Geometric Interpretation via Energy Density Localization
In the nonlinear regime, the amplitude I(x,t)2|I(\vec{x}, t)|^2I(x,t)2 serves as an effective energy density. When localized excitations form, they influence the propagation characteristics of subsequent perturbations---mimicking how curvature arises from energy-momentum in General Relativity.
We propose an effective emergent metric tensor geffg_{\mu\nu}^{\text{eff}}geff, not introduced a priori but rather inferred from the field configuration:
geff=+T[I]g_{\mu\nu}^{\text{eff}} = \eta_{\mu\nu} + \alpha T_{\mu\nu}[I]geff=+T[I]
Here:
\eta_{\mu\nu}: background (Minkowski-like) metric,
T[I]T_{\mu\nu}[I]T[I]: effective stress-energy-like tensor derived from field gradients and energy density,
\alpha: scaling constant depending on the medium.
This tensor arises organically from the information field dynamics. For example, we define:
T00=12(It2+c2I2+2I4)T_{00} = \frac{1}{2} \left( \left| \frac{\partial I}{\partial t} \right|^2 + c^2 |\nabla I|^2 + \frac{\lambda}{2} |I|^4 \right)T00=21(tI2+c2I2+2I4)
Regions with high T00T_{00}T00 (energy density) act as effective attractors or curvature sources, deflecting or trapping information pulses, much like gravitational wells.
Analogy with Scalar Curvature RRR
To approximate the emergent curvature, we consider:
Reff=(It2+c2I2+I4)R \sim \kappa \, \rho_{\text{eff}} = \kappa \left( \left| \frac{\partial I}{\partial t} \right|^2 + c^2 |\nabla I|^2 + \lambda |I|^4 \right)Reff=(tI2+c2I2+I4)
