Small Causal Diamond: The Stage for Variation

“The stage of gravity is not the grand universe, but tiny local causal regions.”

🎯 Core Question

In the previous article, we defined generalized entropy:

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

But where is this entropy varied?

Answer: The small causal diamond (small causal diamond)!

💎 What is a Causal Diamond?

Intuitive Image

Imagine an hourglass:

     ⋰ Future vertex q
    ╱ ╲
   ╱   ╲
  ╱     ╲
 ╱  Waist  ╲  ← Thickest part
╱    S_ℓ   ╲
╲         ╱
 ╲       ╱
  ╲     ╱
   ╲   ╱
    ╲ ╱
     ⋱ Past vertex p

This is the shape of a causal diamond!

Physical meaning:

• All future light cones emitted from past vertex $p$
• Intersected with all past light cones reaching future vertex $q$
• The intersection of the two

Mathematical Definition

On a Lorentzian manifold $(M,g)$, for point $p\in M$, take a sufficiently small scale $\ell \ll L_{\text{curv}}(p)$ ($L_{\text{curv}}$ is the local curvature scale), define the small causal diamond:

$$\boxed{{\mathcal{D}}_{\ell }(p)=J^{+}(p^{-})\cap J^{-}(p^{+})}$$

Where:

• $p^{-}$: past vertex, point at proper time $-\ell$ along some reference timelike direction
• $p^{+}$: future vertex, point at proper time $+\ell$
• $J^{+}(p^{-})$: causal future of $p^{-}$ (all points causally reachable from $p^{-}$)
• $J^{-}(p^{+})$: causal past of $p^{+}$ (all points that can causally reach $p^{+}$)

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

    W --> N1["Null Hypersurface 𝓝⁺"]
    W --> N2["Null Hypersurface 𝓝⁻"]

    N1 --> D["Small Causal Diamond<br/>𝒟_ℓ(p)"]
    N2 --> D

    style W fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style D fill:#e1ffe1,stroke:#ff6b6b,stroke-width:2px
    style Q fill:#ffe1e1
    style P fill:#e1f5ff

🔍 Structure of Small Causal Diamond

Boundary Components

The boundary $\partial {\mathcal{D}}_{\ell }$ of the small causal diamond consists of:

1. Past null hypersurface ${\mathcal{N}}^{+}$:

   • Future light cone emitted from past vertex $p^{-}$
   • Generated by null geodesics
   • Dimension: $d-1$ ($d$ is spacetime dimension)

2. Future null hypersurface ${\mathcal{N}}^{-}$:

   • Past light cone reaching future vertex $p^{+}$
   • Also generated by null geodesics
   • Dimension: $d-1$

3. Waist $S_{\ell }$:

   • Intersection line of the two null cones
   • Boundary of maximum spatial cross-section in the diamond
   • Dimension: $d-2$
   • This is the key to generalized entropy variation!

Importance of Waist

Why is it called “waist”?

Because it is the thickest part in the middle of the diamond, like the waist of an hourglass!

Physical meaning:

graph TB
    subgraph "Inside Small Causal Diamond"
        C["Center Point p"]
        B["Maximum Spatial Cross-Section<br/>B_ℓ (Volume)"]
        S["Boundary Waist<br/>S_ℓ (Area)"]
    end

    B --> S

    S --> A["Geometric Entropy<br/>A(S_ℓ)/4Gℏ"]
    S --> Q["Quantum Field Entropy<br/>S_out(S_ℓ)"]

    A --> SG["Generalized Entropy<br/>S_gen"]
    Q --> SG

    style S fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style SG fill:#e1ffe1,stroke:#ff6b6b,stroke-width:2px

Geometric data of waist:

• Area: $A(S_{\ell })\sim {\ell }^{d-2}$
• Volume of internal maximum spatial cross-section: $V(B_{\ell })\sim {\ell }^{d-1}$
• Curvature radius: $L_{\text{curv}}\gg \ell$

📏 Meaning of “Small”

Small Diamond Limit

What does “small” mean?

In the IGVP derivation, we take the limit $\ell \to 0$, i.e., the small diamond limit:

$$\ell \ll L_{\text{curv}}$$

Why take the small limit?

1. Locality: Gravity is a local physical law, should hold near each point
2. Controllability: In the small limit, curvature corrections are higher-order small quantities $O({\ell }^2/L_{\text{curv}}^2)$
3. Approximate flatness: Inside the small diamond, it approximates the causal diamond of Minkowski spacetime

Geometric Approximation

In normal coordinates, the small causal diamond satisfies:

$$g_{\mu \nu }={\eta }_{\mu \nu }+O\left(\frac{{\ell }^2}{L_{\text{curv}}^2}\right)$$

Where ${\eta }_{\mu \nu }$ is the Minkowski metric.

