Meta Theory of the Zeckendorf-Hilbert Universe

14.2 Derivation of Quantum Null Energy Condition (QNEC): Non-positivity of Second-Order Variation of Generalized Entropy

In Section 14.1, we saw that the classical Null Energy Condition (NEC) is universally violated in quantum field theory, leading to potential instability of semiclassical gravity. To save the causal structure of spacetime, we need a weaker but more robust energy condition against quantum fluctuations.

This section will prove that if we insist on the second-order stability requirement of the Maximum Entanglement Equilibrium Axiom (MEEA)—that generalized entropy near equilibrium (vacuum) is not only first-order stationary but also locally maximal (second-order variation non-positive)—we can directly derive the Quantum Null Energy Condition (QNEC). QNEC relates the lower bound of energy density to the second derivative of entanglement entropy, showing that energy can be negative, but not more negative than the rate of information loss. This is the first pointwise inequality in physics that explicitly connects local energy density with quantum information.

14.2.1 Stability Assumption for Generalized Entropy

In Chapter 12, we used $\delta S_{\text{gen}}=0$ to derive Einstein’s field equations. This corresponds to the equilibrium condition in thermodynamics. However, a stable physical system should have its equilibrium state as a maximum point of entropy (at least locally).

Axiom 14.2.1 (Generalized Entropy Stability)

For any small causal diamond $\Sigma$ in spacetime, if its geometry and matter fields are in the ground state (vacuum or near-vacuum), then the generalized entropy functional $S_{\text{gen}}$ should have non-positive second-order variation relative to deformations of null slices:

$$\frac{d^2{S}_{\text{gen}}}{d{\lambda }^2}\le 0$$

where $\lambda$ is the affine parameter along horizon generating lines (null geodesics).

Physical Picture:

This condition is equivalent to requiring vacuum to be a “mountain peak” regarding geometry and entanglement. Any dynamical evolution deviating from vacuum (whether caused by geometric curvature or matter excitation) will cause the growth rate of generalized entropy to slow (concave function property), thereby ensuring the system does not spontaneously run out of control toward unstable directions of infinite entropy increase.

14.2.2 Second-Order Expansion of Geometric Term: Return of Raychaudhuri Equation

Generalized entropy consists of geometric part $S_{\text{grav}}=A/4G$ and matter part $S_{\text{mat}}$. We first calculate the second derivative of the geometric term.

Consider a bundle of null generating lines of the causal diamond edge (holographic screen), with tangent vector $k^{\mu }=(\frac{d}{d\lambda }{)}^{\mu }$.

The first derivative of area $A(\lambda )$ is given by the expansion scalar $\theta$:

$$\frac{dA}{d\lambda }={\int }_{\partial \Sigma }\theta \sqrt{h}{d}^{d-2}y$$

We choose a cross-section such that at initial time $\lambda =0$ it is an extremal surface (Minimal Surface), i.e., $\theta {|}_{\lambda =0}=0$ (or $\delta \theta =0$).

The second derivative of area is:

$$\frac{d^{2}A}{d{\lambda }^2}{|}_{\lambda =0}={\int }_{\partial \Sigma }\frac{d\theta }{d\lambda }\sqrt{h}{d}^{d-2}y$$

Substituting the Raychaudhuri equation (Section 11.4.2), and neglecting the shear term ${\sigma }^2$ (higher-order small quantity in the small diamond limit, or second-order for vacuum perturbations):

$$\frac{d\theta }{d\lambda }=-R_{\mu \nu }{k}^{\mu }{k}^{\nu }-\frac{1}{d-2}{\theta }^2-{\sigma }^2\approx -R_{kk}$$

where $R_{kk}\equiv R_{\mu \nu }{k}^{\mu }{k}^{\nu }$.

Therefore, the second-order variation of geometric entropy is:

$$\frac{d^2{S}_{\text{grav}}}{d{\lambda }^2}=\frac{1}{4G\hslash }\frac{d^{2}A}{d{\lambda }^2}\approx -\frac{1}{4G\hslash }{\int }_{\partial \Sigma }{R}_{kk}dA$$

14.2.3 Second-Order Shape Derivative of Matter Term

Now consider the change of matter entanglement entropy $S_{\text{mat}}$ with $\lambda$.

In QFT, when the entanglement surface deforms along null directions, the rate of change of entanglement entropy is usually divergent. However, QNEC focuses on the second derivative $S_{\text{mat}}^{\prime \prime }\equiv \frac{d^2{S}_{\text{mat}}}{d{\lambda }^2}$.

