34-EFT.WP.Astro.Acceleration v1.0 | Chapter 3 Reference Frames & Acceleration Kinematics

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Chapter 3 Reference Frames & Acceleration Kinematics

I. Abstract & Scope
This chapter establishes the frame selection and kinematic baseline for the layered acceleration protocol. It defines the local co-moving frame F_flow, the reconnection-sheet frame F_sheet, and the shear-layer frame F_shear; introduces the acceleration rate A_acc(E), the acceleration timescale tau_acc(E), the per-cycle energy gain g_cycle(E), and the cycle time tau_cycle(E); and closes kinematics prior to explicit coupling with escape/loss/transport. All symbols use English notation with backticks; SI units are used; composite expressions are parenthesized.

II. Dependencies & References

• Terms & unified symbols: EFT.WP.Core.Terms v1.0 P10-*; centralized symbols: Chapter 2 Tab. 2-1.
• Metrology & dimensional checks: EFT.WP.Core.Metrology v1.0 Ch.1–3 (check_dim).
• Path & ToA formulations (for observation mapping, not derived here): EFT.WP.Core.Equations v1.1 Ch.2 S20-*.
• Forward links: reconnection (Ch.4, S30-), shear (Ch.5, S40-), spectrum formation (Ch.7), transport & losses (Ch.8).

III. Normative Anchors (added in this chapter)

• S20-0 (Frame Selection): Use the local co-moving frame F_flow (so that v_vec = 0 locally) to evaluate rates and timescales; use F_sheet and F_shear for geometry and boundary conditions.
• S20-1 (Acceleration Rate): A_acc(E) = ( 1 / E ) * ( dE/dt ) |_{F_flow}.
• S20-2 (Mechanism Decomposition): A_acc(E) = A_rec(E) + A_shear(E).
• S20-3 (Acceleration Timescale): tau_acc(E) = ( 1 / A_acc(E) ).
• S20-4 (Per-Cycle Energy Gain): g_cycle(E) = E^{-1} * ⟨ ΔE ⟩_{cycle}.
• S20-5 (Cycle Time): tau_cycle(E) = tau_sc(E) + tau_adv(E).
• S20-6 (Effective Closure): A_acc(E) ≈ g_cycle(E) / tau_cycle(E).
• S20-7 (Net-Growth Criterion): with loss term A_loss(E), require A_acc(E) > A_loss(E).

IV. Body Structure

I. Background & Problem Statement

• Frame choice directly impacts evaluation of energy-gain rates and statistical packaging. F_flow provides a unified evaluation surface, avoiding spurious gains from large-scale shear and drift.
• F_sheet is anchored to the reconnection neutral plane; F_shear aligns with the principal gradient of the velocity field. Mappings among F_flow/F_sheet/F_shear and their use cases are fixed.
• Kinematic definitions are given without mechanism-specific coefficients; detailed dynamics appear in Ch.4 (reconnection) and Ch.5 (shear).

II. Key Equations & Derivations (S-series)

• S20-1: A_acc(E) = ( 1 / E ) * ( dE/dt ) |_{F_flow}.
• S20-2: A_acc(E) = A_rec(E) + A_shear(E).
• S20-3: tau_acc(E) = ( 1 / A_acc(E) ).
• S20-4: g_cycle(E) = E^{-1} * ⟨ ΔE ⟩_{cycle}.
• S20-5: tau_cycle(E) = tau_sc(E) + tau_adv(E).
• S20-6: A_acc(E) ≈ g_cycle(E) / tau_cycle(E).
• S20-7: A_acc(E) > A_loss(E).

III. Methods & Flows (M-series)

• M20-1 (Frame Determination): Input v_vec(x,t) and geometry; output {F_flow, F_sheet, F_shear} local bases and mappings; verify the v_vec = 0 condition under F_flow.
• M20-2 (Shear & Vorticity Metrics): Compute ∇v; define the symmetric tensor S = ( ∇v + ( ∇v )^T ) / 2; set sigma_shear = ( 0.5 * Tr( S^2 ) )^{1/2} and omega_vec = ∇ × v_vec.
• M20-3 (Timescale Evaluation): Given tau_sc(E) and a residence model for tau_adv(E), compute A_acc(E) and tau_acc(E) via S20-6; return energy-binned outputs for Ch.7.
• M20-4 (Observation Mapping Interface): Pass {A_acc(E), tau_acc(E)} forward to Ch.7 (spectrum formation) and Ch.8 (transport & losses). ToA and path terms are handled by common modules in later chapters.

IV. Cross-References within/beyond this Volume

• Timebase & path corrections from frame to observation: EFT.WP.Metrology.TimeBase v1.0, EFT.WP.Metrology.PathCorrection v1.0.
• Mechanism details: A_rec(E) in Ch.4 (S30-); A_shear(E) in Ch.5 (S40-).
• Net-growth and spectrum formation: Ch.7 (S50-); loss definition and composition: Ch.8 (S52-).
• Unified symbols & do-not-confuse: Chapter 2 Tab. 2-1 and P12-*.

V. Validation, Criteria & Counterexamples

1. Positive criteria:
   • Timescale ordering with an energy band where tau_acc(E) < tau_loss(E).
   • Monotonic relation between sigma_shear and A_acc(E) in shear-dominated regions.
   • Positive correlation of A_acc(E) with reconnection rate R_rec in sheet regions.
2. Negative criteria:
   • A_acc(E) does not decrease when k_STG → 0 (or shear metrics vanish).
   • tau_acc(E) shows strong sensitivity to frame choice (indicating mis-specification of F_flow or geometry).
3. Contrasts: Hold A_loss(E) fixed and switch off A_rec(E) or A_shear(E) individually to localize the source of gain.

VI. Summary & Handoff
This chapter fixes the F_flow evaluation baseline and the minimal S20-* kinematic set, with flows and validation criteria for downstream use. Chapter 4 specifies reconnection geometry and rates (S30-*) and constructs a testable form of A_rec(E).

V. Figures & Tables (this chapter)

• Fig. 3-1 Frame & geometry schematic: F_flow, F_sheet, F_shear nesting and mappings.
• Tab. 3-1 Local Symbol Table (this chapter)

• Tab. 3-2 Kinematic criteria checklist: timescale ordering, A_acc response to sigma_shear and R_rec, and related diagnostics.