Master's Level

Coulomb Shearing - formalism, evaluation pipeline, and worked results

This page covers the full mechanism at the level of the primary paper. It assumes familiarity with plasma physics, relativistic accretion, and basic GR notation. The focus is the evaluation pipeline: from force decomposition through the admissibility criterion, species-resolved windows, the pre-committed protocol, and the worked M87* result.

Section 1 - The coupling assumption in classical accretion

Classical Eddington-limited growth treats the infalling plasma as a single effective fluid. The derivation is exact at the electron level - the Thomson-channel radiative force per electron balances gravitational force on the associated baryon mass column at L_Edd = 4πGMm_p c/σ_T - but the extension to a bulk inflow bound requires a coupling closure: the electrostatic restoring force is assumed to communicate the electron-level radiation pressure to the ion population instantaneously and perfectly.

The closure is not enforced by plasma physics. It is a modeling assumption whose validity is conditional on the Coulomb coupling timescale t_c being short compared to the dynamical timescale over which species-selective forcing operates. In the dense disk body (n_e ~ 10¹³–10¹⁵ cm⁻³), this condition is well satisfied. In the hot, dilute polar plasma above the equatorial disk, the ratio t_res/t_c can approach or exceed unity, making the closure locally inadmissible along specific trajectory families.

Section 2 - Trajectory geometry: polar channels in Kerr spacetime

All evaluations are performed in the exterior Kerr spacetime (r > r_H). In Boyer–Lindquist coordinates the off-diagonal dt dφ term scales as sin²θ, making polar and equatorial trajectories dynamically inequivalent at the same coordinate radius. The centrifugal potential scales as L²/(r² sin²θ), so near the spin axis the barrier is suppressed.

The operative transport quantity is the residence time t_res:

t_res = ∫[r_in → r★] dr / v_ff(r)
v_ff(r) = η √(2GM/r),   η ∈ (0, 1]

η parameterizes the infall speed relative to free-fall. η = 1 gives the shortest (conservative) t_res. For M87* at r★ = 5r_g: t_res(η = 1) ≈ 1.4 days.

Section 3 - Species-resolved force decomposition

Along a polar trajectory, gravity cancels exactly in the differential. The driving imbalance comes entirely from the radiative and electromagnetic channels:

Δa∥ ≡ a_e − a_i ≈ μ_s · σ_T L / (4π r² c m_e)   [Track T, E∥ = 0]

At L = L_Edd:   Δa∥ ≈ μ_s · (m_p/m_e) · GM/r²  ≈  1836 μ_s g(r)

Section 4 - Coulomb coupling and the displacement kernel

The electron–ion relative drift Δv satisfies:

dΔv/dt + Δv/t_c = Δa∥(t)

t_c ≈ 3×10⁵ s · (T_e/10⁸ K)^(3/2) · (10⁸ cm⁻³/n_e) · (20/lnΛ) · Z⁻²

For approximately constant Δa∥, integrating twice yields:

Δx(t) ≈ Δa∥ [t_c t − t_c² (1 − e^(−t/t_c))]

Limiting regimes:
  Free-drift   (t ≪ t_c): Δx ≈ ½Δa∥t²
  Drag-limited (t ≫ t_c): Δx ≈ Δa∥ t_c t

Section 5 - Admissibility criterion and the shearing functional

Coulomb Shearing is defined by the displacement-form admissibility condition:

Δx(t_res) ≥ κ λ_D    ↔    χ_s = 1

κ ~ O(1), declared before evaluation, not adjusted per object.

S(r; θ, v, Z, q) ≡ Δx(t_res) / (κ λ_D)

A_Z^seg ≡ { r ∈ [r_in, r_out] : S(r; ...) ≥ 1 }

The window [r_min(Z), r_max(Z)] is a computed output, not a free parameter.

Section 6 - Pre-committed evaluation protocol

Three permitted outcomes per system, all irreversible:

• PASS: Predicted directional inequalities satisfied within locked 90% propagated intervals; required comparator dataset present.
• FAIL: Predicted signatures contradicted by admissible data at declared significance, or predicted signature absent where comparator is decisive.
• DATA-INSUFFICIENT: Missing required inputs or comparators; or uncertainty too large to discriminate.

The evaluation steps are order-locked. Inputs are fixed before the calculation runs. No parameter is adjusted after the fact. The post-hoc prohibition is absolute: changing the distribution assignment rules, credibility level, MC seed, multi-source rule, or any input bound after any outcome exists is a protocol violation.

Section 7 - Observable predictions

All predictions are conditional on χ_s = 1 for at least one species on nonzero phase-space measure. Observable transitions must co-locate with the computed admissibility window.

P1 - Transverse RM stratification:
  |RM|_spine / |RM|_sheath < 1
  |RM|_sheath − |RM|_spine > 3σ_combined

P2 - Depolarization contrast:
  DP_spine / DP_sheath > 1
  (DP_spine − DP_sheath) > 3σ

P3 - Polarization–Faraday anti-correlation:
  ρ_P3 = Spearman(I_pol, |RM|) < −0.5
  p-value < 0.01 (two-sided)

Section 8 - Worked result: M87*

Input packet (benchmark run, Track EM):

Admissibility ratio (free-drift regime):

t_fail ≡ √(2κλ_D / |Δa∥|) ≈ 6.0 × 10⁻⁷ s
Δx/(κλ_D) ≈ 4 × 10²²  ≫ 1

At ADAF density scaling n_e(r) ∝ r⁻¹·³:
  S(3r_g) ≈ 120,  S(5r_g) ≈ 366,  S(10r_g) ≈ 647

Species-resolved results:

Protocol outcomes: PASS on P1–P3 (VLBA 8/15 GHz). PASS on P5. P4: DATA-INSUFFICIENT (no matched-cadence multi-epoch RM series at required resolution).

For the full formalism including the pair-loading problem, GRMHD context, and explicit falsification conditions, see the Doctorate level page or the paper.