Meta Theory of the Zeckendorf-Hilbert Universe

11.2 Generalized Entropy Functional: Definition of Geometric Area Term and Matter Entanglement Entropy Term

In Section 11.1, we established small causal diamonds as fundamental probes for detecting spacetime geometry and proved that the area deficit of their boundary $\partial \Sigma$ directly encodes the Einstein tensor $G_{00}$. However, in the framework of holographic thermodynamics, geometric area is not merely a squared length—it is the carrier of entropy.

To establish a variational principle for gravitational dynamics, we must construct a total entropy functional that governs both “geometry” and “matter.” This leads to the concept of Generalized Entropy ($S_{\text{gen}}$). This section will rigorously define the generalized entropy functional, argue for its status as the fundamental free energy of quantum gravity, and resolve the ultraviolet divergence problem in quantum field entanglement entropy, revealing the renormalization nature of the gravitational constant $G$.

11.2.1 Physical Motivation for Generalized Entropy: Rescuing the Second Law of Thermodynamics

In classical thermodynamics, entropy is extensive. But in black hole physics, Bekenstein discovered that the black hole horizon area $A$ behaves like entropy. Hawking radiation further established the black hole entropy $S_{BH}=\frac{A}{4G\hslash }$.

However, when matter falls into a black hole, the matter entropy $S_{\text{mat}}$ appears to vanish, violating the second law of thermodynamics. Bekenstein proposed that the quantity that monotonically increases with time is not pure matter entropy, nor pure black hole area, but their sum:

$$S_{\text{total}}=S_{BH}+S_{\text{outside}}$$

This idea was generalized by Jacobson, Sorkin, and other scholars to arbitrary causal horizons (not limited to black hole horizons), forming the concept of generalized entropy. In our QCA discrete ontology, this is a natural result: the total information of the universe is divided into “information encoded in geometry” (area term) and “explicit quantum state information” (matter term).

11.2.2 Definition and Formal Construction

Consider a Cauchy slice $\Sigma$ in a Lorentzian manifold $\mathcal{M}$, and an entanglement surface $\partial \Sigma$ (the edge of a causal diamond) that divides it into two parts (system and environment).

Definition 11.2.1 (Generalized Entropy Functional)

For any given geometric division surface $\sigma =\partial \Sigma$, its generalized entropy $S_{\text{gen}}[\sigma ]$ is defined as the sum of geometric entropy and matter entanglement entropy:

$$S_{\text{gen}}[\sigma ]\equiv S_{\text{grav}}[\sigma ]+S_{\text{mat}}[\sigma ]=\frac{A[\sigma ]}{4G\hslash }+S_{\text{VN}}({\rho }_{\Sigma })$$

where:

1. Geometric Term $S_{\text{grav}}$: Given by the Bekenstein-Hawking formula, where $A[\sigma ]$ is the intrinsic area of surface $\sigma$, and $G$ is Newton’s gravitational constant. From the microscopic QCA perspective, this represents the Shannon entropy of the discrete degrees of freedom (“atoms”) that constitute spacetime background itself.

2. Matter Term $S_{\text{mat}}$: The von Neumann entanglement entropy $S_{\text{VN}}=-\text{Tr}({\rho }_{\Sigma }\ln {\rho }_{\Sigma })$ of the reduced density matrix ${\rho }_{\Sigma }={\text{Tr}}_{\bar{\Sigma }}|\Psi ⟩⟨\Psi |$ of quantum fields (matter fields and gravitational wave perturbations) inside region $\Sigma$.

Physical Interpretation:

$S_{\text{gen}}$ measures the total information deficit of an external observer regarding the internal state of region $\Sigma$. Part of the information is hidden in the geometric structure of the horizon (similar to the heat capacity of a thermal reservoir), and another part is hidden in the entanglement correlations of quantum fields.

11.2.3 Ultraviolet Divergence and Renormalization: Finiteness of Generalized Entropy

In quantum field theory (QFT) calculations, entanglement entropy $S_{\text{mat}}$ exhibits the famous area law ultraviolet divergence.

For a QFT with ultraviolet cutoff $ϵ$ (such as lattice spacing), the entanglement entropy across a boundary of area $A$ expands as:

$$S_{\text{mat}}=\frac{c_{1}A}{{ϵ}^2}+c_2\ln (ϵ)+S_{\text{finite}}$$

As $ϵ\to 0$, the first term diverges. This seems to invalidate the definition of generalized entropy.

However, from the perspective of induced gravity and renormalization group flow, this divergence is precisely absorbed by the renormalization of the gravitational constant $G$.

Let the bare gravitational constant be $G_{\text{bare}}$, and the renormalized physical gravitational constant be $G_{\text{ren}}$. The coefficient $1/4G_{\text{bare}}$ of the Einstein-Hilbert action also receives quantum corrections.

Susskind and Uglum (1994) proved that the area term and matter entropy divergence in generalized entropy automatically cancel:

$$S_{\text{gen}}=\frac{A}{4G_{\text{bare}}}+\left(\frac{c_{1}A}{{ϵ}^2}+S_{\text{finite}}\right)=A\left(\frac{1}{4G_{\text{bare}}}+\frac{c_1}{{ϵ}^2}\right)+S_{\text{finite}}$$

Define the physical gravitational constant to satisfy:

$$\frac{1}{4G_{\text{ren}}}=\frac{1}{4G_{\text{bare}}}+\frac{c_1}{{ϵ}^2}$$

Then generalized entropy becomes finite:

$$S_{\text{gen}}=\frac{A}{4G_{\text{ren}}}+S_{\text{finite}}$$

Theorem 11.2.2 (Finiteness Theorem for Generalized Entropy)

In a consistent theory including gravity (such as the continuum limit of QCA), generalized entropy $S_{\text{gen}}$ is a UV-finite physical quantity, independent of the microscopic cutoff scale $ϵ$. The distinction between geometric entropy and matter entanglement entropy depends only on the choice of renormalization scale $\mu$, but their sum $S_{\text{gen}}$ is a renormalization group invariant (RG Invariant).

Significance in QCA Universe:

In our discrete ontology, $ϵ$ is the physical Planck length $l_P$, and $G$ is a fundamental constant. At this point, there is no divergence, and $S_{\text{gen}}$ directly counts the number of QCA links on the horizon (geometric part) plus the number of entangled bits in internal excited states (matter part). This natural cutoff eliminates the mathematical pathologies in continuous field theory.

11.2.4 Generalized Entropy as Effective Description of Microstate Counting

Generalized entropy is not only a thermodynamic quantity, it is also the potential function of geometric dynamics.

In Chapter 12, we will propose the Entropic Variational Principle (IGVP): the dynamical evolution of spacetime geometry (Einstein equations) is essentially the maximization condition (or equilibrium condition) of generalized entropy $S_{\text{gen}}$ on small causal diamonds.

• Vacuum Equilibrium: For pure vacuum, geometric entropy is maximized (area maximized for fixed volume), corresponding to flat space.

• Matter Perturbation: When matter introduces changes in entanglement entropy $S_{\text{mat}}$, to maintain the extremal property (or equilibrium) of $S_{\text{gen}}$, the geometric part must respond (changing area $A$), leading to spacetime curvature.

Corollary 11.2.3 (Gravity as Entropic Force)

The establishment of the generalized entropy functional allows us to completely rewrite gravitational interactions in the language of information geometry. Force $F$ is no longer fundamental, but rather the gradient of entropy $F\sim T\nabla S_{\text{gen}}$. This explains why gravity equals inertial mass (equivalence principle)—because they both arise from the same statistical potential function.

Summary

This section defined generalized entropy $S_{\text{gen}}=A/4G+S_{\text{mat}}$ and clarified its finiteness and renormalization properties in the context of quantum field theory. This is the core physical quantity of Volume III.

In the next section 11.3, we will delve into the First Law of Entanglement, which is the mathematical bridge transforming the variation of generalized entropy into the energy-momentum tensor $T_{\mu \nu }$.