Appendix X — Derivation 24: Full Coherence Kernel and Anisotropic Generalisation

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In earlier notes we used the spherically symmetric kernel

Γ_{iso} (r) = \frac{1}{4 π α} \frac{e^{- k r}}{r}, k = \sqrt{β / α} .

Real sources (galaxies, spinning nucleons) are not isotropic.
This appendix shows how the same Helmholtz operator yields a
general two-point Green’s function $Γ (x, x^{'})$
that automatically encodes dipole, quadrupole, or higher-order coherence
patterns.

1 Anchoring equation with an arbitrary source

Total static cost (Foundations §1) gives the Euler–Lagrange equation

\begin{matrix} (1.1) & α \nabla^{2} B (x) - β B (x) + J (x) = 0, \end{matrix}

where the anchoring demand of a source distribution $S (x)$ is

\begin{matrix} (1.2) & J (x) = \int Γ (x, x^{'}) S (x^{'}) d^{3} x^{'} . \end{matrix}

2 Definition of the full kernel

\begin{matrix} (2.1) & (α \nabla^{2} - β) Γ (x, x^{'}) = - δ^{3} (x - x^{'}) . \end{matrix}

Equation (2.1) is identical to the isotropic Helmholtz Green’s equation,
but we now keep angular dependence explicit.

3 Spherical-harmonic expansion

Let $r = x - x^{'}$ , $r = | r |$ ,
$\hat{r} = r / r$ . Expand

\begin{matrix} (3.1) & Γ (x, x^{'}) = \sum_{ℓ = 0}^{\infty} \sum_{m = - ℓ}^{ℓ} Γ_{ℓ} (r) Y_{ℓ m} (\hat{r}) Y_{ℓ m}^{*} ({\hat{r}}^{'}) . \end{matrix}

Each radial coefficient satisfies

\frac{1}{r^{2}} \partial_{r} (r^{2} \partial_{r} Γ_{ℓ}) - \frac{ℓ (ℓ + 1)}{r^{2}} Γ_{ℓ} - k^{2} Γ_{ℓ} = - \frac{δ (r)}{α} \frac{δ_{ℓ 0}}{4 π r^{2}} .

Solution (for $r > 0$ ):

Γ_{ℓ} (r) = \frac{1}{4 π α} \frac{e^{- k r}}{r} I_{ℓ} (k r), I_{0} \equiv 1; I_{ℓ} (k r) = \sum_{n = 0}^{ℓ} \frac{(ℓ + n)!}{n! (ℓ - n)!} (k r)^{- n} .

For $ℓ = 1, 2$ this reproduces dipole/quadrupole tails $\sim r^{- 2}, r^{- 3}$
modulated by $e^{- k r}$ .

4 Structured source examples

4.1 Dipole phase emitter

A spinning envelope with axial angle $θ$ has

S (x^{'}) = f (r^{'}) \cos θ^{'} .

Convolution with (3.1) keeps only the $ℓ = 1$ term, yielding an
equatorial coherence excess → frame-drag–like bias on passing modes.

4.2 Quadrupole galaxy disk

Take $S (x^{'}) = f (r^{'}) P_{2} (\cos θ^{'})$ . Resulting field has the
expected “peanut-shaped” potential; ray-tracing through this kernel
predicts asymmetric weak-lensing shear.

5 Practical simulation recipe

Pre-compute $Γ_{ℓ} (r)$ on a radial grid up to needed $ℓ_{max}$ .
Decompose the source $S (x)$ into spherical harmonics
$S_{ℓ m} (r)$ .
Convolve in harmonic space: $B_{ℓ m} (r) = \int Γ_{ℓ} (| r - r^{'} |) S_{ℓ m} (r^{'}) r^{' 2} d r^{'} .$
Re-sum to real space $B (x)$ for motion or lensing.

No extra parameters: only the locked constants
$α, β$ enter via $k$ .

6 Implications

Directional lensing – anisotropic $Γ$ predicts shear patterns
beyond GR’s monopole + quadrupole.
Spin–orbit coupling – dipole twist in $Γ$ adds a small
precession to nearby orbits (testable near rapidly rotating pulsars).
Photon polarisation – phase‐dependent paths through anisotropic
coherence field give tiny birefringence signatures.

Bottom line

The full two-point kernel $Γ (x, x^{'})$ is a
geometry-dependent Green’s function of the same Helmholtz operator.
Once the source is expanded in harmonics, every anisotropic effect
follows from the locked constants $α, β$ — no fresh couplings
or force fields required.

Appendix W | [Index](./Appendix Master) | Appendix Y - Continuum Mechanics