Meta Theory of the Zeckendorf-Hilbert Universe

14.3 Information Flow and Geometric Focusing: Proving Generalized Horizon Monotonicity Using QNEC

In Section 14.2, we derived the Quantum Null Energy Condition (QNEC) from second-order stability of generalized entropy. This condition $⟨T_{kk}⟩\ge \frac{\hslash }{2\pi }{S}_{\text{mat}}^{\prime \prime }$ allows energy density to be negative, but sets strict information-theoretic lower bounds on its negative values. This section will prove that QNEC is not merely a static inequality—it is the microscopic guarantee of the Generalized Second Law (GSL) in spacetime geometric dynamics.

We will introduce the Quantum Focusing Conjecture (QFC), which is the quantum generalization of the classical focusing theorem. We will show that QNEC guarantees that even when classical energy conditions fail (gravitational repulsion), the “generalized horizon region” including quantum corrections still follows monotonic non-decreasing evolution laws. This not only saves black hole thermodynamics, but also reveals the essence of gravity: spacetime curves to conform to the irreversibility of information flow.

14.3.1 Failure of Classical Focusing and Generalized Expansion Scalar

Recalling the Raychaudhuri equation (Section 11.4), the classical focusing theorem $\frac{d\theta }{d\lambda }\le 0$ depends on $R_{kk}\ge 0$ (i.e., NEC, $T_{kk}\ge 0$). When negative energy such as the Casimir effect exists, $R_{kk}<0$, causing $\frac{d\theta }{d\lambda }>0$, beams may diverge, and horizon area may decrease. This seems to violate the second law of thermodynamics.

To resolve this contradiction, we must combine “geometric area” with “matter entropy.” We define the Generalized Expansion Scalar $\Theta$.

Definition 14.3.1 (Generalized Expansion Scalar $\Theta$)

For a bundle of null geodesics, let the cross-sectional area element be $\mathcal{A}$. The generalized expansion scalar $\Theta$ is defined as the first-order variation (normalized) of generalized entropy $S_{\text{gen}}$ with respect to affine parameter $\lambda$:

$$\Theta \equiv \frac{4G\hslash }{\mathcal{A}}\frac{d{S}_{\text{gen}}}{d\lambda }=\theta +\frac{4G\hslash }{\mathcal{A}}\frac{d{S}_{\text{mat}}}{d\lambda }$$

where $\theta =\frac{1}{\mathcal{A}}\frac{d\mathcal{A}}{d\lambda }$ is the classical geometric expansion rate, and $S_{\text{mat}}^{\prime }$ is the rate of change of matter entanglement entropy along the beam.

Physical Significance:

$\Theta$ describes the rate of change of the “total information cross-section.” Even if the geometric cross-section is contracting ($\theta <0$), if internal entanglement entropy is rapidly increasing ($S_{\text{mat}}^{\prime }>0$), the total generalized horizon may still be “expanding.”

14.3.2 Quantum Focusing Conjecture (QFC)

The classical focusing theorem asserts ${\theta }^{\prime }\le 0$. As its holographic generalization, we propose the quantum focusing conjecture.

Axiom 14.3.2 (Quantum Focusing Conjecture / QFC)

For any bundle of null geodesics, the generalized expansion scalar $\Theta$ is monotonically decreasing along the affine parameter:

$$\frac{d\Theta }{d\lambda }\le 0$$

That is:

$$\frac{d\theta }{d\lambda }+\frac{4G\hslash }{\mathcal{A}}\frac{d^2{S}_{\text{mat}}}{d{\lambda }^2}\le 0$$

(ignoring coupling of higher-order terms such as $\theta S^{\prime }$).

