28-EFT.WP.Propagation.PathRedshift v1.0 | Chapter 3 — Kinematic Redshift (Doppler / Sagnac / Non-inertial)

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Chapter 3 — Kinematic Redshift (Doppler / Sagnac / Non-inertial)

One-sentence goal: Establish a convention for how relative motion, rotation/Sagnac, and non-inertial effects impact frequency and arrival time, provide the computable decomposition
z_kin = z_Doppler ⊕ z_Sagnac ⊕ z_acc, and specify the companion relationship to the two-form T_arr and the publication rules.

I. Scope & Objects

1. Inputs
   • TX/RX state: state_emit / state_obs = { r(t), v(t), a(t), att(t), Omega } (position / velocity / acceleration / attitude / angular velocity), with reference frame frame ∈ { ECI, ECEF, body }.
   • Path & geometry: propagation path gamma(ell) (see Chapter 2); loop geometry (oriented area) A.
   • Observation stream: carrier / spectral-line frequency f_obs(t) from PLL / CFO / line fitting (see Chapter 9).
   • Reference conditions: RefCond = { ephemeris.hash, gravity.hash, met.hash, timebase.hash, tz, … }.
2. Outputs
   • Kinematic components: z_Doppler, z_Sagnac, z_acc, and the composition z_kin;
   • Arrival-time companions: ΔT_kin and a physical explanation of the two-form gap delta_form;
   • Manifest: manifest.redshift.kin.* with uncertainties u / U.
3. Boundary
   Default to weak-field, small-velocity engineering (while retaining the relativistic factor). Gravitational terms are in Chapter 4 (z_grav), media terms in Chapter 5 (z_med).

II. Terms & Variables

• Geometry & spacetime: t, tau, r, v, a, n_hat (instantaneous LOS unit vector TX→RX), A (loop area), Omega (rotational angular velocity).
• Doppler & Lorentz: beta = v_los / c_ref, gamma_L = 1 / sqrt( 1 − beta^2 ), v_los = v • n_hat.
• Frequency & redshift: f_emit, f_obs, 1+z = f_emit / f_obs (unit(z) = "1").
• Arrival time: T_arr^{form1}, T_arr^{form2}, delta_form, ΔT_kin (arrival-time companion induced by motion/rotation).
• Dimensions: unit(v) = "[L]/[T]", unit(Omega) = "[1]/[T]", unit(A) = "[L]^2".

III. Postulates P65-3x

• P65-301 (Component composition):
• 1 + z_kin = (1+z_Doppler)(1+z_Sagnac)(1+z_acc)

Small-signal approximation z_kin ≈ z_Doppler + z_Sagnac + z_acc.

• P65-302 (Explicit path/measure): any time/path averaging uses explicit measures ( ∫_{t∈W} • dt ), ( ∫_{gamma(ell)} • d ell ); loop geometry is given by area A or equivalent line integral.
• P65-303 (Two-form pairing): publication of z_kin must include T_arr^{form1/form2} and an ΔT_kin consistency note; enforce delta_form ≤ tol_Tarr (see Chapter 2).
• P65-304 (Dimensions & timebase): all fields pass check_dim( y − f(x) ); compute on tau_mono, publish at ts.
• P65-305 (Frame traceability): frame selection, ECI/ECEF/body transforms, and ephemeris sources are recorded in RefCond.

IV. Minimal Equations S65-3x

1. Relativistic Doppler (1-D LOS engineering form)

• S65-301. With instantaneous LOS n_hat and relative LOS speed v_los,
• 1 + z_Doppler = γ_L ( 1 - β ), where β = v_los / c_ref, γ_L = 1/√(1-β^2)

Small-speed expansion:

z_Doppler ≈ - v_los / c_ref + (1/2)(v^2 / c_ref^2)

(includes the transverse Doppler (time dilation) second-order term).

• S65-302 (Two-end motion). If both ends move, use relative LOS speed or multiply end factors:
• 1 + z_Doppler ≈ ( γ_L_emit (1 - β_emit) ) / ( γ_L_obs (1 - β_obs) )

2. Sagnac (rotating frames)

• S65-303 (Time bias). Sagnac time bias on a closed loop:
• Δt_Sag = ( 2 * Omega • A ) / c_ref^2

(Omega and oriented area A follow the right-hand rule). For open paths, use the line integral

Δt_Sag = ( 2 / c_ref^2 ) ∮ ( Omega × r ) • d r

• S65-304 (Frequency mapping). Average apparent frequency shift over the measurement window T_meas:
• z_Sagnac ≈ - Δt_Sag / T_meas

For round-trip links, the sign follows bidirectional geometry (often ≈ twofold).

3. Non-inertial / acceleration term (narrow-window approximation)

• S65-305. If LOS acceleration is approximately constant over window W,
• d(ln f)/dt ≈ - a_los / c_ref ⇒ z_acc(t) ≈ - (1/c_ref) ∫_{t∈W} a_los(t) dt

This term is not mixed with z_grav (gravity is Chapter 4); it only covers apparent chirp from non-inertial frames / maneuvers.

