Status: CANONICAL Paper: 23 — Additive Decomposition of the Observed Light-Travel Time
The total observed light-travel time between two points can be decomposed additively:
Δt_total = Δt_flat + Δt_grav
where:
- Δt_flat = travel time in flat spacetime (Euclidean distance / c)
- Δt_grav = gravitational delay from segmentation
Δt_grav = (1/c) ∫ (s(r) - 1) dr = (1/c) ∫ Ξ(r) dr
This integral is taken along the light path. The delay is caused by the effective refractive index n_eff = s(r) = 1 + Ξ(r).
The Shapiro delay is a special case of the additive decomposition:
Δt_Shapiro = (1/c) ∫ Ξ(r) dr (Ξ-only, g_tt contribution)
CRITICAL: For the full GR-equivalent Shapiro delay, PPN completion is required:
Δt_full = (1 + γ) · Δt_Ξ = 2 · Δt_Ξ (γ=1)
The factor 2 comes from PPN (g_tt + g_rr), not from double counting.
Δt_Shapiro = (1+γ) · (r_s/c) · ln(4r₁r₂/d²)
= 2 · (r_s/c) · ln(4r₁r₂/d²)
ALWAYS distinguish:
- One-way signal delay vs round-trip radar echo (extra factor 2)
- This is separate from PPN (1+γ). Never conflate them.
The Cassini spacecraft provided the most precise measurement:
γ = 1.000021 ± 0.000023
SSZ predicts γ = 1 exactly → within measurement uncertainty.
| Test | Repository |
|---|---|
| test_shapiro_delay.py | frequency-curvature-validation |
| test_cassini.py | frequency-curvature-validation |
© 2025–2026 Carmen N. Wrede, Lino P. Casu