28-EFT.WP.Propagation.PathRedshift v1.0 | Chapter 4 — Gravitational Redshift & Path Terms (Weak Field / PN / ISW)

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Chapter 4 — Gravitational Redshift & Path Terms (Weak Field / PN / ISW)

One-sentence goal: Provide a computable engineering convention for gravitational redshift and path-dependent terms under weak-field / post-Newtonian (PN) assumptions—including Shapiro delay and the (Integrated) Sachs–Wolfe (ISW) contribution—together with the companion relationship to the two-form arrival time T_arr^{form1/form2}, the required contracts, and manifest publication.

I. Scope & Objects

1. Inputs
   • Potentials & ephemerides: phi_grav(x,t) (Earth + celestial bodies, weak field), ephemeris, (optional) frame-dragging / tidal terms.
   • Worldlines & paths: Gamma(τ) or segmented path gamma(ell) (fiber / free space / deep space), observation window W = [t_0, t_1].
   • Observation streams: f_obs(t) (PLL / CFO / spectral-line fitting, see Chapter 9), local clock offset / skew / J.
   • References: RefCond = { ephemeris.hash, gravity.hash, timebase.hash, tz, … }.
2. Outputs
   • Gravitational components & path term: z_grav = z_φ ⊕ z_path (z_φ from potential difference; z_path from line-of-sight integrals including Shapiro / ISW), composed upstream into the gross z_path (total).
   • Arrival-time companion: ΔT_grav (including Shapiro delay), with a physical attribution for the two-form gap delta_form.
   • Manifest: manifest.redshift.grav.* with uncertainties u / U.
3. Boundary
   Weak-field / PN engineering convention by default; strong-field or higher-order corrections are recorded as extensions with sources and assumptions.

II. Terms & Variables

• Potentials & parameters: phi_grav(x,t) (Newtonian potential), U = phi_grav / c_ref^2 (dimensionless), GM (standard gravitational parameter).
• Spacetime / path: t, τ, x(•), n_hat (instantaneous LOS), impulsive path length L_gamma = ( ∫_gamma 1 d ell ).
• Redshift & frequency: 1+z = f_emit / f_obs; gravitational term z_grav; path-induced term z_path(grav).
• Arrival time: T_arr^{form1}, T_arr^{form2}, ΔT_grav, delta_form.
• Dimensions: unit(phi_grav) = [L^2/T^2], unit(z) = 1, unit(ΔT_grav) = [T].

III. Postulates P65-4x

• P65-401 (Weak-field convention): When |U| ≪ 1, use the first-order engineering form (potential difference + path integral). If PN / frame-dragging / tides are included, state the PN order and model version.
• P65-402 (Two-form pairing): Publishing z_grav / ΔT_grav must include T_arr^{form1/form2} and the recorded delta_form; enforce delta_form ≤ tol_Tarr (see Chapter 2).
• P65-403 (Explicit measures): Any along-path or time integral declares ( ∫_{gamma(ell)} • d ell ) / ( ∫_{t∈W} • dt ) and the coordinate frame (ECI/ECEF).
• P65-404 (Dimensions / timebase): All fields pass check_dim( y − f(x) ); compute on tau_mono, publish at ts.
• P65-405 (RefCond traceability): Sources, resolution, epoch of potentials/ephemerides and tides/polar motion are recorded in RefCond with hash / validity.

IV. Minimal Equations S65-4x

1. Gravitational redshift from potential difference (weak field)

• S65-401
• z_φ ≈ - Δphi_grav / c_ref^2 = - ( phi_grav(obs) - phi_grav(emit) ) / c_ref^2

If TX/RX are referenced in the same frame and time-averaged:

⟨ z_φ ⟩_W ≈ - ⟨ Δphi_grav ⟩_W / c_ref^2

2. Shapiro path delay & equivalent frequency shift

• S65-402 (Delay): for a single gravitating body M (engineering approximation)
• Δt_Sh ≈ ( 2GM / c_ref^3 ) ln( (r_e + r_o + R) / (r_e + r_o - R) )

where r_e, r_o are TX/RX distances to the mass, and R is the geometric separation term.

• S65-403 (Instantaneous apparent shift):
• z_Sh(t) ≈ - d(Δt_Sh)/dt

(often negligible under window averaging, or folded into ΔT_grav for timing-only corrections).

3. Integrated Sachs–Wolfe (ISW, engineering approximation)

• S65-404: in time-varying potentials phi_grav(x,t):
• z_ISW ≈ ( 2 / c_ref^2 ) ( ∫_{t_emit}^{t_obs} ∂phi_grav/∂t • dt )

(relevant for deep-space / cosmology; near-Earth may be ignored or treated as tide corrections).

4. PN / frame-dragging (optional)

• S65-405: with 1PN, a compact correction is
• z_PN ≈ z_φ + (v^2/2 - phi_grav)/c_ref^2 + O(c^{-3})

Frame-dragging (Lense–Thirring) contributes along-path as

z_LT = O( J⋅r / r^3 c_ref^3 )

which must be recorded with source and upper bound if used.

