Transverse wave speed |
|---|---|
| Rubber | ~30 m/s |
| Water | none — a fluid carries no shear wave |
| Steel | 3,200 m/s |
| Earth's mantle (S-waves) | 4,500–7,000 m/s |
| The Aether | 2.998 × 108 m/s |
3. This is Maxwell's derivation, not a new one
In On Physical Lines of Force, published across 1861 and 1862, Maxwell built an explicit mechanical model of the electromagnetic medium: molecular vortices separated by small particles he identified with electricity. Displacement current — the term that completes his equations and makes electromagnetic waves possible — came out of that machinery.
He then computed the wave speed of his medium the way anyone computes it for an elastic solid: as the square root of rigidity over density. That is equation (1), and it is his.
What clinched it was a numerical coincidence. Weber and Kohlrausch had measured the ratio of electromagnetic to electrostatic units in 1856 and obtained about 3.107 × 108 m/s. Fizeau had measured the speed of light in 1849 and obtained about 3.149 × 108. Within about one per cent of each other, and Maxwell concluded that light must be a disturbance of the same medium.
What changed in 1865, and what did not
Three years later, in A Dynamical Theory of the Electromagnetic Field, Maxwell dropped the vortices and the idle wheels and wrote the field equations directly. Those are the equations still in use.
But he did not drop the medium. His 1878 Encyclopædia Britannica article on the ether argues for it explicitly. What was removed was the scaffolding, not the substance.
And the two constants that replaced his mechanical ones — the permittivity and permeability of free space — are exactly what his density and rigidity became when the machinery was stripped out. Which is why c2 = 1/(μ0ε0) and c2 = G/ρ are the same equation written twice.
4. c is defined, not measured
Since 1983 the metre has been defined as the distance light travels in 1/299,792,458 of a second. The speed of light is therefore a defined constant with no uncertainty: measuring it more precisely now measures the metre, not the speed.
This matters for how equation (1) is read. It cannot be tested by measuring c more accurately. What it can be tested against is variation — whether the ratio G/ρ is the same everywhere and at all times.
ProofEquation (1) is not a free choice
For an isotropic elastic medium the transverse wave speed is √(G/ρ). This is standard continuum mechanics and holds for any such medium.
Memo 1.4 establishes that light is the transverse wave of this medium. Substituting the measured speed of light therefore fixes the ratio G/ρ exactly, with no freedom remaining.
The separate values of G and ρ are not fixed by this — only their ratio. Memo 1.3 pins G by adopting a value for ρ, and that adoption is an assumption, recorded as such. The ratio is forced; the two numbers individually are not.
Chain: standard isotropic elasticity → memo 1.4 (light is transverse) → the measured c. Premise: light is a wave in this medium.
5. Where light does change speed
Light slows in glass, in water, in air. On this account that is not light behaving differently — it is the medium being different, and the refractive index reads the difference directly.
So the refractive index squared gives the local Aether density as a multiple of the void value. Air comes out at 1.0006 — six hundredths of a per cent more Aether than empty space. Water at 1.78. Diamond at 5.84.
That reading is developed in memo 4.6, where the Lorentz–Lorenz relation is shown to make refractometry a direct measurement of the medium for every material ever measured. It is noted here only because it follows immediately from equation (1) and belongs with it.
6. What would falsify this
- Variation in c with direction. A rigid medium has a rest frame, and motion through it should show as anisotropy. Rotating optical resonator experiments bound this below one part in 1017, which is the framework's most serious unanswered problem — memo 5.7.
- Variation in c with time. If the lattice expanded with the universe, G and ρ would both change and c with them. Quasar absorption spectra bound the fine structure constant, which depends on c, to within about one part in 105 out to redshift 3. This rules out a freely expanding lattice and is why memo 1.2 requires the force law to have an equilibrium spacing.
- A refractive index below 1 for a real material at the frequencies where equation (2) is claimed to apply would mean less Aether than the void, which the framework has no mechanism to produce.
7. Sources
Reference codes read source.work.passage and resolve on the Master Source Register, which carries every source used across this series.
- J. C. Maxwell. On Physical Lines of Force, Philosophical Magazine Series 4, vols 21 (1861) and 23 (1862). Part III derives the wave speed of the medium as the square root of rigidity over density and identifies it with light. Equation (1) is his. 79
- Bureau International des Poids et Mesures. The International System of Units (SI), 9th edition (2019). The speed of light as a defined constant, and the 1983 definition of the metre. 72
- L. D. Landau & E. M. Lifshitz. Theory of Elasticity, Course of Theoretical Physics Vol. 7. The transverse wave speed of an isotropic medium. 75
- W. Weber & R. Kohlrausch (1856), and H. Fizeau (1849). The two measurements Maxwell compared. Figures in section 3 are as Maxwell reported them. Carried as reported from secondary sources; not verified against the originals.
Verification register: the Weber–Kohlrausch and Fizeau figures are as Maxwell quoted them and have not been checked against the original papers. The seismic and material wave speeds in section 2 are standard handbook values and are not individually sourced.
Ledger — memo 1.6
- Derived
- That the ratio G/ρ is fixed exactly by the measured speed of light, with no freedom left in it
- That the refractive index squared gives the local Aether density as a multiple of the void value
- That a freely expanding lattice is excluded, because c would have varied over cosmic time
- Measured
- Speed of light, 299,792,458 m/s — defined, not measured 72
- Refractive indices of air, water and diamond
- Bound on the variation of the fine structure constant to redshift 3
- Bound on the anisotropy of c, one part in 1017
- Assumed
- That light is a wave in a medium, and specifically the transverse wave of this one (memo 1.4). Without that, equation (1) is an analogy rather than a derivation.
- Void density ρ = 0.9 kg/m3, carried from memo 1.1. Only the ratio G/ρ is fixed by c. Splitting it into two numbers requires this assumption, and every absolute value in Part 1 inherits it.
- That the medium is the same everywhere, so that c measured here applies elsewhere. This is assumed rather than shown, and it is what the anisotropy bound in section 6 is testing.
- Open
- The rest frame. Equation (1) describes a medium, and a medium has a frame. It has not been found, to 0.95 m/s. Memo 5.7
- Whether ε0 and μ0 being reinterpretations of density and rigidity is a derivation or a restatement. Memo 4.4 argues it is a derivation once the action is written; this memo does not settle it
- The absolute value of G depends on a void density that is inferred rather than measured
- Prior art
- J. C. Maxwell, 1861–62 — equation (1) applied to the electromagnetic medium, and the comparison with Fizeau that established light as an electromagnetic wave. The derivation in this memo is his. What is added is the identification of the rigidity with a specific lattice
- A. Fresnel, 1820s — the transverse character of light, on which the use of the shear rather than the bulk modulus depends
- Weber and Kohlrausch, 1856 — the unit ratio Maxwell compared against