⚛️ My RST-Corrected Relativistic Mass Equation

⚛️ My RST-Corrected Relativistic Mass Equation

When I think about how relativistic mass should be expressed in the Reactive Substrate Theory (RST), I want the equation to reflect the physical properties of the Substrate Field (Σ) itself. Instead of treating the speed of light as an abstract constant, I define it as the maximum propagation speed of Σ tension, determined by the field’s elasticity and density. This corrected form fixes the dimensional inconsistency and shows how mass grows as a soliton approaches the substrate’s intrinsic speed limit.

m_rel = m0 / sqrt(1 - v^2 / Lambda_Sigma^2)

Lambda_Sigma = sqrt(E_Sigma / rho_Sigma)

Explanation: This equation says that the relativistic mass (m_rel) of a Σ soliton increases toward infinity as its velocity (v) approaches the substrate’s maximum wave speed (Lambda_Sigma). That speed limit is not arbitrary—it comes directly from the Σ field’s elasticity (E_Sigma, its restoring tension) and density (rho_Sigma, its inertial resistance). In other words, mass is the energetic “debt” required to keep a soliton’s knot intact while dragging against the substrate. As v → Lambda_Sigma, the substrate cannot propagate tension fast enough, and the energy cost diverges, enforcing the universal speed limit.

⚛️ Minimum Motion Condition for Soliton Stability (RST)

In Reactive Substrate Theory (RST), matter is a Σ–soliton: a localized knot of tension sustained by continuous motion through the substrate (Σ). Absolute rest would eliminate the dynamical balance (oscillation + translation) that keeps the knot coherent, causing it to dissolve back into the substrate potential. The condition for stability can be captured as a compact “minimum motion” threshold.

Omega^2 + (v^2 / L^2) >= Omega_crit^2

Meaning: A soliton with internal oscillation frequency (Omega) and center‑of‑mass speed (v) remains stable only if the combined motion exceeds a critical threshold (Omega_crit). Here, L is the soliton’s characteristic length scale (knot size). If both Omega → 0 and v → 0, the inequality fails and the knot unravels into the substrate.

Omega_crit^2 = alpha * (E_Sigma / rho_Sigma) * (1 / L^2) - beta_nl * A^2

Parameters: E_Sigma and rho_Sigma are the substrate’s elasticity and density; alpha is a geometric factor set by the knot’s curvature; beta_nl is the effective nonlinear locking strength; A is the soliton’s amplitude. Larger nonlinearity (topological locking) lowers the required motion; weaker nonlinearity raises it.

Energy-form equivalent (same idea, different lens)

E_osc + E_trans >= E_bind

E_osc = (1/2) * I_eff * Omega^2 E_trans = (1/2) * m_eff * v^2

Meaning: The sum of internal oscillation energy and translational kinetic energy must meet or exceed the binding energy provided by Σ’s nonlinearity and topology. If the available motion‑energy drops below the binding requirement, the soliton cannot remain localized and “pops back” into the substrate potential.

In short: motion is the glue. A Σ–soliton persists only when its oscillation and/or drift through Σ stay above a threshold set by the field’s material properties and the knot’s topology.

📖 Glossary of Terms (RST Equations)

Symbol Meaning
m_rel Relativistic mass — total energy required to sustain the soliton’s knot at velocity v.
m0 Rest mass — intrinsic energy of the stable soliton knot at v = 0.
v Velocity of the soliton relative to the substrate field Σ.
Lambda_Sigma Maximum propagation speed of Σ tension (RST’s version of c).
E_Sigma Elasticity of the substrate field Σ — its restoring tension or stiffness.
rho_Sigma Density of the substrate field Σ — its inertial resistance to shear.
Omega Internal oscillation frequency of the soliton (its vibration rate).
L Characteristic length scale of the soliton (knot size).
Omega_crit Critical threshold frequency required for soliton stability.
alpha Geometric factor set by the knot’s curvature.
beta_nl Nonlinear locking strength — how strongly the knot resists unraveling.
A Amplitude of the soliton’s oscillation.
E_osc Oscillation energy of the soliton.
E_trans Translational kinetic energy of the soliton.
E_bind Binding energy provided by Σ’s nonlinearity and topology.

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