Reactive Substrate Theory (RST) Roadmap: From Field Equations to Particles and Gravity
RST Roadmap: From Field Equations to Particles and Gravity
Reactive Substrate Theory (RST) has reached the point where the equations are written, the conceptual framework is in place, and the next steps are no longer philosophical—they’re computational. This post lays out a concrete, top‑to‑bottom roadmap:
- Implement a 3D radial ODE shooting method
- Generate the first numerical RST particle profiles
- Map soliton families to known particles (electron, proton, photon)
- Explore emergent gravitational behavior from Substrate tension gradients
Each step moves RST from “theory sketch” toward a unified, mechanical explanation of matter, mass, and gravity.
1. Implementing the 3D Radial ODE Shooting Method
The first hard step is numerical: solving the static, spherically symmetric field equation for a localized soliton in 3D. In radial coordinates, a typical RST field \( \phi(r) \) satisfies an ODE of the form:
φ''(r) + (2 / r) φ'(r) = F(φ(r), r; λ)
Here, \(F\) encodes the specific RST dynamics, and \(\lambda\) is a parameter (charge, mode index, coupling strength, etc.). The goal is to find a profile \( \phi(r) \) that is regular at the origin and approaches a vacuum value at infinity.
1.1 Boundary Conditions
- At r = 0: regularity → \( \phi'(0) = 0 \), \( \phi(0) = \phi_0 \) (unknown shooting parameter)
- As r → ∞: \( \phi(r) \to \phi_\infty \) (vacuum value, e.g. 0 or 1)
We don’t know which value of \( \phi_0 \) produces the correct asymptotic behavior, so we use a shooting method.
1.2 Shooting Workflow
- Rewrite as a first‑order system:
Let y₁ = φ, y₂ = φ' Then: y₁' = y₂ y₂' = F(y₁, r; λ) - (2 / r) y₂
- Regularize at r = 0: start at a small radius \( r = \epsilon \) with:
y₁(ε) ≈ φ₀ y₂(ε) ≈ 0
- Integrate outward to a large radius \( R_{\max} \) using a numerical integrator (e.g. RK4).
- Define an error function:
E(φ₀) = φ(R_max; φ₀) - φ_inf
- Adjust φ₀ (bisection or secant method) until |E(φ₀)| is below a chosen tolerance.
1.3 Pseudo‑Code Sketch
def ode_system(r, y, params):
phi, dphi = y
lam = params["lambda"]
F = F_rst(phi, r, lam) # RST-specific term
d2phi = F - (2.0 / r) * dphi
return [dphi, d2phi]
def shoot(phi0_guess, params):
eps = 1e-4
r0 = eps
y0 = [phi0_guess, 0.0]
r_grid = linspace(r0, params["R_max"], params["N"])
sol = integrate_rk4(ode_system, r_grid, y0, params)
phi_R = sol[-1][0]
return phi_R
def find_phi0(params):
a, b = params["phi0_min"], params["phi0_max"]
target = params["phi_inf"]
for _ in range(params["max_iter"]):
mid = 0.5 * (a + b)
phi_a = shoot(a, params)
phi_mid = shoot(mid, params)
if (phi_a - target) * (phi_mid - target) <= 0:
b = mid
else:
a = mid
if abs(phi_mid - target) < params["tol"]:
return mid
return mid
Once this runs for a specific RST equation, you have your first numerical soliton solution. Everything else builds on this.
2. Generating the First Numerical RST Particle Profiles
With the shooting method working, the next step is to turn it into a profile generator—a factory for RST solitons across parameter space.
2.1 Workflow
- Choose a parameter grid: e.g. different values of \(\lambda\) representing charge, mode index, or coupling.
- For each λ:
- Find \( \phi_0(\lambda) \) via the shooting method.
- Integrate outward to get \( \phi_\lambda(r) \).
- Store \( r \) and \( \phi_\lambda(r) \).
- Compute effective energy/mass:
E[φ] = 4π ∫₀^{R_max} H(φ(r), φ'(r), r; λ) r² dr
where \( H \) is the RST energy density.