5. Arrival-time companion

• S65-406
• ΔT_grav = Δt_Sh + ( 1 / c_ref ) ( ∫_{gamma} Δn_grav d ell )

In vacuum Δn_grav ≈ 0. Fiber elevation changes can map via segment-level variations in φ_grav into the n_eff thermo-baric sensitivity (corr_env)—often negligible but recordable.

6. Composition & small-signal approximation

• S65-407
• 1 + z_grav = (1+z_φ)(1+z_Sh)(1+z_ISW)(1+z_PN)…

Small-signal: z_grav ≈ z_φ + z_Sh + z_ISW (+ z_PN).

V. Metrology Pipeline M65-4 (Ready → Modeling → Estimation → Verification → Persistence)

1. Ready: unify coordinates/epoch; load ephemeris and gravity models (Earth gravity degree/order, celestial GM/positions), and RefCond.
2. Model / estimate:
   • Compute phi_grav(emit/obs) → z_φ;
   • From geometry/ephemerides compute Δt_Sh and z_Sh; compute z_ISW if needed;
   • Compute T_arr^{form1/form2} in parallel and ΔT_grav.
3. Verify:
   • check_dim(z) = 1, check_dim(ΔT_grav) = [T]; enforce delta_form ≤ tol_Tarr;
   • Zero-baseline / loopback: co-elevation / iso-potential windows yield z_grav ≈ 0; validate Δt_Sh during limb/close-approach;
   • Record uncertainties u(phi_grav), u(ephemeris), u(Δt_Sh) and propagate to u/U (see Chapter 13).
4. Persist:
   manifest.redshift.grav = { gravity.hash, ephemeris.hash, z_parts:{ z_φ, z_Sh, z_ISW, z_PN? }, z_grav, Δt_Sh, ΔT_grav, T_arr_forms, delta_form, u/U, RefCond, contracts.*, signature }.

VI. Contracts & Assertions C65-4x (suggested thresholds)

• C65-401 (Two-form gap): delta_form_p95 ≤ tol_Tarr.
• C65-402 (Freshness of fields): age(gravity_model) ≤ Δt_max, age(ephemeris) ≤ Δt_max, coverage ≥ cov_min.
• C65-403 (Zero-baseline consistency): in iso-potential co-located windows, | z_grav |_p95 ≤ z_quiet_max.
• C65-404 (Shapiro/ISW annotation): if z_Sh / z_ISW are published, the manifest must state whether they are folded into ΔT_grav or applied as pure z corrections.
• C65-405 (Dimensional compliance): unit(z) = 1, unit(Δt_Sh) = [T]; never mix dB↔linear power quantities in this chapter.

VII. Implementation Bindings I65-4* (interfaces, I/O, invariants)

• I65-41 compute_grav_z(potential_model, worldlines, order) -> { z_φ, meta }
• I65-42 shapiro_delay(body_params, geometry) -> { Δt_Sh, z_Sh, meta }
• I65-43 isw_term(phi_time_series, path, window) -> { z_ISW, meta }
• I65-44 pn_corrections(state, phi, order) -> { z_PN, meta }
• I65-45 map_grav_to_tarr(z_grav, path) -> { ΔT_grav, T_arr_forms }
• I65-46 assert_grav_contracts(ds, rules) -> report
• I65-47 emit_grav_manifest(results, policy) -> manifest.redshift.grav
  Invariants: two_forms_present = true; check_dim(*) passes; gravity/ephemeris versions traceable; presence/absence of ISW/PN is explicitly annotated.

VIII. Cross-References

• Redshift baseline & two forms: Chapter 2;
• Kinematic term: Chapter 3; medium term: Chapter 5;
• Path integrals: Chapter 8; observations / PLL / spectral fits: Chapter 9;
• Fusion & calibration: Chapters 10 / 11; uncertainty & publication: Chapter 13 and Appendices C / E.

IX. Quality & Risk Control

• SLI / SLO: | z_grav |_p95 (zero baseline), z_resid_p95 = | z_meas − ( z_kin + z_grav + … ) |_p95, delta_form_p95, panel_freshness, RefCond_coverage.
• Fallback strategies: stale or inconsistent gravity/ephemerides → revert to last version and expand guardbands; z_resid breach → shorten windows, increase gravity model degree/order, or suppress ISW/PN extensions; two-form over-limit → standardize to form2 and re-check media/path parameters.
• Audit: gravity/ephemeris sources & resolution, Δt_Sh evidence, zero-baseline / limb validation sets, and the manifest.redshift.grav signature chain and replay scripts.

Summary

• This chapter establishes the engineering baseline for gravitational redshift and path terms: weak-field potential difference (z_φ), Shapiro / ISW path terms, and optional PN corrections, paired with the two-form arrival-time structure.
• Using M65-4, C65-4x, and I65-4*, one can robustly estimate, verify, and manifest z_grav and ΔT_grav, providing a consistent anchor for the subsequent media / worldline / observation / fusion chapters.