Nonlinear Dynamics of the Substrate Field

Nonlinear Dynamics of the Substrate Field: Blow-Up, Stability, and Lagrangian Structure

At this stage, RST must demonstrate that its Substrate Field S forms a mathematically coherent nonlinear field theory. We begin with the uncoupled evolution equation:

∂²S/∂t² − c²∇²S + βS³ = σΨ

For blow-up analysis, we first remove forcing:

∂²S/∂t² − c²∇²S + βS³ = 0

This is the nonlinear Klein–Gordon equation with cubic nonlinearity. Its stability depends entirely on the sign of β.

I. Can the Cubic Term Prevent Blow-Up?

Case A: β > 0 (Defocusing)

The energy functional is:

E = ∫ d³x [ ½(∂_tS)² + ½c²|∇S|² + (β/4)S⁴ ]

With β > 0, the potential is positive definite. Energy controls the L⁴ norm of S, and in 3+1 dimensions the defocusing cubic Klein–Gordon equation is globally well-posed for finite-energy initial data.

Conclusion: β > 0 strongly suppresses finite-time blow-up and stabilizes evolution.

Case B: β < 0 (Focusing)

The potential becomes unbounded below:

V(S) = −|β| S⁴

This allows:

• Collapse solutions
• Finite-time blow-up
• Instability under large amplitudes

Conclusion: RST must choose β > 0 if it intends to prevent singular behavior.

II. Do Collapse Solutions Exist?

With β > 0, true point-collapse is unlikely for finite-energy data in 3+1D. However, localized high-density structures (oscillons, soliton-like states) can form.

When coupling to Ψ is included:

κ S Ψ

the Ψ-field can act as a source term. Strong coupling may induce steep gradients.

Conclusion: RST cannot claim “singularities are impossible,” but it can claim that nonlinear structure strongly resists unbounded growth under stabilizing parameters.

III. Can RST Produce Inflation-Like Behavior?

Inflation requires:

• Accelerated expansion
• Vacuum-dominated energy
• Nearly flat potential

The current RST potential:

V(S) = (β/4) S⁴

does not naturally yield slow-roll inflation. A modified potential:

V(S) = (m²/2)S² + (β/4)S⁴

with m² < 0 produces a symmetry-breaking (Mexican-hat) structure capable of inflation-like behavior. However, the current RST equation lacks:

• Hubble friction term (3H Ṡ)
• Coupling to spacetime expansion

Conclusion: RST requires gravitational backreaction to generate inflation.

IV. Constructing a Consistent Lagrangian

We seek a Lagrangian L that yields the coupled system:

∂²S − c²∇²S + βS³ = σΨ
∂²Ψ − v²∇²Ψ + μΨ + λ|Ψ|²Ψ = κSΨ

A consistent Lagrangian is:

L = ½(∂_μS)(∂_μS) + ½(∂_μΨ)(∂_μΨ) − V(S,Ψ)

with potential:

V(S,Ψ) = (β/4)S⁴ + (μ/2)Ψ² + (λ/4)Ψ⁴ − (κ/2)SΨ²

Euler–Lagrange equations reproduce the RST field equations exactly (up to normalization).

V. Reality Check

RST is currently:

• A nonlinear coupled scalar field theory
• Mathematically viable
• Stabilized by β > 0
• Capable of soliton-like structures

RST is not yet:

• A full theory of gravity
• A proof against singularities
• A cosmological model
• A solution to entropy or inflation

VI. Summary of Findings

• β > 0 suppresses blow-up; β < 0 destabilizes.
• Localized structures exist; true collapse is unlikely but not forbidden.
• Inflation requires gravitational coupling.
• A clean Lagrangian formulation exists.
• RST is mathematically consistent as a nonlinear field theory.