Generalized Entropy: Unification of Geometry and Quantum

“Entropy comes not only from matter, but also from spacetime itself.”

🎯 Core Question

In the Foundation chapter, we learned Boltzmann entropy:

$$S=k_B\ln \Omega$$

But in the presence of gravity, this definition is incomplete!

Why?

Because spacetime itself carries entropy!

🌟 Revelation of Bekenstein-Hawking Entropy

Black Hole Entropy

In 1973, Bekenstein proposed: black holes should have entropy.

In 1974, Hawking calculated:

$$\boxed{S_{\text{BH}}=\frac{A}{4G\hslash }}$$

Where:

• $A$: horizon area
• $G$: Newton’s gravitational constant
• $\hslash$: Planck constant

Key observations:

1. Entropy is proportional to area, not volume!
2. Entropy contains gravitational constant $G$
3. Entropy contains quantum constant $\hslash$

This means: Entropy is the intersection of gravity, quantum, and thermodynamics!

graph TB
    BH["Black Hole Entropy<br/>S_BH = A/4Gℏ"]

    BH --> G["Gravity<br/>G"]
    BH --> Q["Quantum<br/>ℏ"]
    BH --> T["Thermodynamics<br/>k_B"]
    BH --> GEO["Geometry<br/>Area A"]

    style BH fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style G fill:#e1f5ff
    style Q fill:#e1ffe1
    style T fill:#ffe1e1
    style GEO fill:#f5e1ff

📐 Definition of Generalized Entropy

Sum of Two Terms

In quantum gravity, generalized entropy is defined as:

$$\boxed{S_{\text{gen}}(\Sigma )=\underset{\text{Geometric Entropy}}{\underbrace{\frac{A(\Sigma )}{4G\hslash }}}+\underset{\text{Quantum Field Entropy}}{\underbrace{S_{\text{out}}(\Sigma )}}}$$

Where:

• $\Sigma$: spatial hypersurface (Cauchy slice)
• $A(\Sigma )$: area of boundary of $\Sigma$
• $S_{\text{out}}(\Sigma )$: von Neumann entropy of quantum fields outside the boundary

Physical Meaning

graph TB
    subgraph "Spatial Hypersurface Σ"
        B["Boundary ∂Σ<br/>Area A"]
    end

    subgraph "Outside Region"
        O["Quantum Field State ρ<br/>Entropy S_out = -tr(ρ ln ρ)"]
    end

    S["Generalized Entropy<br/>S_gen"] --> G["Geometric Contribution<br/>A/4Gℏ"]
    S --> Q["Quantum Contribution<br/>S_out"]

    B -.-> G
    O -.-> Q

    style S fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style G fill:#e1f5ff
    style Q fill:#ffe1e1

First term $A/(4G\hslash )$ (geometric entropy):

• From degrees of freedom of spacetime geometry
• Same form as black hole entropy
• Reflects “gravitational entropy” of boundary

Second term $S_{\text{out}}$ (quantum field entropy):

• From quantum entanglement of matter fields
• von Neumann entropy: $S_{\text{out}}=-\text{tr}({\rho }_{\text{out}}\ln {\rho }_{\text{out}})$
• Reflects “information entropy” of fields

💡 Balloon Analogy

Imagine a balloon:

🎈

Total “information” = information on balloon surface + information of internal gas

Key insight:

We cannot only look at the gas (matter fields), but also the balloon itself (spacetime geometry)!

🔬 Generalized Entropy on Small Causal Diamond

In the IGVP framework, we define generalized entropy on the small causal diamond ${\mathcal{D}}_{\ell }(p)$.

Structure of Small Causal Diamond

graph TB
    Q["Future Vertex<br/>q"] --> |"Past Null Cone"| W["Waist S_ℓ<br/>Boundary of Maximum Spatial Cross-Section"]
    P["Past Vertex<br/>p"] --> |"Future Null Cone"| W

    W --> A["Area A(S_ℓ)"]
    W --> V["Volume V(B_ℓ)"]

    style W fill:#fff4e1,stroke:#ff6b6b,stroke-width:2px
    style Q fill:#ffe1e1
    style P fill:#e1f5ff

Waist $S_{\ell }$:

• Boundary of maximum spatial cross-section in the diamond
• Dimension: $d-2$ ($d$ is spacetime dimension)
• Area: $A\sim {\ell }^{d-2}$

Volume $V(B_{\ell })$:

• Volume of maximum spatial cross-section
• Volume: $V\sim {\ell }^{d-1}$

Explicit Form of Generalized Entropy

On the small causal diamond:

$$S_{\text{gen}}=\frac{A(S_{\ell })}{4G\hslash }+S_{\text{out}}^{\text{ren}}+S_{\text{ct}}^{\text{UV}}-\frac{\Lambda }{8\pi G}\frac{V(B_{\ell })}{T}$$

Meaning of each term:

1. $A/(4G\hslash )$: geometric entropy of waist
2. $S_{\text{out}}^{\text{ren}}$: renormalized field entropy
3. $S_{\text{ct}}^{\text{UV}}$: UV counterterm (handles divergence)
4. $-\Lambda V/(8\pi GT)$: volume dual term (Lagrange multiplier)

Temperature $T$:

$$T=\frac{\hslash |{\kappa }_{\chi }|}{2\pi }$$

Where ${\kappa }_{\chi }$ is the surface gravity of the approximate Killing field.

