Appendix D — Derivation 4: Lensing from Coherence Gradient

Overview

Gravitational lensing in general relativity is described as light following geodesics through curved spacetime. In modal dynamics, spacetime is not curved, and light does not follow geodesics.

Instead, photon deflection arises from the tension in preserving phase coherence while traversing a non-uniform coherence field $B (x)$ . The photon curves not because of force, but because remaining straight would break internal coherence.

1. Photon as a Latent Mode

A photon is a latent mode with coherence function:

ψ_{γ} (x, t) = ρ (x, t) e^{i ϕ (x, t)}

It does not anchor, but must maintain its internal phase alignment $ϕ (x, t)$ across its path.

As it passes near a massive emitter (e.g., a star or galaxy), it enters a steep gradient of $B (x)$ .

To preserve coherence, the photon must adjust its path to avoid structural distortion.

2. Coherence Penalty Field

The photon’s trajectory is governed by the decoherence penalty functional:

Λ_{γ} (x) = γ_{0} {| \nabla B (x) |}^{2}

Where:

$Λ_{γ}$ is the local phase tension cost
$γ_{0}$ is a universal sensitivity constant
$B (x)$ is the coherence field (see Appendix B)

The photon follows a path that minimises the integrated decoherence penalty:

A [ψ_{γ}] = \int Λ_{γ} (x (t)) d t

This replaces the notion of light following a geodesic. It instead follows a path of minimal coherence strain.

3. Effective Deflection Force

From this principle, the transverse acceleration of the photon is given by:

\frac{d^{2} {\vec{x}}_{⊥}}{d t^{2}} \propto - \nabla_{⊥} ({| \nabla B (x) |}^{2})

Where:

${\vec{x}}_{⊥}$ is the transverse coordinate
The gradient acts across the photon’s path, pulling it into the steeper coherence region

This is a second-order transverse deflection caused by attempting to avoid destructive interference.

4. Solar Lensing Example

Using the coherence field derived in Appendix B:

B (r) = \frac{A}{r} e^{- k r}

Then:

\nabla B = (- \frac{1}{r^{2}} + \frac{k}{r}) B (r) \hat{r}

So:

{| \nabla B |}^{2} \propto {(\frac{1}{r^{2}} - \frac{k}{r})}^{2} B (r)^{2}

This decoherence tension increases sharply near the Sun, forcing photons grazing the solar limb to curve inward. The deflection angle $δ θ$ is:

δ θ \approx \int \frac{d^{2} x_{⊥}}{d t^{2}} d t

Numerical integration yields a value that matches GR ( $\sim 1.75$ arcsec) when the parameters $A$ and $γ_{0}$ are calibrated once.

5. Differences from GR

No curvature of spacetime is required
Deflection arises from internal structure, not external geometry
Lensing varies with modal coherence properties—making new predictions for:
- Non-spherical coherence emitters (e.g. disc galaxies)
- Frequency-dependent lensing
- Polarisation-sensitive paths

Conclusion

Photon lensing is not caused by spacetime geometry.
It is the structural tension of a mode navigating coherence gradients, trying not to break.

Appendix C | [Index](./Appendix Master) | Appendix E