Status: ALWAYS FOLLOW — VERBINDLICH
(1) "Null" refers to the geodesic type (ds²=0), not "no effect." (2) Statements like "Ξ → 1" are asymptotic — Ξ saturates near 1 in the transition zone (blend), although strong-field is formally used for r/r_s < 1.8. (3) Energy rules concern observed transformation factors (multiplicative), not statements about global GR energy conservation.
Never use a single method for all observables.
Always classify the observable first (null vs timelike vs orbit) and select the correct method (PPN vs Ξ) accordingly.
Examples: Lensing, Shapiro delay, VLBI/group delay, lens time-delays
Method: PPN completion — multiply Ξ-only result by (1+γ)
result = Ξ_only × (1 + γ)
- For GR-equivalent: γ = 1.0 → factor 2
- Ξ-only = g_tt contribution only (half of full GR for these observables)
Examples: Time dilation, gravitational redshift, GPS, Pound-Rebka
Method: Ξ-based formulation directly
D(r) = 1/(1 + Ξ(r))
- Keep local c invariant by construction
- Differences are comparative (between clocks/frames)
Examples: Perihelion advance, precession, frame dragging
Method: PPN orbit machinery (β, γ, ...)
- NOT Ξ-only shortcuts
Observable Type → Method
─────────────────────────────────
NULL (light) → PPN (1+γ)
TIMELIKE (clocks) → Ξ-proxy
TIMELIKE (orbits) → PPN (γ,β)
If a null-observable comes out at 50% of GR:
- This is NOT a bug by default
- It indicates "Ξ-only = g_tt piece"
- Full GR requires PPN: (1+γ), γ=1
Ξ-integration captures only g_tt (temporal)
PPN captures g_tt + g_rr (temporal + spatial)
α_total = α_tt + α_rr = r_s/b + r_s/b = 2r_s/b ✅
Before using any Ξ formula, determine regime via r/r_s:
| Regime | r/r_s | Operative Xi branch |
|---|---|---|
| very_close | < 1.8 | g2 / inner exponential |
| blended | 1.8–2.2 | C² Hermite blend |
| photon_sphere | 2.2–3.0 | physical photon-sphere regime; g1 branch in current calculators |
| strong | 3.0–10.0 | physical strong-field regime; g1 branch in current calculators |
| weak | > 10.0 | g1 / weak field |
Never mix multiple "strong-field Ξ" formulas without an explicit blend rule.
Formula domain → Formula
─────────────────────────────────
r/r_s > 2.2 → Ξ = r_s/(2r)
r/r_s < 1.8 → inner exponential / g2
1.8 ≤ r/r_s ≤ 2.2 → Hermite blend
Above r/r_s=2.2, "g1 branch" does not mean "physically weak everywhere": photon-sphere and compact-object strong-field contexts still extend up to r/r_s=10.
For oscillatory fields (large k·ρ):
- DO NOT validate using finite-difference second derivatives
- USE analytic chain rule
When transforming dρ = s(r)·dr, ensure the second-derivative operator includes the extra term:
- In 1D: d²/dr² introduces an s·s' term
- In 3D: include the correct divergence/geometry term
Missing this term is a technical error, not physics.
Always distinguish:
- One-way signal delay vs Round-trip radar echo (extra factor 2)
This is separate from PPN (1+γ). Never conflate them.
NEVER add SR + GR + "extra" energies linearly (avoid triple counting).
USE multiplicative factor bookkeeping:
E_obs = E_rest × (transform factors)
Phase/Action links are handled via path/geometry, not "changing local laws."
Any deprecated/old Ξ(r) expression is FORBIDDEN in new derivations.
Deprecated:
❌ Ξ = (r_s/r)² × exp(-r/r_φ) ← VERBOTEN
Treat as hard fail in any computation.
- Add "Method selection & scope" text:
- Ξ-only for timelike observables
- PPN completion for null observables
- Add explicit factor-2 remark for lensing/Shapiro:
- Ξ-only = half
- Full GR recovered with (1+γ), γ=1
- Keep "local c invariant" explicit
- Emphasize: effects arise from geometry/path + clock mapping, not local speed changes
Observable → Class → Method → Scope → Then calculate.
This document contains the binding rules for SSZ calculations. On violation: check and correct result.
© 2025–2026 Carmen N. Wrede, Lino P. Casu