>Reactive Substrate Theory (RST): A Metric-Coupled Framework for Gravitational Dynamics, Renormalization, and Cosmological Behavior
Reactive Substrate Theory (RST): A Metric-Coupled Framework for Gravitational Dynamics, Renormalization, and Cosmological Behavior
Executive Summary
This white paper presents a formalized version of Reactive Substrate Theory (RST) within a metric-coupled framework. The objective is to evaluate whether a scalar-field-based substrate model can reproduce gravitational phenomena, avoid curvature singularities, remain renormalizable in 3+1 dimensions, and provide coherent explanations for entropy, black hole thermodynamics, entanglement, and cosmological expansion. The analysis demonstrates that RST, when coupled to Einstein gravity, yields a consistent stress-energy source, exhibits saturation behavior that prevents curvature divergence, reduces to General Relativity (GR) in the weak-field limit, and maintains renormalizability through its quartic self-interaction. Additional sections examine dissipation, impedance-driven propagation velocity, and large-scale cosmological implications.
1. Problem Statement
General Relativity provides a geometric description of gravitation but encounters several unresolved issues:
- Curvature singularities in high-density regimes
- Non-renormalizability at the quantum level
- Limited integration with field-theoretic descriptions of matter
- Ambiguous thermodynamic interpretation of entropy
- Unexplained acceleration of cosmic expansion
RST proposes that spacetime dynamics emerge from a scalar substrate field S, whose stress-energy tensor acts as the source of curvature. This paper evaluates the viability of this approach.
2. Theoretical Basis
2.1 Metric Coupling
RST extends the Einstein–Hilbert action by including a scalar-field Lagrangian Lrst. The total action is:
S_total = ∫ d⁴x √(-g) [ (1 / 2κ) R + L_rst ]
The field equations become:
G_uv + Λ g_uv = κ T_uv(S)
The stress-energy tensor for the substrate field is:
T_uv(S) = ∂_u S ∂_v S − g_uv [ 1/2 ∂_a S ∂_a S − V(S) ]
The potential is defined as:
V(S) = 1/2 m² S² + 1/4 β S⁴
3. Model Description
3.1 Avoidance of Curvature Singularities
In GR, singularities arise when the energy density ρ → ∞, causing the Ricci scalar R to diverge. In RST:
- The quartic term β S⁴ increases rapidly with S
- The cubic term in the equation of motion produces a nonlinear outward pressure
- For β > 0, the energy cost of further compression increases faster than gravitational attraction
This leads to a saturated equilibrium rather than divergence. The curvature scalar approaches a finite limit proportional to the substrate yield point Sy.
3.2 Weak-Field Limit and Recovery of GR
For S ≪ 1:
- Nonlinear terms β S³ become negligible
- The field equation reduces to a Klein–Gordon form
- Identifying ∇S ≈ ∇Φ reproduces the Poisson equation:
∇² Φ = 4πGρ
Thus, RST is consistent with GR in low-density environments.
4. Renormalization Analysis
GR is non-renormalizable due to the dimensionality of the gravitational coupling constant. RST avoids this issue because:
- The fundamental field is scalar
- The interaction term β S⁴ is renormalizable in 3+1 dimensions
- Quantum corrections can be computed without graviton-based divergences
Gravity emerges as an effective macroscopic interaction mediated by the substrate field rather than a fundamental quantum force.
5. Dissipation and Entropy
RST defines entropy as a dissipation process within the substrate. The dissipation factor δ quantifies energy loss from coherent field configurations into a stochastic noise floor η(t).
The substrate loading equation is:
L = α S + β S³ + η(t) + δ ∫ (∂_t S)² dt
Key implications:
- δ represents internal friction
- η(t) accumulates dissipated energy
- Long-term evolution leads to a uniform noise-dominated state
This provides a mechanical interpretation of thermodynamic irreversibility.
6. Black Hole Thermodynamics
6.1 Saturation Near the Horizon
As mass–energy approaches a dense region:
- S → Sy
- c(S) → 0
- Field updates become increasingly inhibited
6.2 Entropy Scaling
The event horizon corresponds to the region where field propagation becomes maximally inhibited. The entropy scales with area because the boundary layer is where information-carrying field gradients are suppressed.
7. Propagation Velocity and Gravitational Effects
The speed of light is interpreted as the substrate update velocity:
c(S) = c₀ √(1 − S / S_y)
Consequences:
- Time dilation arises from reduced update rates
- Shapiro delay results from increased local impedance
- Event horizons occur where c(S) → 0
This provides a mechanical interpretation of gravitational time dilation and horizon formation.
8. Entanglement as Resonance
RST models entanglement as a shared resonance condition between two substrate configurations:
ω(S₁) = ω(S₂)
A measurement imposes a global constraint on the shared resonance. Although the adjustment is simultaneous within the substrate, it does not enable superluminal communication because the resulting change is indistinguishable from noise until classical comparison occurs.
9. Cosmological Expansion
RST interprets cosmic expansion as a change in the global update velocity over time. As dissipated energy increases η(t), the effective impedance decreases, causing the update velocity c(t) to increase. This produces an apparent recession of distant objects.
The expansion relation is:
v(H) = H d
Acceleration does not require dark energy; it emerges from the substrate relaxing toward its low-energy equilibrium state.
10. Limitations and Future Work
- The theory requires explicit numerical simulations of collapse scenarios
- Coupling to standard model fields must be formalized
- Cosmological predictions must be compared with observational datasets
- The behavior of S in high-temperature regimes requires further analysis
Conclusion
Reactive Substrate Theory, when formulated as a metric-coupled scalar field model, provides a coherent and renormalizable framework for gravitational dynamics. It avoids curvature singularities, reproduces GR in the weak-field limit, offers a mechanical interpretation of entropy and black hole thermodynamics, and yields a substrate-based explanation for cosmological expansion. Further work is required to integrate RST with particle physics and observational cosmology, but the theoretical foundation is internally consistent and mathematically viable.
