3. Mathematical Framework

3.1 State Variables and Fields

The framework is defined on a continuous spatial domain with explicit time dependence. The fundamental state of the system is specified by three fields:

• The scalar density field

interpreted as an informational or mass-like density.

• The velocity field

which carries inertial degrees of freedom and enables momentum storage and transport.

• The emergent potential

which is not postulated as a gravitational potential a priori, but is generated dynamically from the density field.

No additional force fields, metrics, or geometric structures are assumed. All subsequent dynamics arise solely from the interaction of these variables.

3.2 Dynamical Equations of the Inertial Emergent Gravity Model

The evolution of the system is governed by three coupled equations. These equations constitute the complete dynamical content of the model.

Continuity Equation

Conservation of density is enforced through the standard transport equation:

This equation ensures that the density field evolves solely through advection by the velocity field.

Equation of Motion with Damping

The velocity field evolves according to an inertial equation with linear damping:

Here, $\gamma$ is a damping coefficient controlling the transition between overdamped and inertial regimes. The presence of the time derivative $\partial_t \mathbf{v}$ is essential: without it, orbital and cyclic dynamics are strictly forbidden.

Screened Poisson Equation

The emergent potential is determined instantaneously from the density field via a screened Poisson equation:

The subtraction of the spatial mean $\langle \rho \rangle$ enforces global neutrality, while the screening scale $\mu^{-1}$ controls the effective range of interaction. This equation introduces no force law; it merely defines how $\Phi$ responds to density fluctuations.

3.3 Extracted Observables (Non-Assumed Quantities)

All diagnostic quantities used to classify gravitational behavior are extracted from the evolving fields. None are imposed at the level of the equations.

Centers of Mass

For each localized density component $\rho_i(\mathbf{x},t)$, the center of mass is defined as

Binary Separation

The instantaneous separation between two density concentrations is

Radial Velocity

The radial velocity is obtained by temporal differentiation:

Zero crossings of $\dot d(t)$ serve as indicators of turning points in the relative motion.

Angular Momentum Proxy

An effective angular momentum diagnostic is defined by

where $\mathbf{v}_{\mathrm{eff}}(t)$ is the spatially averaged velocity within a localized density region. This quantity is not conserved a priori, but its persistence or decay provides a direct measure of inertial structure.

The time-averaged magnitude,

plays a central role in classifying dynamical phases in later sections.