Status: CANONICAL
D_SSZ(r) = 1 / (1 + Ξ(r))
This is the ratio of proper time (τ) to coordinate time (t):
dτ = D_SSZ(r) · dt
D_GR(r) = √(1 - r_s/r)
D_SSZ(r) = 1/(1 + Ξ(r))
| Property | D_GR | D_SSZ |
|---|---|---|
| r → ∞ | 1 | 1 |
| r = r_s | 0 (singular) | 0.555 (finite) |
| r < r_s | Imaginary | Still finite |
| Monotonic | Yes | Yes |
| Smooth | Yes (except r_s) | Yes (everywhere) |
| Object | r/r_s | D_GR | D_SSZ | Δ(%) | Regime |
|---|---|---|---|---|---|
| Earth (surface) | ~7×10⁸ | 0.9999999993 | 0.9999999993 | ~0% | weak |
| Sun (surface) | ~5×10⁵ | 0.999999 | 0.999999 | ~0% | weak |
| White dwarf | ~10³ | 0.9995 | 0.9995 | <0.01% | weak |
| GPS satellite | ~10⁹ | 0.99999999994 | 0.99999999994 | ~0% | weak |
| Mercury perihelion | ~10⁷ | 0.99999997 | 0.99999997 | ~0% | weak |
| S2 star perihelion | ~2800 | 0.99982 | 0.99982 | <0.01% | weak |
| PSR J0030+0451 | 3.06 | 0.819 | 0.790 | -3.5% | strong |
| PSR J0740+6620 | 2.23 | 0.742 | 0.697 | -6.1% | strong |
| PSR J0348+0432 | 2.10 | 0.724 | 0.674 | -6.9% | strong |
| D-intersection (decay/global) | 1.595 | 0.611 | 0.611 | 0.0% | comparison |
| D-intersection (saturation/local) | 1.387 | 0.528 | 0.528 | 0.0% | comparison |
| Schwarzschild radius | 1.00 | 0.000 | 0.555 | ∞ | very_close |
z_SSZ(r) = 1/D(r) - 1 = Ξ(r)
For two observers at r₁ and r₂:
1 + z = D(r_obs) / D(r_emit) = (1 + Ξ_emit) / (1 + Ξ_obs)
The scaling factor s(r) is the reciprocal of time dilation:
s(r) = 1/D(r) = 1 + Ξ(r)
This appears in:
- Radial scaling gauge: dρ = s(r)·dr
- Maxwell equations: E' = s·E, B' = s·B
- Light-travel time: Δt_grav = ∫ Ξ(r) dr/c
For Ξ ≪ 1 (weak field):
D_SSZ ≈ 1 - Ξ + Ξ² - Ξ³ + ...
D_GR ≈ 1 - r_s/(2r) + ...
With Ξ_weak = r_s/(2r):
D_SSZ ≈ 1 - r_s/(2r) + O(r_s²/r²)
Identical to GR's first-order expansion → SSZ recovers GR in the weak field.
D_GR(r_s) = √(1 - 1) = 0 → Infinite redshift, frozen time
Ξ(r_s) = 1 - e^(-φ) = 0.80171
D_SSZ(r_s) = 1/(1 + 0.80171) = 0.55503
D is finite! Consequences:
- Time does not freeze at the horizon
- Redshift is finite (z = 0.802, not ∞)
- No information paradox from frozen surfaces
- No singularity in the time-dilation function
| Test File | Repository | What it tests |
|---|---|---|
| test_time_dilation.py | segmented-calculation-suite | D(r) computation |
| test_dilation_finite.py | segmented-calculation-suite | D(r_s) > 0 |
| test_validation.py | ssz-qubits | GPS, Pound-Rebka, S2 |
- Segment density Ξ(r) — The field that determines D
- Regime definitions — 5 regimes, where each Ξ formula applies
- Regime vs formula domains — Formula boundaries ≠ regime boundaries
- Scaling factor s(r) — s = 1+Ξ = 1/D
- Special values — D(rₛ)=0.555, Ξ(rₛ)=0.802
- Formula compendium — Complete formula list
- PPN formulas — Lensing/Shapiro use PPN, not Ξ
© 2025–2026 Carmen N. Wrede, Lino P. Casu