Let's define characteristic scales for experimental adaptation:
Energy scale:
Using EI04L0d\mathcal{E} \sim \lambda I_0^4 L_0^dEI04L0d, for system dimensionality ddd, this gives insight into the energy budget required for emergent excitation.
Frequency scale:
Set by 01/T0I0\omega_0 \sim 1/T_0 \sim \sqrt{\lambda} I_001/T0I0, predicting oscillation modes or resonance peaks.
Length scale:
From balance I2Ic22I\lambda |I|^2 I \sim c^2 \nabla^2 II2Ic22I, we obtain:
L0cI0L_0 \sim \frac{c}{\sqrt{\lambda} I_0}L0I0c
which dictates the minimum spatial resolution or curvature envelope in emergent proto-geometry.
Scaling Implications for Physical Systems
Depending on platform:
In magnonic condensates: I0103TI_0 \sim 10^{-3} \, \text{T}I0103T, c103m/sc \sim 10^3 \, \text{m/s}c103m/s, 104s2T2\lambda \sim 10^4 \, \text{s}^{-2} \text{T}^{-2}104s2T2
In optical BECs: c105m/sc \sim 10^5 \, \text{m/s}c105m/s, and L0106mL_0 \sim 10^{-6} \, \text{m}L0106m
The scaling allows estimation of pulse duration, cavity dimensions, and input power, crucial for building table-top analogs.
Summary of Scaling Laws
These insights provide a unified scaling framework to relate lab-based analogs with cosmological or high-energy analogues. In the next section, we extend the model into 2D and explore the geometric implications of curvature emergence.
C. 2D/3D Extensions and Emergent Metric Tensor
Analogy with Scalar Curvature from Energy-Density Localization
Extending the 1D nonlinear information field equation to higher dimensions enables a richer set of phenomena, including the emergence of spatial curvature analogs and topologically distinct structures. We consider the generalized form of the governing equation in ddd-dimensions:
2I(x,t)t2+Itc22I+I2I=B(x,t)\frac{\partial^2 I(\vec{x}, t)}{\partial t^2} + \gamma \frac{\partial I}{\partial t} - c^2 \nabla^2 I + \lambda |I|^2 I = B(\vec{x}, t)t22I(x,t)+tIc22I+I2I=B(x,t)
with xR2\vec{x} \in \mathbb{R}^2xR2 or R3\mathbb{R}^3R3, and 2\nabla^22 representing the Laplacian in 2D or 3D space.