This means: At sufficiently small scales, spacetime locally “looks like” flat spacetime!

graph LR
    C["Curved Spacetime<br/>Small Causal Diamond"] --> |"ℓ → 0 Limit"| M["Minkowski Spacetime<br/>Causal Diamond"]

    C --> E["Error<br/>O(ℓ²/L²_curv)"]

    style C fill:#ffe1e1
    style M fill:#e1f5ff
    style E fill:#f0f0f0,stroke-dasharray: 5 5

🌊 Why Use Small Causal Diamond?

Reason 1: Jacobson’s Inspiration

In 1995, when Jacobson first derived Einstein’s equations from thermodynamics, he used local causal horizons.

The small causal diamond is a precise mathematical realization of this idea:

• Waist $S_{\ell }$ is similar to a local horizon
• Generalized entropy is defined on this “horizon”
• Variation is performed with fixed volume $V(B_{\ell })$

Reason 2: Principle of Locality

Physical laws should be local:

Key logic:

From local entropy extremum → through Radon-type closure → derive pointwise Einstein equations

Reason 3: Error Control

In the small limit, all error terms are controllable higher-order small quantities:

1. Geometric error: $O({\ell }^2/L_{\text{curv}}^2)$
2. Quantum field theory error: $O({\epsilon }^2)$ ($\epsilon$ is a small parameter)
3. Boundary effects: $o({\ell }^{d-2})$

This guarantees the rigor of the derivation!

🎨 Causal Diamond in Flat Spacetime

Example in Minkowski Spacetime

In flat spacetime $({\mathbb{R}}^{1,d-1},\eta )$, take origin $p=0$, reference timelike direction as $t$-axis, then:

Causal diamond:

$${\mathcal{D}}_{\ell }(0)=\{(t,\mathbf{x}):|t|+|\mathbf{x}|\le \ell \}$$

Waist (boundary of $t=0$ cross-section):

$$S_{\ell }=\{(0,\mathbf{x}):|\mathbf{x}|=\ell \}$$

This is a $(d-2)$-dimensional sphere of radius $\ell$!

Area:

$$A(S_{\ell })={\Omega }_{d-2}{\ell }^{d-2}$$

Where ${\Omega }_{d-2}$ is the volume of a unit $(d-2)$-sphere.

Maximum spatial cross-section (ball at $t=0$):

$$B_{\ell }=\{(0,\mathbf{x}):|\mathbf{x}|\le \ell \}$$

Volume:

$$V(B_{\ell })=\frac{{\Omega }_{d-2}}{d-1}{\ell }^{d-1}$$

Specific Calculation in Four-Dimensional Spacetime

When $d=4$:

• Waist $S_{\ell }$ is a 2-sphere (ordinary sphere)
• Area: $A=4\pi {\ell }^2$
• Volume: $V=\frac{4\pi }{3}{\ell }^3$

Geometric entropy:

$$S_{\text{geom}}=\frac{A}{4G\hslash }=\frac{4\pi {\ell }^2}{4G\hslash }=\frac{\pi {\ell }^2}{G\hslash }$$

Relationship with Planck area ${\ell }_P^2=G\hslash /c^3$:

$$S_{\text{geom}}=\frac{\pi c^3}{\hslash }{\left(\frac{\ell }{{\ell }_P}\right)}^2$$

Physical interpretation: Geometric entropy is proportional to area (measured in Planck units)!

🔄 Approximate Killing Field

On the small causal diamond, there exists an approximate Killing field ${\chi }^a$:

$${\mathcal{L}}_{\chi }{g}_{ab}=O\left(\frac{{\ell }^2}{L_{\text{curv}}^2}\right)$$

Physical meaning:

At small scales, there exists approximate symmetry, corresponding to:

• Approximate time translation invariance
• Approximate boost symmetry (along null direction)

Surface gravity:

$${\kappa }_{\chi }=\frac{2}{\ell }+O\left(\frac{\ell }{L_{\text{curv}}^2}\right)$$

Unruh temperature:

$$T=\frac{\hslash |{\kappa }_{\chi }|}{2\pi }=\frac{\hslash }{\pi \ell }+O\left(\frac{\ell }{L_{\text{curv}}^2}\right)$$

Key insight: The small causal diamond possesses an intrinsic temperature determined by geometry.