4. Arrival-time companion

• S65-306. Motion/rotation-induced arrival-time correction
• ΔT_kin = ( ∫_{gamma} ( Δn_kin / c_ref ) d ell ) + Δt_Sag + Δt_acc

with the engineering approximation

Δt_acc = - ∫_{t∈W} ( a_los / c_ref ) dt • ( L_eff / c_ref )

where L_eff is the effective path length.

5. Composition & small-signal

• S65-307.
• 1 + z_kin = (1+z_Doppler)(1+z_Sagnac)(1+z_acc)

Small-signal: z_kin ≈ z_Doppler + z_Sagnac + z_acc.

V. Metrology Pipeline M65-3 (Ready → Modeling → Estimation → Verification → Persistence)

1. Ready
   Unify coordinates & timeline; acquire r, v, a, att, Omega from ephemerides and sensors; choose frame and persist base transforms (ECI/ECEF/body).
2. Model / estimate
   • Compute n_hat(t), v_los, β, γ_L → z_Doppler(t);
   • From path geometry and Omega compute Δt_Sag → z_Sagnac;
   • From IMU/navigation derive a_los → z_acc;
   • From PLL / CFO / line-fit obtain f_obs(t) → z_meas(t) (see Chapter 9).
3. Verify
   • check_dim(z) = 1; compute T_arr^{form1/form2} in parallel and record delta_form (see Chapter 2);
   • Baselines/loopbacks: validate z_kin ≈ 0 in stationary collinear windows; validate Δt_Sag on loops;
   • Record uncertainties u(v_los), u(Omega•A), u(a_los) and propagate u(z_kin) (see Chapter 13).
4. Persist
   manifest.redshift.kin = { frame, n_hat.hash, A, Omega, z_parts:{ z_Doppler, z_Sagnac, z_acc }, z_kin, Δt_Sag, ΔT_kin, T_arr_forms, delta_form, u/U, RefCond, contracts.*, signature }.

VI. Contracts & Assertions C65-3x (suggested thresholds)

• C65-301 (Two-form gap): delta_form_p95 ≤ tol_Tarr.
• C65-302 (Loop-geometry consistency): | (Omega•A)_{meas} − (Omega•A)_{cfg} | ≤ τ_{ΩA}; reject if line-integral vs area convention disagree.
• C65-303 (Static/aligned baseline): in stationary collinear windows | z_kin |_p95 ≤ z_quiet_max.
• C65-304 (Maneuver window consistency): z_acc consistency between IMU and trajectory estimates passes KS / χ^2.
• C65-305 (Dimensional compliance): unit(z) = "1", unit(Δt_Sag) = "[T]", unit(Omega•A/c_ref^2) = "[T]".

VII. Implementation Bindings I65-3* (interfaces, I/O, invariants)

• I65-31 compute_kinematic_z(state_emit, state_obs, los, frame) -> { z_Doppler, meta }
• I65-32 sagnac_delay(Omega, A | path) -> { Δt_Sag, z_Sagnac, meta } (supports area and line-integral conventions)
• I65-33 accel_to_chirp(a_los_time_series, window) -> { z_acc, meta }
• I65-34 compose_z_kin(z_Doppler, z_Sagnac, z_acc, mode) -> z_kin (mode ∈ { product, linear })
• I65-35 map_kin_to_tarr(z_kin, path, timebase) -> { ΔT_kin, T_arr_forms }
• I65-36 assert_kin_contracts(ds, rules) -> report
• I65-37 emit_kin_manifest(results, policy) -> manifest.redshift.kin
  Invariants: two_forms_present = true; check_dim(*) passes; frame / RefCond / ephemeris.hash traceable; area and line-integral conventions must reconcile.

VIII. Cross-References

• Redshift baseline & two forms: Chapter 2;
• Gravitational term: Chapter 4; media term: Chapter 5;
• Worldlines & path integrals: Chapter 8;
• Observations / PLL / CFO / line fitting: Chapter 9;
• Fusion & calibration: Chapters 10 / 11;
• Uncertainty & publication: Chapter 13 and Appendices C / E.

IX. Quality & Risk Control

• SLI / SLO: | z_kin |_p95 (static baseline), z_kin_resid_p95 = | z_meas − z_kin |_p95, delta_form_p95, panel_freshness.
• Fallback: if z_kin_resid breaches thresholds → (1) verify frame/LOS and A; (2) switch line-integral vs area convention; (3) shorten window and re-estimate; (4) fall back to linear approximation.
• Audit: evidence for Omega / A computation, LOS/ephemeris source hashes, static & maneuver validation sets, and the manifest.redshift.kin signature chain with replay scripts.

Summary

• This chapter decomposes kinematic redshift into Doppler / Sagnac / non-inertial parts, supplies computable S65-3x equations, and specifies the companion relationship to the two-form T_arr.
• Via M65-3, C65-3x, and I65-3*, z_kin and ΔT_kin can be robustly estimated, verified, and published for engineering scenarios—and closed with the gravitational / media / observation / fusion chapters that follow.