🌊 Why Renormalization?

Divergence Problem

In quantum field theory, entanglement entropy diverges at short distances:

$$S_{\text{out}}\sim \frac{A}{{ϵ}^{d-2}}+\cdots$$

($ϵ$ is UV cutoff)

Solution:

1. Point-splitting renormalization: Use fine regularization scheme
2. Subtract divergent terms: $S_{\text{out}}^{\text{ren}}=S_{\text{out}}^{\text{raw}}-S_{\text{ct}}^{\text{UV}}$
3. Keep finite part: $S_{\text{gen}}$ is finite as $ϵ\to 0$

Physical Meaning of Renormalization

graph LR
    R["Raw Field Entropy<br/>S_out^raw"] --> |"Subtract UV Divergence"| C["Counterterm<br/>S_ct^UV"]
    R --> |"Subtract"| REN["Renormalized Entropy<br/>S_out^ren"]
    C -.-> REN

    style R fill:#ffe1e1
    style C fill:#f0f0f0,stroke-dasharray: 5 5
    style REN fill:#e1ffe1

Physical interpretation:

• UV divergent terms are absorbed by geometric term $A/(4G\hslash )$
• Finite part $S_{\text{out}}^{\text{ren}}$ is physical
• This is similar to mass renormalization

📊 Properties of Generalized Entropy

Property 1: Monotonicity

During evolution, generalized entropy is considered to satisfy the second law:

$$\frac{d{S}_{\text{gen}}}{d\lambda }\ge 0$$

Along affine parameter $\lambda$ of null geodesics.

This suggests the existence of a thermodynamic arrow of time.

Property 2: Extremality

On solutions of Einstein’s equations, generalized entropy takes an extremum at the waist of the small causal diamond:

$$\delta S_{\text{gen}}=0(\text{fixed volume})$$

This is the first-order variational condition of IGVP.

Property 3: Concavity

Second-order variation is typically non-negative:

$${\delta }^2{S}_{\text{rel}}\ge 0$$

This provides a theoretical guarantee for the stability of solutions.

🔗 Connection to Core Insights

Entropy is the Arrow

Monotonicity of generalized entropy $d{S}_{\text{gen}}/d\lambda \ge 0$ defines the arrow of time:

$$p\prec q⟺S_{\text{gen}}(p)\le S_{\text{gen}}(q)$$

Boundary is Reality

Geometric entropy term $A/(4G\hslash )$ emphasizes the ontological status of boundary area.

Time is Geometry

Temperature $T=\hslash |{\kappa }_{\chi }|/(2\pi )$ connects thermal time with geometric time.

📝 Key Formulas Summary

🎓 Further Reading

• Original paper: J.D. Bekenstein, “Black holes and entropy” (Phys. Rev. D 7, 2333, 1973)
• Hawking radiation: S.W. Hawking, “Black hole explosions?” (Nature 248, 30, 1974)
• Generalized entropy: T. Faulkner et al., “Gravitation from entanglement” (JHEP 03, 051, 2014)
• GLS documentation: igvp-einstein-complete.md
• Next: 02-causal-diamond_en.md - Small Causal Diamond

🤔 Exercises

1. Conceptual understanding:

   • Why is black hole entropy proportional to area rather than volume?
   • What physical degrees of freedom do the two terms of generalized entropy come from?
   • Why is renormalization needed?

2. Order-of-magnitude estimates:

   • What is the horizon area of a solar-mass black hole? What is the Bekenstein-Hawking entropy?
   • How does it compare with the thermodynamic entropy of gas inside the Sun?

3. Physical applications:

   • How does Hawking radiation maintain monotonicity of generalized entropy?
   • How does the Page curve reflect evolution of generalized entropy?
   • What is the relationship between black hole information paradox and generalized entropy?

4. Advanced thinking:

   • What problems would arise if the geometric entropy term is not included?
   • How does Wald entropy (higher-order gravity theories) generalize $A/(4G\hslash )$?
   • What is the connection between holographic entanglement entropy and generalized entropy?

Next step: After understanding generalized entropy, we will learn where it is varied—the small causal diamond!