📐 Variation Setup

In IGVP, we perform the following variation on the small causal diamond:

Variation Parameters

1. Waist position: Change embedding of $S_{\ell }$
2. Quantum state: Change quantum state $\rho$ of fields

Constraints

1. Fixed endpoints: $p^{-}$ and $p^{+}$ unchanged
2. Fixed volume: $\delta V(B_{\ell })=0$
3. Fixed temperature: $\delta T=0$ (at first-order variation level)

Variation Object

Generalized entropy:

$$S_{\text{gen}}(S_{\ell })=\frac{A(S_{\ell })}{4G\hslash }+S_{\text{out}}(S_{\ell })$$

First-order condition:

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

This is one of the core assumptions of IGVP.

graph TB
    D["Small Causal Diamond 𝒟_ℓ"] --> V["Variation: Change Waist Position<br/>δS_ℓ"]
    D --> C["Constraint: Fixed Volume<br/>δV = 0"]

    V --> S["Generalized Entropy Variation<br/>δS_gen = δ(A/4Gℏ) + δS_out"]
    C --> S

    S --> E["Extremum Condition<br/>δS_gen = 0"]

    E --> R["Derive<br/>Einstein Equations"]

    style D fill:#e1f5ff
    style S fill:#fff4e1,stroke:#ff6b6b,stroke-width:2px
    style E fill:#ffe1e1
    style R fill:#e1ffe1,stroke:#ff6b6b,stroke-width:3px

🌟 From Local to Global

Radon-Type Closure

Core idea: If an integral condition holds for all small causal diamonds, can we derive a pointwise equation?

Answer: Yes, under appropriate conditions.

Steps:

1. Localization: For arbitrary test function $\phi \in C_c^{\infty }(S_{\ell })$ on waist $S_{\ell }$
2. Integral condition: ${\int }_{S_{\ell }}\phi {\int }_0^{{\lambda }_{\ast }}\lambda (R_{kk}-8\pi G{T}_{kk})d\lambda dA=o({\ell }^2)$
3. Closure: By local invertibility of ray transform, derive that $R_{kk}=8\pi G{T}_{kk}$ holds at each point

This constitutes the logical bridge from “family constraint” to “pointwise equation”.

Physical Meaning of Family Constraint

Family constraint: For a family of small causal diamonds, entropy extremum condition holds

$$\forall p\in M,\forall \ell \ll L_{\text{curv}}(p):\delta S_{\text{gen}}({\mathcal{D}}_{\ell }(p))=0$$

Pointwise equation: At each point, Einstein equations hold

$$\forall p\in M:G_{ab}(p)+\Lambda g_{ab}(p)=8\pi G{T}_{ab}(p)$$

Logic chain:

Family Constraint (for all small diamonds)
    ↓
Integral Identity (along null geodesics)
    ↓
Radon-Type Closure
    ↓
Pointwise Equation (at each point)

🎓 Comparison with Other Methods

Advantages of IGVP:

1. Completely local (no need for global horizon)
2. Mathematically rigorous (explicit error control)
3. Complete derivation (first-order + second-order)
4. Widely applicable (not limited to flat backgrounds)

📝 Key Formulas Summary

🎓 Further Reading

• Jacobson’s original paper: T. Jacobson, “Thermodynamics of spacetime” (Phys. Rev. Lett. 75, 1260, 1995)
• Small diamond geometry: T. Jacobson, “Entanglement Equilibrium and the Einstein Equation” (Phys. Rev. Lett. 116, 201101, 2016)
• GLS complete derivation: igvp-einstein-complete.md
• Previous: 01-generalized-entropy_en.md - Generalized Entropy Definition
• Next: 03-raychaudhuri-equation_en.md - Raychaudhuri Equation

🤔 Exercises

1. Conceptual understanding:

   • Why is a causal diamond called a “diamond”? Draw it in Minkowski spacetime
   • What is the waist? Why is it the “boundary of maximum spatial cross-section”?
   • What does “small” mean? Why take the $\ell \to 0$ limit?

2. Geometric calculations:

   • In three-dimensional spacetime $(t,x,y)$, write the explicit expression for a causal diamond with $\ell =1$
   • Calculate the area of the waist and internal volume
   • Verify $A\sim {\ell }^{d-2}$ and $V\sim {\ell }^{d-1}$

3. Physical applications:

   • Why is fixed volume $\delta V=0$ a reasonable constraint?
   • How to understand Unruh temperature $T=\hslash /(\pi \ell )$?
   • What is the relationship between small causal diamond and Rindler wedge?

4. Advanced thinking:

   • What problems would arise if we don’t take the small limit?
   • How does family constraint become a pointwise equation through Radon-type closure? (Hint: ray transform)
   • Why is the boundary of small causal diamond a null hypersurface? What is the physical meaning?

Next step: After understanding the stage of variation, we will learn how geometry responds—the Raychaudhuri equation!