Relationship between QFC and QNEC:

Substituting the Raychaudhuri equation ${\theta }^{\prime }=-R_{kk}-\frac{1}{d-2}{\theta }^2-{\sigma }^2$ into the QFC inequality:

$$-R_{kk}-\frac{1}{d-2}{\theta }^2-{\sigma }^2+\frac{4G\hslash }{\mathcal{A}}{S}_{\text{mat}}^{\prime \prime }\le 0$$

In the local limit (taking $\mathcal{A}\to 0$ or considering plane wavefronts such that $\theta \approx 0,\sigma \approx 0$), the above reduces to:

$$R_{kk}\ge \frac{4G\hslash }{\mathcal{A}}{S}_{\text{mat}}^{\prime \prime }$$

Using Einstein’s equation $R_{kk}=8\pi G{T}_{kk}$, this is precisely QNEC ($T_{kk}\ge \frac{\hslash }{2\pi \mathcal{A}}{S}_{\text{mat}}^{\prime \prime }$).

Therefore, QNEC is the local limit of QFC. Conversely, if QNEC holds, then QFC also holds in the semiclassical approximation (because ${\theta }^2$ and ${\sigma }^2$ are always non-negative, only strengthening the inequality).

14.3.3 Generalized Horizon Monotonicity Theorem

Using QFC, we can prove a strong version of the Generalized Second Law (GSL).

Theorem 14.3.3 (Generalized Horizon Monotonicity)

Consider a causal horizon (such as a black hole event horizon or de Sitter horizon) whose generating lines are null geodesics. If the horizon tends to a steady state in the distant future ($\Theta \to 0$ as $\lambda \to \infty$), then at any time $\lambda$, the generalized expansion rate must be non-negative:

$$\Theta (\lambda )\ge 0$$

This means generalized entropy $S_{\text{gen}}$ increases monotonically with time.

Proof:

According to QFC, $\Theta (\lambda )$ is a monotonically decreasing function (${\Theta }^{\prime }\le 0$).

If in the future $\Theta (\infty )=0$ (thermal equilibrium state), then for any finite time $\lambda$, we must have $\Theta (\lambda )\ge \Theta (\infty )=0$.

$$\frac{d{S}_{\text{gen}}}{d\lambda }=\frac{\mathcal{A}}{4G\hslash }\Theta \ge 0$$

$\square$

Physical Corollary:

Even when a black hole is evaporating (Hawking radiation causes $\theta <0$, decreasing geometric horizon area), the radiated entanglement entropy flow $S_{\text{mat}}^{\prime }$ must be sufficiently large such that $\Theta =\theta +4G{S}^{\prime }\ge 0$.

This proves that the Generalized Second Law (GSL) strictly holds in semiclassical gravity, provided matter obeys QNEC.

14.3.4 Information Flow Drives Geometric Focusing

The conclusion of this section completely reverses causality:

• Traditional View: Mass curves spacetime $\to$ light rays focus $\to$ information is captured.

• Entropic Gravity View: Information flow structure ($S_{\text{mat}}^{\prime \prime }$) sets inequality constraints $\to$ energy density $T_{kk}$ must satisfy QNEC $\to$ spacetime geometry must curve to maintain QFC.

Essence of Geometric Focusing:

Why does gravity cause focusing (attraction)? Because information processing is irreversible.

When we trace a quantum system along light rays, the rate at which we gain information ($S^{\prime }$) tends to slow down ($S^{\prime \prime }<0$ has a lower bound, or is constrained by QFC). To compensate for this “spreading” or “forgetting” of information, the geometric cross-section must contract ($\theta <0$), to “focus” physical degrees of freedom into smaller phase space volumes, thereby maintaining the validity of holographic bounds.

Gravitational collapse is the geometric manifestation of information compression.

14.3.5 Summary

This section used QNEC and QFC to prove generalized horizon monotonicity.

1. Unity: QFC unifies geometric focusing (Raychaudhuri) with entropy concavity (information theory) into a single inequality.

2. Stability: Proved that as long as microscopic matter satisfies QNEC, macroscopic spacetime will not violate the second law of thermodynamics, even in the presence of negative energy.

3. Directionality: The direction of spacetime evolution (gravitational attraction) is locked by the irreversibility of information flow (generalized entropy increase).

In the next section 14.4, we will explore the ultimate consequence of this focusing—singularities. We will prove that singularities are not the collapse of spacetime, but thermodynamic phase transition points (geometric caustics) caused by information density saturation.