Causal Diamond: Atom of Spacetime

“Causal diamonds are atoms of spacetime; all physics is defined on their boundaries.”

🎯 Core of This Article

In GLS theory, the most fundamental geometric object is not spacetime manifold $M$, but causal diamond (also called small causal lozenge):

$$\boxed{D(p,q)=J^{+}(p)\cap J^{-}(q)}$$

where $p\prec q$ are two causally related events.

Core Insight:

• All observables localized on boundary of causal diamond
• Bulk information completely determined by boundary null surfaces
• Modular Hamiltonian completely defined on boundary

This is manifestation of holographic principle at causal level!

🔷 Analogy: Facets of a Diamond

Imagine a diamond:

graph TB
    subgraph "Diamond Structure"
        TOP["Vertex q<br/>(Future)"] --> FACET1["Upper Facet"]
        FACET1 --> MID["Waist<br/>(Maximum Cross-Section)"]
        MID --> FACET2["Lower Facet"]
        FACET2 --> BOTTOM["Bottom p<br/>(Past)"]
    end

    FACET1 -.->|"Null Hypersurface"| NULL["Boundary E⁺"]
    FACET2 -.->|"Null Hypersurface"| NULL2["Boundary E⁻"]

    style TOP fill:#ffe1e1
    style BOTTOM fill:#e1f5ff
    style MID fill:#fff4e1

Physical Analogy of Diamond:

• Vertex $q$: Future observation event
• Bottom $p$: Past preparation event
• Upper Facet $E^{+}$: Future-directed null hypersurface (forward light cone)
• Lower Facet $E^{-}$: Past-directed null hypersurface (backward light cone)
• Waist: Maximum spacelike cross-section (Cauchy hypersurface)

Key Insight:

• Diamond’s value not in bulk, but in facets (boundary)
• Physical information encoded on null boundaries $E^{+}\cup E^{-}$
• Bulk is just reconstruction from boundary data

📐 Precise Definition

Causal Future and Causal Past

Given a point $p$ in spacetime, define:

Causal Future: $$J^{+}(p):=\{x\in M\mid p\prec x\}=\{x\mid \exists \gamma :p\to x,{\dot{\gamma }}^2\le 0\}$$

Causal Past: $$J^{-}(q):=\{x\in M\mid x\prec q\}=\{x\mid \exists \gamma :x\to q,{\dot{\gamma }}^2\le 0\}$$

where $\gamma$ is a non-spacelike curve (null or timelike).

graph TB
    subgraph "Light Cone Structure"
        Q["q"] --> FUTURE["J⁻(q)<br/>(Causal Past of q)"]
        PAST["J⁺(p)<br/>(Causal Future of p)"] --> P["p"]
        FUTURE -.->|"Intersection"| DIAMOND["Causal Diamond<br/>D(p,q)"]
        PAST -.->|"Intersection"| DIAMOND
    end

    style Q fill:#ffe1e1
    style P fill:#e1f5ff
    style DIAMOND fill:#fff4e1

Causal Diamond Definition

For causally related points $p\prec q$, define causal diamond:

$$D(p,q):=J^{+}(p)\cap J^{-}(q)$$

Geometric Meaning:

• $D(p,q)$ is set of all events simultaneously in future of $p$ and past of $q$
• This is a compact causally related region
• Boundary consists of two null hypersurfaces

Null Boundaries

Boundary of causal diamond divides into two parts:

Future Boundary (Future Null Boundary): $$E^{+}(p,q):=\partial J^{+}(p)\cap D(p,q)$$

Past Boundary (Past Null Boundary): $$E^{-}(p,q):=\partial J^{-}(q)\cap D(p,q)$$

Complete Boundary: $$\partial D(p,q)=E^{+}(p,q)\cup E^{-}(p,q)$$

graph TB
    Q["Future Endpoint q"] --> EPLUS["Future Boundary E⁺<br/>(Light Cone from p)"]
    EPLUS --> MID["Maximum Cross-Section Σ"]
    MID --> EMINUS["Past Boundary E⁻<br/>(Light Cone Converging to q)"]
    EMINUS --> P["Past Endpoint p"]

    EPLUS -.->|"Null"| NULLPLUS["ℓ·ℓ = 0"]
    EMINUS -.->|"Null"| NULLMINUS["k·k = 0"]

    style Q fill:#ffe1e1
    style P fill:#e1f5ff
    style EPLUS fill:#fff4e1
    style EMINUS fill:#e1ffe1