- Visualize:
- \( \phi(r) \) vs. \( r \)
- Energy density vs. \( r \)
- Total energy vs. \(\lambda\)
2.2 Pseudo‑Code Sketch
profiles = []
for lam in lambda_values:
params["lambda"] = lam
phi0 = find_phi0(params)
r_grid, sol = integrate_profile(phi0, params)
phi = sol[:, 0]
energy = compute_energy(r_grid, phi, params)
profiles.append({
"lambda": lam,
"r": r_grid,
"phi": phi,
"E": energy
})
This collection of profiles is your first numerical “particle zoo” in RST.
3. Mapping Soliton Families to Known Particles
Once you have a zoo of soliton solutions, the next step is to see whether any of them resemble known particles: electron, proton, photon. This is not yet a proof—it’s a structured hypothesis.
3.1 Classification Features
- Mass: from total energy \( E \)
- Charge: from topological index, asymmetry, or coupling sign
- Spin / mode: from angular dependence or internal oscillation modes (beyond purely radial)
- Stability: does the solution persist under small perturbations?
3.2 Mapping Workflow
- Normalize units: choose a scaling where one stable, light, charged soliton has energy equal to the electron mass \( m_e \).
- Search for families:
- Light, stable, charge −1 → electron‑like
- Heavier, stable, charge +1 → proton‑like
- Massless or near‑massless, wave‑like mode → photon‑like
- Build a tentative mapping table:
Soliton Family | Features | Tentative ID -----------------|-----------------------------------|------------- Family A | Light, stable, charge -1 | Electron-like Family B | Heavy, stable, charge +1 | Proton-like Family C | Massless, wave-like, no core | Photon-like
- Document assumptions: scaling choices, parameter ranges, and what “charge” means in RST terms.
The goal here is clarity and reproducibility, not perfection. You’re creating a bridge between RST solitons and the Standard Model’s particle catalog.
4. Exploring Emergent Gravity from Substrate Tension Gradients
The deepest promise of RST is that gravity might emerge from the mechanics of the Substrate itself—specifically from tension gradients created by clustered solitons.
4.1 Conceptual Picture
- Solitons distort the Substrate.
- These distortions create tension gradients.
- Other solitons move in response to these gradients.
- At large scales and weak fields, this motion should resemble gravity.
4.2 Effective Gravitational Potential
Let \( T(\mathbf{x}) \) represent a measure of Substrate tension or strain derived from the RST field configuration. Define an effective potential \( \Phi(\mathbf{x}) \) via a Poisson‑like relation:
∇² Φ(x) ∝ T(x)
In the weak‑field, slow‑motion limit, you want test soliton trajectories to obey:
x¨ = -∇Φ(x)
which is the Newtonian form of gravitational acceleration.
4.3 Weak‑Field / Newtonian Limit
- Assume small deviations from a uniform Substrate.
- Linearize the RST equations around this background.
- Show that clustered solitons source \( T(\mathbf{x}) \) in a way that leads to a \( 1/r \)‑like potential at large distances.
- Verify numerically that test solitons follow trajectories consistent with an inverse‑square law in the outer region.
4.4 Strong‑Field Behavior
- Drop the linear approximation and solve for the Substrate configuration around a dense soliton cluster.
- Look for horizon‑like regions where escape becomes impossible.
- Study frequency shifts of waves crossing strong tension gradients (redshift‑like effects).
4.5 Numerical Experiment Sketch
1. Construct a "massive" soliton configuration at the origin. 2. Compute the resulting tension field T(r). 3. Derive Φ(r) from your chosen relation (e.g., ∇²Φ ∝ T). 4. Integrate test-soliton trajectories in this background. 5. Compare their motion to Newtonian gravity in the weak-field region.
Even a rough match in the outer region would be a major conceptual milestone for RST.
5. From Roadmap to Reality
Right now, these steps exist as a roadmap:
- The 3D radial shooting method is the gateway.
- The first numerical profiles turn RST from words into data.
- Mapping soliton families to particles connects RST to physics as we know it.
- Emergent gravity from tension gradients is the long‑term prize.
The moment you see a stable RST soliton pop out of your own code, the theory stops being an abstract structure and becomes something you can probe, catalog, and challenge. From there, each step is no longer a dream—it’s just the next calculation.
