Quantum Tunneling Through Dual‑Mode Substrate Propagation

Why Tunneling Appears Superluminal in a Pre‑Stressed Substrate

Within Reactive Substrate Theory (RST), the propagation of disturbances depends on the internal state of the Substrate itself. When the Substrate is pre‑stressed, it supports two distinct propagation modes. The first is a transverse mode, limited to the speed of light c, corresponding to the familiar electromagnetic response. The second is a longitudinal pressure mode, which can propagate at approximately √2 · c within a tensioned medium.

This faster longitudinal mode does not transmit information or signals beyond the relativistic limit. Instead, it represents a rapid internal adjustment of the Substrate’s pressure state. Because this adjustment can occur ahead of the transverse disturbance, certain tunneling and wave‑packet experiments may exhibit apparent superluminal behavior. In RST, such effects arise naturally from the dual‑mode response of a pre‑stressed Substrate rather than from any violation of relativistic causality.

RST Core Field Equation

(∂t2S − c2∇2S + βS3) = σ(x,t) · FR(C[Ψ])

Field S(x,t):
S is the Substrate field, representing the local state of tension or displacement in the reactive medium.

Temporal curvature ∂t2S:
The second time derivative describes how the Substrate field accelerates or curves in time.

Spatial curvature −c2∇2S:
The Laplacian term encodes spatial curvature of the field, scaled by c2, linking it to the transverse propagation speed.

Nonlinear self‑interaction βS3:
The cubic term models self‑interaction of the Substrate, allowing soliton‑like, self‑stabilizing structures to form.

Source term σ(x,t):
σ(x,t) represents localized sources, sinks, or external driving of the Substrate at position x and time t.

Reactive functional FR(C[Ψ]):
FR is the reactive response of the Substrate to configuration C[Ψ], where Ψ encodes the matter/field configuration (e.g., particle, soliton, or quantum state). This term couples the Substrate dynamics to the effective configuration space.

Interpretation:
The left side describes the intrinsic dynamics of the Substrate (wave propagation + nonlinearity). The right side describes how configurations Ψ drive and shape the Substrate through a reactive coupling, allowing RST to model particles, fields, and solitons as emergent structures in a single unified medium.

Key Concepts

  • Pre‑Stressed Substrate: The Substrate carries built‑in tension, allowing rapid internal adjustments.
  • Dual Propagation Modes:
    • Transverse mode limited to c
    • Longitudinal pressure mode near √2 · c
  • No Superluminal Signaling: The faster mode adjusts the Substrate but does not transmit information.
  • Tunneling Interpretation: Apparent “superluminal” effects arise from the longitudinal mode preparing the Substrate ahead of the transverse wave.
  • RST Framework: These modes emerge naturally from a reactive, tension‑bearing Substrate rather than from violations of relativity.

RST Equation Summary

  • General propagation in the Substrate:
    v = √(T / ρ)
  • Transverse electromagnetic mode:
    vT = c
    (Defines the speed of light)
  • Longitudinal pressure mode:
    TL = 2TT
    vL = √2 · c
    (Substrate pressure adjustment, not a signal)
  • Barrier pre‑conditioning:
    ΔP(x) ∝ e−kx
    (Longitudinal mode reduces effective tunneling delay)
  • Apparent tunneling velocity:
    veff > c
    (Due to early Substrate adjustment)
  • Actual information speed:
    vsignal = c

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