Bousso, Fisher, Leichenauer, and Wall (2016) proved that for relativistic quantum field theory, the second derivative of entanglement entropy is well-defined and closely related to the energy-momentum tensor.

Combining both parts, the stability condition becomes:

$$\frac{d^2{S}_{\text{gen}}}{d{\lambda }^2}=-\frac{1}{4G\hslash }\int R_{kk}+\frac{d^2{S}_{\text{mat}}}{d{\lambda }^2}\le 0$$

For pointwise cases (taking the integration region infinitesimal), we obtain:

$$R_{kk}\ge 4G\hslash \frac{d^2{S}_{\text{mat}}}{d{\lambda }^2}{|}_{\text{density}}$$

(Note: Here $S^{\prime \prime }$ is understood as entropy variation rate density per unit area, or defined through limit ${\lim }_{A\to 0}\frac{1}{A}{S}^{\prime \prime }$).

14.2.4 Derivation of Quantum Null Energy Condition

Using Einstein’s field equation $R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }=8\pi G{T}_{\mu \nu }$.

For null vector $k^{\mu }$, since $g_{\mu \nu }{k}^{\mu }{k}^{\nu }=0$, the equation contracts to:

$$R_{kk}=8\pi G{T}_{kk}$$

Substituting into the stability inequality:

$$8\pi G{T}_{kk}\ge 4G\hslash S_{\text{mat}}^{\prime \prime }$$

Canceling $4G$, we obtain the famous Quantum Null Energy Condition (QNEC):

Theorem 14.2.2 (QNEC)

For any point $p$ and any null direction $k$ in relativistic quantum field theory, the renormalized energy-momentum tensor expectation value $⟨T_{\mu \nu }⟩$ is bounded below by the second derivative of entanglement entropy:

$$⟨T_{kk}⟩\ge \frac{\hslash }{2\pi }\frac{d^2{S}_{\text{mat}}}{d{\lambda }^2}$$

(Note: The coefficient $\hslash /2\pi$ comes from normalization of Unruh temperature effects, usually written as $S^{\prime \prime }/2\pi$ in geometric units).

14.2.5 Physical Meaning: Energy Limited by Information

QNEC is the first non-classical energy condition in the history of physics. It profoundly changes our understanding of energy and gravity:

1. Allows Negative Energy: QNEC does not require $T_{kk}\ge 0$. If the second derivative of entanglement entropy $S^{\prime \prime }$ is negative (i.e., entanglement entropy “accelerates decrease” with deformation), then energy density can be negative.

2. Cost of Negative Energy: Energy cannot be “arbitrarily” negative. It is limited by the rate of information loss (curvature of entropy). To create negative energy (such as maintaining wormholes), one must construct a special quantum state such that when probing along light rays, the rate of information gain decreases extremely rapidly.

3. Causal Protection: QNEC is just weak enough to allow quantum fluctuations (such as the Casimir effect), but strong enough to prohibit macroscopic time machines and horizon contraction violating the second law of thermodynamics. It is nature’s delicate compromise between “allowing quantum miracles” and “maintaining macroscopic causality.”

14.2.6 QCA Perspective: Flow Limitations in Discrete Networks

In QCA discrete ontology, QNEC has an intuitive network flow interpretation.

• Energy Flow $T_{kk}$: Corresponds to transmission flux of information on QCA lattice points (bits/step).

• Entropy Variation Rate $S^{\prime \prime }$: Corresponds to convexity of network connection structure.

QNEC $T\ge S^{\prime \prime }$ is actually a dynamic version of the Max-flow Min-cut theorem: the maximum information flow (energy) through an interface is limited by the rate of correlation change that the entanglement structure on both sides can support. If one tries to extract too much energy from a region where entanglement rapidly decays, the network will break, leading to failure of geometric definitions.

Conclusion

This section rigorously derived QNEC through stability analysis of generalized entropy. This result not only resolves the stability crisis of semiclassical gravity, but also establishes the dominance of information (entropy) over matter (energy): the energy density of matter is no longer a freely set parameter; it must obey information-theoretic inequalities prescribed by spacetime entanglement structure. In the next section, we will use QNEC to prove the Generalized Second Law (GSL) and horizon monotonicity theorem, further consolidating the thermodynamic arrow of spacetime.