This appendix presents the formalism and results for geodesic propagation through a multi-layered spacetime geometry, including analytical and numerical treatment of lensing and time-delay phenomena induced by inter-layer discontinuities and refractive gravitational potentials.
E.1. Framework Overview
In the fractal-layered cosmology, spacetime is composed of hierarchically nested layers, each with slightly differing metric properties and gravitational potential curvatures. The transition interfaces between layers act as semi-permeable gravitational boundaries, giving rise to:
Phase discontinuities in the metric tensor g(i)g_{\mu\nu}^{(i)}
Localized deflections of null geodesics (lensing)
Differential Shapiro delays (effective echoes)
Each layer ii is characterized by its own curvature kik_i, scale factor ai(t)a_i(t), and gravitational refractive index nin_i, defined as:
ni(r,t)1+i(r,t)/c2n_i(r, t) \equiv 1 + \Phi_i(r,t)/c^2
where i\Phi_i is the effective Newtonian potential associated with that layer's matter and vacuum content.
E.2. Geodesic Equation with Layer Interfaces
The geodesic equation in each region (layer ii) remains standard:
d2xd2+dxddxd=0\frac{d^2 x^\mu}{d \lambda^2} + \Gamma^{\mu}_{\alpha \beta} \frac{dx^\alpha}{d \lambda} \frac{dx^\beta}{d \lambda} = 0
However, across a layer boundary at radius r=rijr = r_{ij}, matching conditions must satisfy:
Tangential continuity: xx^\mu continuous across layers
Normal discontinuity: x\partial_\nu x^\mu may jump, leading to lensing-like angular shifts
Deflection angle ij\Delta\theta_{ij} at layer interface is given (to first order) by:
