Chapter 11 Section 2: Information-Geometric Variational Principle (IGVP)

“Generalized entropy is not only a thermodynamic quantity, but also a deep connection between spacetime geometry and quantum information.”

Section Overview

In the previous section, we constructed the cosmic consistency functional. This section will delve into its core component—the Information-Geometric Variational Principle (IGVP), which is the key bridge from abstract consistency conditions to concrete Einstein equations.

1. What is Information Geometry?

1.1 Classical Geometry vs. Information Geometry

Classical Differential Geometry studies:

• Shapes of points, lines, surfaces in space
• Metrics, curvature, geodesics
• Examples: sphere, torus, hyperboloid

Information Geometry studies:

• Spaces formed by probability distributions
• “Distance” and “curvature” between distributions
• Examples: all normal distributions, all Boltzmann distributions

graph LR
    A["Classical Geometry"] --> B["Point = Position in Space"]
    A --> C["Metric = Definition of Distance"]

    D["Information Geometry"] --> E["Point = A Probability Distribution"]
    D --> F["Metric = Fisher-Rao Metric"]

    B -.->|"Analogy"| E
    C -.->|"Analogy"| F

    style A fill:#e1f5ff,stroke:#333,stroke-width:2px
    style D fill:#ffe1e1,stroke:#333,stroke-width:2px

1.2 Fisher-Rao Metric

For a parameterized family of probability distributions $p(x;\theta )$, the Fisher-Rao metric is defined as:

$$g_{ij}^{\mathrm{FR}}(\theta )=\mathbb{E}\left[\frac{\partial \ln p}{\partial {\theta }^i}\frac{\partial \ln p}{\partial {\theta }^j}\right]$$

Physical Meaning:

• It measures the effect of small parameter changes on the distribution
• It is the unique invariant metric under data processing inequality (Čencov theorem)
• Its second-order variation gives Fisher information matrix

Analogy:

Imagine adjusting a radio’s frequency knob. Fisher-Rao metric tells you: rotating the knob a little, how much does the signal (probability distribution) change. Some directions (parameters) change a lot with small rotation (high curvature), some directions change little even with large rotation (flat).

1.3 Relative Entropy and Umegaki Entropy

For two quantum states $\rho ,\sigma$, relative entropy (Umegaki entropy) is defined as:

$$S(\rho \Vert \sigma )=\mathrm{Tr}(\rho \ln \rho )-\mathrm{Tr}(\rho \ln \sigma )$$

Core Properties:

1. Non-negativity: $S(\rho \Vert \sigma )\ge 0$, equality holds if and only if $\rho =\sigma$
2. Convexity: Convex function in $\rho$
3. Monotonicity: Non-increasing under quantum channels

Analogy:

Relative entropy measures the “distance” between two quantum states. If $\sigma$ is viewed as “standard state,” $S(\rho \Vert \sigma )$ tells you how far $\rho$ deviates from standard. Like using a standard ruler ($\sigma$) to measure the error of an actual ruler ($\rho$).

2. Generalized Entropy: Area + Bulk Entanglement

2.1 Definition of Generalized Entropy

For a small causal diamond $D_{p,r}$ in spacetime, generalized entropy is defined as:

$$S_{\mathrm{gen}}(D)=\underset{\text{Geometric Term: Area}}{\underbrace{\frac{A(\partial D)}{4G\hslash }}}+\underset{\text{Quantum Term: Entanglement Entropy}}{\underbrace{S_{\mathrm{out}}(D)}}$$

graph TD
    A["Small Causal Diamond D"] --> B["Boundary Area A"]
    A --> C["Bulk Quantum State rho"]

    B --> D["Geometric Contribution<br/>A/4Gh"]
    C --> E["Entanglement Entropy<br/>S_out"]

    D --> F["Generalized Entropy<br/>S_gen"]
    E --> F

    style A fill:#f9f,stroke:#333,stroke-width:3px
    style F fill:#9f9,stroke:#333,stroke-width:4px

2.2 Why “Area + Entanglement”?

Source of Area Term:

• Bekenstein-Hawking black hole entropy formula: $S_{\mathrm{BH}}=\frac{A}{4G}$
• Holographic principle: Boundary area encodes bulk information
• Ryu-Takayanagi formula: Holographic dual of entanglement entropy

Source of Entanglement Term:

• von Neumann entropy of bulk quantum fields
• UV finite under Hadamard states (after renormalization)
• Represents real quantum entanglement

Unified Physical Picture:

graph LR
    A["Total Cosmic Information"] --> B["Stored on Boundary<br/>Geometric Encoding"]
    A --> C["Stored in Bulk<br/>Quantum Entanglement"]

    B --> D["Area Term<br/>A/4Gh"]
    C --> E["Entanglement Term<br/>S_out"]

    style A fill:#ffcccc,stroke:#333,stroke-width:3px
    style D fill:#ccffcc,stroke:#333,stroke-width:2px
    style E fill:#ccccff,stroke:#333,stroke-width:2px

Analogy:

Imagine a library (universe). Information has two storage methods:

1. Catalog cards (area): Each book occupies one card in catalog, number of cards proportional to shelf area
2. Book content (entanglement): Text inside books, representing actual knowledge

Generalized entropy = Catalog information + Content information.

2.3 Role of Unified Time Scale

Key Question: Why use unified time scale $\kappa (\omega )$?

Answer:

1. Calibrate different clocks: Modular time, thermal time, geometric time must align
2. Connect scattering and geometry: $\kappa (\omega )=\frac{1}{2\pi }\mathrm{tr}\mathsf{Q}(\omega )$ connects Wigner-Smith group delay with area variation
3. Ensure causal consistency: All observers’ time readings comparable on $\kappa$

$$\kappa (\omega )=\frac{{\phi }^{\prime }(\omega )}{\pi }={\rho }_{\mathrm{rel}}(\omega )=\frac{1}{2\pi }\mathrm{tr}\mathsf{Q}(\omega )$$

3. Mathematical Formulation of IGVP

3.1 Variational Setup

Scenario: At each point $p$ on manifold, take small causal diamond $D_{p,r}$ (radius $r\ll L_{\mathrm{curv}}$)

Fixed Data:

• Volume of waist surface $V(B_{\ell })$
• Unified time scale $\kappa (\omega )$
• Boundary conditions (Dirichlet type)

Variable Data:

• Shape of waist surface (changing metric $g_{\mu \nu }$)
• Bulk quantum state ${\omega }_{\mathrm{bulk}}$

3.2 IGVP Principle

Core Proposition:

$$\boxed{\delta S_{\mathrm{gen}}=0\text{under fixed volume constraint}}$$

Equivalently, introducing Lagrange multiplier $\mu$:

$$\delta \left(S_{\mathrm{gen}}-\mu V\right)=0$$

Physical Interpretation:

Under the premise of keeping the small diamond’s “size” (volume) constant, generalized entropy reaches extremum. This is similar to finding free energy minimum in “isothermal isochoric” process in thermodynamics.

3.3 Two-Level Variation

IGVP contains two levels of variational conditions:

First Level: Extremum condition

$$\delta S_{\mathrm{gen}}=\frac{\delta A}{4G\hslash }+\delta S_{\mathrm{out}}=0$$

This will derive linearization of Einstein equations.

Second Level: Stability condition

$${\delta }^2{S}_{\mathrm{gen}}\ge 0$$

This corresponds to non-negativity of relative entropy and Quantum Null Energy Condition (QNEC).

graph TD
    A["IGVP"] --> B["First-Order Variation<br/>delta S_gen = 0"]
    A --> C["Second-Order Variation<br/>delta^2 S_gen >= 0"]

    B --> D["Einstein Equations<br/>R_kk = 8pi G T_kk"]
    C --> E["Quantum Energy Condition<br/>QNEC"]

    D --> F["Spacetime Geometry"]
    E --> F

    style A fill:#f9f,stroke:#333,stroke-width:4px
    style D fill:#9f9,stroke:#333,stroke-width:3px
    style E fill:#ff9,stroke:#333,stroke-width:3px

4. First Law of Entanglement

4.1 Modular Hamiltonian

On waist surface of small causal diamond, choose approximate Killing vector ${\chi }^a$ (boost generator), define modular Hamiltonian:

$$H_{\mathrm{mod}}=2\pi {\int }_{B_{\ell }}{\xi }^{\mu }{T}_{\mu \nu }d{\Sigma }^{\nu }$$

where ${\xi }^{\mu }$ is localized boost vector field, satisfying:

$${\xi }^{\mu }=\frac{{\ell }^2-r^2}{2\ell }{u}^{\mu }+O({\ell }^3)$$

4.2 First Law of Entanglement

Theorem 4.1 (First Law of Entanglement):

In small diamond limit and Hadamard states,

$$\delta S_{\mathrm{out}}=\delta ⟨H_{\mathrm{mod}}⟩$$

accurate to $O({\epsilon }^2)$, where $\epsilon =r/L_{\mathrm{curv}}$.

Physical Meaning:

Change in entanglement entropy equals change in modular Hamiltonian. This is similar to thermodynamic first law $dS=\frac{dQ}{T}$, but here “temperature” is determined by local geometry ($T=\hslash |{\kappa }_{\chi }|/2\pi$).

Proof Sketch (details in appendix):

1. Use Bisognano-Wichmann theorem: Vacuum state in Rindler wedge is thermal
2. Modular flow and KMS condition
3. Variational properties of relative entropy

4.3 Expansion of Modular Hamiltonian

In small diamond limit,

$$\delta ⟨H_{\mathrm{mod}}⟩=\frac{2\pi }{\hslash }{\int }_{\mathcal{H}}\lambda T_{kk}d\lambda dA+O({\epsilon }^2)$$

where:

• $\mathcal{H}$: Null geodesic bundle (null generators from waist surface)
• $\lambda$: Affine parameter along null generators
• $T_{kk}:=T_{\mu \nu }{k}^{\mu }{k}^{\nu }$ (stress in null direction)

Analogy:

Modular Hamiltonian measures “energy flow along light ray directions as seen from waist surface.” Like standing on mountaintop (waist surface), measuring water flow (energy) along different directions (null generators).

5. Area Variation and Raychaudhuri Equation

5.1 Null Geodesic Bundle and Expansion

Null generators $k^{\mu }$ from waist surface $\partial B_{\ell }$ satisfy:

$$k^{\mu }{\nabla }_{\mu }{k}^{\nu }=0\text{(affine parameterization)}$$

Define expansion:

$$\theta :={\nabla }_{\mu }{k}^{\mu }$$

It measures “opening” or “contraction” of null bundle.

5.2 Raychaudhuri Equation

Along null generators, expansion satisfies:

$$\frac{d\theta }{d\lambda }=-\frac{1}{d-2}{\theta }^2-{\sigma }^2+{\omega }^2-R_{kk}$$

where:

• $\sigma$: Shear
• $\omega$: Twist
• $R_{kk}=R_{\mu \nu }{k}^{\mu }{k}^{\nu }$ (projection of Ricci curvature in null direction)

Key Observation: At waist surface $\theta (0)=0$ (maximum volume condition), and $\omega =0$ (Frobenius integrability), hence:

$${\theta }^{\prime }(0)=-R_{kk}(0)-{\sigma }^2(0)$$

5.3 Area Variation Formula

Through integration of Raychaudhuri equation:

$$\frac{\delta A}{A}=-{\int }_0^{{\lambda }_{\ast }}\theta d\lambda ={\int }_0^{{\lambda }_{\ast }}\lambda R_{kk}d\lambda +O({\epsilon }^3)$$

(ignoring higher-order contributions of ${\sigma }^2$)

Physical Meaning:

Change in boundary area is determined by integral of spacetime curvature along null directions. Larger curvature, faster null bundle convergence, faster area decrease.

graph TD
    A["Waist Surface<br/>theta=0"] -->|"null generators"| B["Future Vertex"]

    A -.->|"expansion theta"| C["Area Change<br/>delta A"]

    D["Ricci Curvature<br/>R_kk"] -.->|"Raychaudhuri"| C

    style A fill:#ffe1e1,stroke:#333,stroke-width:3px
    style C fill:#e1ffe1,stroke:#333,stroke-width:2px
    style D fill:#e1f5ff,stroke:#333,stroke-width:2px

6. Closure of First-Order Variation

6.1 Combining Area and Entropy

Combining area variation and first law of entanglement:

$$\delta S_{\mathrm{gen}}=\frac{\delta A}{4G\hslash }+\delta S_{\mathrm{out}}$$

Substituting:

$$\delta S_{\mathrm{gen}}=\frac{1}{4G\hslash }{\int }_{\mathcal{H}}\lambda R_{kk}d\lambda dA+\frac{2\pi }{\hslash }{\int }_{\mathcal{H}}\lambda T_{kk}d\lambda dA+O({\epsilon }^2)$$

6.2 Extremum Condition

Requiring $\delta S_{\mathrm{gen}}=0$, we get:

$$\frac{1}{4G\hslash }{\int }_{\mathcal{H}}\lambda R_{kk}d\lambda dA+\frac{2\pi }{\hslash }{\int }_{\mathcal{H}}\lambda T_{kk}d\lambda dA=0$$

Simplifying:

$${\int }_{\mathcal{H}}\lambda \left(R_{kk}-8\pi G{T}_{kk}\right)d\lambda dA=0$$

6.3 Localization to Pointwise

Through Radon-type closure (details in Section 3):

• For all test functions $\phi (x)$ on waist surface
• For all null directions $\hat{k}$

The above integral being zero implies at each point, each null direction:

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

Analogy:

Imagine measuring “average height” of a surface. If integrals are zero for all measurement lines (null generators), all starting points (waist surface), then surface height must be zero at each point.

7. From Null Directions to Full Tensor

7.1 Null Cone Characterization Lemma

Lemma 7.1 (necessary for $d\ge 3$ dimensions):

If symmetric tensor $X_{ab}$ satisfies $X_{ab}{k}^a{k}^b=0$ for all null vectors $k^a$, then:

$$X_{ab}=\Phi g_{ab}$$

for some scalar function $\Phi$.

Proof (sketch): Use dimension of null cone and degree of freedom counting of symmetric tensors.

7.2 Application to Einstein Tensor

Define:

$$X_{ab}:=R_{ab}-8\pi G{T}_{ab}$$

Since $R_{kk}=8\pi G{T}_{kk}$ holds in all null directions, we get:

$$X_{ab}=\Phi g_{ab}$$

7.3 Using Bianchi Identity

Contracted Bianchi identity:

$${\nabla }^a{R}_{ab}=\frac{1}{2}{\nabla }_{b}R$$

Energy-momentum conservation:

$${\nabla }^a{T}_{ab}=0$$

Therefore:

$${\nabla }^a{X}_{ab}={\nabla }^a{R}_{ab}-8\pi G{\nabla }^a{T}_{ab}=\frac{1}{2}{\nabla }_{b}R$$

On the other hand, from $X_{ab}=\Phi g_{ab}$:

$${\nabla }^a{X}_{ab}={\nabla }_b\Phi$$

Hence:

$${\nabla }_b\left(\Phi -\frac{R}{2}\right)=0$$

That is, $\Phi -\frac{R}{2}=\Lambda$ (constant).

7.4 Einstein Equations

Rearranging:

$$R_{ab}-8\pi G{T}_{ab}=\Phi g_{ab}=\left(\frac{R}{2}+\Lambda \right)g_{ab}$$

That is:

$$R_{ab}-\frac{R}{2}{g}_{ab}+\Lambda g_{ab}=8\pi G{T}_{ab}$$

$$\boxed{G_{ab}+\Lambda g_{ab}=8\pi G{T}_{ab}}$$

This is the Einstein field equations!

graph TD
    A["IGVP<br/>delta S_gen = 0"] --> B["Null Directions<br/>R_kk = 8pi G T_kk"]

    B --> C["Null Cone Characterization<br/>X_ab = Phi g_ab"]

    C --> D["Bianchi Identity<br/>nabla Phi = nabla R/2"]

    D --> E["Einstein Equations<br/>G_ab + Lambda g_ab = 8pi G T_ab"]

    style A fill:#f9f,stroke:#333,stroke-width:4px
    style E fill:#9f9,stroke:#333,stroke-width:4px

8. Second-Order Variation and Quantum Energy Condition

8.1 Convexity of Relative Entropy

Relative entropy $S(\rho \Vert \sigma )$ is convex in $\rho$, hence:

$${\delta }^{2}S(\rho \Vert \sigma )\ge 0$$

8.2 Quantum Null Energy Condition (QNEC)

Under second-order deformation along null direction, define:

$$s_{\mathrm{out}}^{\prime \prime }:=\underset{\mathcal{A}\to 0}{\lim }\frac{1}{\mathcal{A}}\frac{d^2{S}_{\mathrm{out}}}{d{\lambda }^2}$$

Theorem 8.1 (QNEC):

$$⟨T_{kk}⟩\ge \frac{\hslash }{2\pi }{s}_{\mathrm{out}}^{\prime \prime }$$

Physical Meaning:

Energy density in null direction has a quantum lower bound, determined by second derivative of entanglement entropy. This is quantum generalization of classical energy conditions.

8.3 Consistency with IGVP

Second-order condition ${\delta }^2{S}_{\mathrm{gen}}\ge 0$ of IGVP is equivalent to QNEC in small diamond limit. This ensures:

• Solutions of Einstein equations are stable
• No uncontrolled negative energy
• Unidirectionality of arrow of time

9. Key Points Review

graph TD
    A["Information Geometry"] --> B["Fisher-Rao Metric<br/>Relative Entropy"]

    B --> C["Generalized Entropy<br/>S_gen = A/4Gh + S_out"]

    C --> D["IGVP Principle<br/>delta S_gen = 0"]

    D --> E["First Law of Entanglement<br/>delta S_out = delta H_mod"]
    D --> F["Area Variation<br/>Raychaudhuri"]

    E --> G["Null Direction Equation<br/>R_kk = 8pi G T_kk"]
    F --> G

    G --> H["Null Cone Characterization<br/>Bianchi Identity"]

    H --> I["Einstein Equations<br/>G_ab + Lambda g_ab = 8pi G T_ab"]

    style A fill:#e1f5ff,stroke:#333,stroke-width:2px
    style C fill:#ffe1e1,stroke:#333,stroke-width:3px
    style D fill:#f9f,stroke:#333,stroke-width:4px
    style I fill:#9f9,stroke:#333,stroke-width:4px

Core Insight:

Information-Geometric Variational Principle (IGVP) connects extremum condition of generalized entropy with Einstein equations. This is not accidental, but deep necessity: spacetime geometry is determined by information structure, Einstein equations are necessary consequences of information consistency.

Specific derivation chain:

$$\text{IGVP: }\delta S_{\mathrm{gen}}=0\Rightarrow \frac{\delta A}{4G\hslash }+\delta S_{\mathrm{out}}=0$$

$$\Rightarrow \int \lambda (R_{kk}-8\pi G{T}_{kk})d\lambda dA=0$$

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

$$\Rightarrow G_{ab}+\Lambda g_{ab}=8\pi G{T}_{ab}$$

10. Philosophical Reflection

10.1 Nature of Gravity

Traditional view: Gravity is spacetime curvature.

IGVP view: Gravity is geometric manifestation of information entropy extremum.

Spacetime is not an a priori existing stage, but an emergent structure of quantum information and entanglement. Einstein equations are not assumptions about gravity, but necessary consequences of information consistency.

10.2 Why Generalized Entropy?

Question: Why use $S_{\mathrm{gen}}=\frac{A}{4G\hslash }+S_{\mathrm{out}}$ instead of area or entanglement entropy alone?

Answer:

1. Area alone: Cannot include quantum corrections, violates quantum information conservation
2. Entanglement entropy alone: UV divergent, and ignores geometric degrees of freedom
3. Generalized entropy: Unifies geometry and quantum, UV finite, satisfies quantum focusing conjecture

10.3 Bridge from Information to Geometry

IGVP reveals a profound correspondence:

Information geometry is not an analogy to physical geometry, but two formulations of the same structure.

Next Section Preview: In Section 3, we will apply the IGVP framework established in this section to specific small causal diamonds, step by step deriving every detail of Einstein equations, including area expansion, exact form of Raychaudhuri equation, mathematical proof of Radon-type closure, and rigorous argument from null directions to full tensor.