ij1c2(ji)r\Delta\theta_{ij} \approx \frac{1}{c^2} \left( \nabla_\perp \Phi_j - \nabla_\perp \Phi_i \right) \Delta r
E.3. Lens Equation in Multilayer Spacetime
In the thin-lens approximation generalized to layered media, the lensing potential ()\Psi(\theta) becomes a sum over interface contributions:
total()=i=1N(i)()\Psi_{\text{total}}(\theta) = \sum_{i=1}^{N} \Psi^{(i)}(\theta)
Each layer contributes an effective convergence (i)\kappa^{(i)} and shear (i)\gamma^{(i)}, determined by the projected interfacial curvature gradients:
(i)()(i)()crit,(i)()2proj(i)\kappa^{(i)}(\theta) \equiv \frac{\Sigma^{(i)}(\theta)}{\Sigma_{\text{crit}}}, \quad \Sigma^{(i)}(\theta) \propto \nabla^2 \Phi^{(i)}_{\text{proj}}
The resulting deflection field ()\vec{\alpha}(\theta) is:
()=total()\vec{\alpha}(\theta) = \nabla_\theta \Psi_{\text{total}}(\theta)
E.4. Gravitational Time Delays and Echoes
Propagation through layers with varying effective potentials introduces layered Shapiro delays, measurable as:
