Emergent Effective Geometry from Nonlinear Complex Scalar Dynamics with Controlled Gain–Loss Imbalance

This work demonstrates that an effective Lorentzian geometry can emerge from nonlinear complex scalar field dynamics with quartic self-interaction and controlled gain–loss imbalance. Starting from a master equation that reduces exactly to the linear Schrödinger equation in the conservative limit, we perform a systematic amplitude–phase decomposition and derive the corresponding continuity and phase equations. Linear perturbations around a homogeneous symmetry-broken background yield a second-order hyperbolic equation for phase fluctuations. This equation can be written in covariant form with respect to an emergent effective metric constructed directly from the background density and flow. The resulting dispersion relation exhibits a density-dependent sound speed and recovers relativistic-like propagation in the conservative regime. In the presence of gain–loss imbalance, the effective geometry acquires a complex deformation, introducing controlled non-Hermitian corrections to the dispersion relation. The framework thus separates conservative emergent geometry from open-system geometric deformation in a mathematically consistent way. The construction does not modify general relativity. Instead, it shows that geometric structure can arise as an effective description of collective excitations in nonlinear scalar dynamics. The model yields concrete, falsifiable predictions for dispersion shifts and instability thresholds in nonlinear optical media and driven condensate systems. This paper extends a unified nonlinear scalar field framework previously developed for quantum conservative dynamics, entanglement structure, thermodynamic stability, and classical emergence. DOI: https://doi.org/10.5281/zenodo.18642787