Black Hole Interior as an EPT in QFunity

Black Hole Interior as an EPT in QFunity

Reinterpreting singularities as stable Emergent Pre-Temporal structures

From Stellar Collapse to EPT Formation

QFunity redefines the interior of a black hole, formed from stellar collapse, as an Emergent Pre-Temporal (EPT) structure rather than a GR singularity. This page details the process using QFunity’s equations, showing how scale-dependent dynamics prevent infinite density.

1. Initiation of Collapse and Scale Transition

Stellar collapse compresses matter beyond \(r_s = \frac{2GM}{c^2}\). QFunity’s scale-dependent metric governs the transition:

\[ g_{\mu\nu}(\epsilon) = g_{\mu\nu}^{GR} + \frac{\ell_P^2}{\epsilon^2} g_{\mu\nu}^{\text{LQG}} + \alpha’ g_{\mu\nu}^{\text{strings}} \]

Scale Dynamics:

The critical scale is \(\epsilon_{\text{crit}} = \sqrt{\frac{\hbar G}{c^3 \omega}}\), with \(\omega \sim \frac{c^3}{G M}\). As \(\epsilon \to \ell_P\), quantum corrections dominate, halting infinite curvature.

2. Activation of the Master Equation

The collapse triggers the master equation:

\[ \lim_{\epsilon \to 0^\pm} \left[ \hat{\mathbb{B}}_\epsilon \hat{\mathbb{V}}_\epsilon – \hat{\mathbb{V}}_\epsilon \hat{\mathbb{B}}_\epsilon^2 \right] \Psi = \Lambda \cdot \frac{\Psi}{\|\Psi\|^2 + \epsilon^2} \]

Operator Definitions:

\(\hat{\mathbb{B}}_\epsilon = \epsilon^2 (\nabla \times \boldsymbol{\omega})\), with \(\boldsymbol{\omega} = \kappa \rho_{\text{vac}} \mathbf{v}\).

\[ \hat{\mathbb{V}}_\epsilon = -\frac{\hbar^2}{2\epsilon^2} \nabla^2 + \frac{\rho_{\text{vac}}(\epsilon)}{\epsilon^2}, \quad \rho_{\text{vac}}(\epsilon) = \rho_0 \epsilon^{-4} e^{-\epsilon/\ell_P} \]

Commutator: \(\left[ \hat{\mathbb{B}}_\epsilon, \hat{\mathbb{V}}_\epsilon \right] \approx \epsilon^4 (\nabla \times \boldsymbol{\omega}) \nabla^2 – \frac{\hbar^2}{2\epsilon^2} (\nabla \times \boldsymbol{\omega})^2\).

3. Formation of the EPT Core

The EPT forms a stable vortex:

\[ \nabla \times \Omega_{\text{QF}} = \kappa \rho_{\text{vac}} \mathbf{v} \times \hat{\mathbf{s}}, \quad \Omega_{\text{QF}} \sim \frac{\kappa G M \omega}{c^3 \epsilon^2} \]

Stability:

\(E_{\text{rot}} = \frac{1}{2} I \omega^2\), \(I \sim M r_s^2\), vs. \(E_{\text{grav}} = \frac{G M^2}{r}\). Superposition state:

\[ \Psi_{\text{EPT}} = \sum_n c_n |\text{State}_n\rangle, \quad |c_n|^2 = \frac{|\langle \text{State}_n | \hat{\mathbb{V}}_\epsilon | \Psi \rangle|^2}{|\langle \Psi | \Psi \rangle|^2 + \epsilon^2} \]

4. Stability and Non-Singularity

Density remains finite:

\[ \rho_{\text{EPT}} = \frac{\rho_{\text{vac}}(\epsilon)}{\epsilon^2} + \frac{E_{\text{rot}}}{\text{Vol}_{\text{fractal}}}, \quad \text{Vol}_{\text{fractal}} \sim r_s^3 \left(\frac{\epsilon}{\ell_P}\right)^{D_f – 3} \]

Fractal Dimension:

\(D_f = 2 + \frac{\log(\epsilon/\epsilon_0)}{\log N}\). Continuity:

\[ \frac{d}{d\epsilon} \left( \hat{\mathbb{B}}_\epsilon \Psi_{\text{EPT}} \right) = \frac{\partial \hat{\mathbb{B}}_\epsilon}{\partial \epsilon} \Psi_{\text{EPT}} + \hat{\mathbb{B}}_\epsilon \frac{\partial \Psi_{\text{EPT}}}{\partial \epsilon} \]

5. Event Horizon as Perceptual Boundary

The horizon scales with \(\epsilon\):

\[ r_s(\epsilon) = \frac{2GM}{c^2} \left(1 + \frac{\ell_P^2}{\epsilon^2}\right)^{-1} \]

Testable Implication: Gravitational Waves

Detect \(\epsilon\)-dependent shifts:

\[ f_{\text{GW}} \propto \frac{\omega}{\epsilon^2}, \quad \Delta f \sim \frac{\partial f_{\text{GW}}}{\partial \epsilon} \Delta \epsilon \]

Test with LIGO/Virgo merger data.