Photon Dispersion & Time Delay

The ether’s EM microstructure modifies the photon dispersion relation at high energies (Eq 3.46): ω² = c²k²(1 + ξ&sub2;(kℓ_e)² + …). This produces energy-dependent group velocities and measurable time delays between photons of different energies from distant gamma-ray bursts. Current GRB observations constrain ℓ_e < 6.2 × 10⁻¹³ m.

High-energy photon E₁ = 31 GeV

Low-energy photon E₂ = 0.1 GeV

Source distance d = 1250 Mpc

Microstructure scale log₁₀(ℓ_e/m) = -13.0 → ℓ_e = 1.0e-13 m

Computed Values

Time delay Δt: 45939054691673497600.0 days
Group velocity deviation: δv_g/c = -3.09e+7
ℓ_e / ℓ_P: 6.19e+21
ℓ_e vs bound (6.2e-13 m): Within bound

Time Delay vs Microstructure Scale

Reference GRB Observations

Source	E_high	E_low	d (Mpc)	Observed Δt	Predicted Δt at ℓ_e
GRB 090510	31 GeV	0.1 GeV	1250	0.859 s	45939054691673497600.0 days(excluded)
Mrk 501 (MAGIC)	1.2 GeV	0.25 GeV	143	240 s	7533222897888697.0 days(excluded)

Verification Against Monograph

ℓ_e upper bound ≈ 6.2 × 10⁻¹³ m (Eq 3.50)(got 6.20e-13, rel err 0.1%)

Δt = 0 when E₁ = E₂(got 0.00e+0, rel err 0.0%)

Δt = 0 when ℓ_e = 0(got 0.00e+0, rel err 0.0%)

Δt at ℓ_P, 100 GeV, z~1: ~10⁻¹⁸ s (Eq 3.51)(got 8.39e-19, rel err 16.1%)

Physical Origin

The ether’s EM microstructure on scale ℓ_e imprints a lattice-type correction on the photon dispersion relation. The leading correction ξ₂ = −1/12 is identical to the continuum limit of a simple cubic lattice (Eq 3.46).

The effect is quadratic in energy: higher-energy photons travel slightly slower than lower-energy ones. For two photons from a distant source, the accumulated time delay scales as ℓ_e²(E₁² − E₂²)d.

GRB observations constrain |ξ₂|ℓ_e² < 3.2e-26 m², giving ℓ_e < 6.2e-13 m. This places the microstructure scale at or below ~femtometre scales — well above the Planck length, but orders of magnitude below direct experimental resolution.

Read §3.8: Photon Dispersion Parameter Explorer All Predictions

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