🌍 Causal Diamonds in Minkowski Spacetime

Coordinate Representation

In Minkowski spacetime $(t,\mathbf{x})$, set:

• $p=(0,0)$ (origin)
• $q=(T,0)$ (on time axis)

Then causal diamond:

$$D(p,q)=\{(t,\mathbf{x})\mid 0\le t\le T,|\mathbf{x}|\le \min (t,T-t)\}$$

Geometric Shape

In $d$-dimensional Minkowski spacetime:

• 2D ($1+1$): Diamond
• 3D ($2+1$): Double cone
• 4D ($3+1$): Double hypercone

graph TB
    subgraph "2D Minkowski Spacetime (t,x)"
        TOP["q = (T,0)"] -->|"x = T-t"| RIGHT["Right Boundary"]
        TOP -->|"x = -(T-t)"| LEFT["Left Boundary"]
        RIGHT --> MID["(T/2, T/2)"]
        LEFT --> MID
        MID --> RIGHT2["Right Boundary"]
        MID --> LEFT2["Left Boundary"]
        RIGHT2 -->|"x = t"| BOTTOM["p = (0,0)"]
        LEFT2 -->|"x = -t"| BOTTOM
    end

    style TOP fill:#ffe1e1
    style BOTTOM fill:#e1f5ff
    style MID fill:#fff4e1

Null Generating Vectors

Generating vector of future boundary $E^{+}$: $${\ell }^{\mu }=(1,\hat{\mathbf{n}})(\text{unit time direction}+\text{spatial radial})$$

Generating vector of past boundary $E^{-}$: $$k^{\mu }=(1,-\hat{\mathbf{n}})(\text{unit time direction}-\text{spatial radial})$$

Verify null property: $$\ell \cdot \ell =-1+1=0,k\cdot k=-1+1=0$$

📊 Boundary Area and Bulk Volume

Area Calculation

For causal diamond $D(p,q)$ in Minkowski spacetime, let proper time interval between $p$ and $q$ be $\tau$.

Boundary Area (at maximum cross-section): $$A(\partial D)={\Omega }_{d-2}{\tau }^{d-2}$$

where ${\Omega }_{d-2}$ is volume of $(d-2)$-dimensional sphere.

Bulk Volume: $$V(D)\sim {\tau }^{d-1}$$

Holographic Scaling

Key observation:

$$\frac{V(D)}{A(\partial D)}\sim \tau$$

Physical Meaning:

• Bulk volume $\sim$ boundary area $\times$ characteristic length
• Growth rate of boundary area one dimension lower than bulk volume
• This is geometric manifestation of holographic principle

In quantum gravity (Planck units ${\ell }_P=\sqrt{G}$):

$$\frac{\text{Bulk Degrees of Freedom}}{\text{Boundary Degrees of Freedom}}\sim \frac{V/{\ell }_P^{d-1}}{A/{\ell }_P^{d-2}}\sim \frac{\tau }{{\ell }_P}\gg 1(\text{classical limit})$$

But at Planck scale $\tau \sim {\ell }_P$:

$$\text{Bulk Degrees of Freedom}\sim \text{Boundary Degrees of Freedom}$$

This suggests boundary degrees of freedom can encode bulk information!

🔬 Causal Diamonds in Black Holes

Schwarzschild Black Hole

Consider Schwarzschild spacetime: $$\mathrm{d}{s}^2=-\left(1-\frac{2M}{r}\right)\mathrm{d}{t}^2+{\left(1-\frac{2M}{r}\right)}^{-1}\mathrm{d}{r}^2+r^2\mathrm{d}{\Omega }^2$$

Set:

• $p$: Some point outside horizon
• $q$: Another point in future of $p$

Causal Diamond Near Horizon:

• Boundaries $E^{+}$, $E^{-}$ approach horizon $r=2M$
• Horizon is null hypersurface, similar to causal diamond boundary
• Black hole horizon can be seen as boundary of “limiting causal diamond”

Rindler Horizon

In accelerating observer (Rindler coordinates): $$\mathrm{d}{s}^2=-{\rho }^2\mathrm{d}{\eta }^2+\mathrm{d}{\rho }^2+\mathrm{d}{x}_{\perp }^2$$

Rindler Horizon $\rho =0$ is special case of causal diamond boundary:

• Accelerating observer can only access causal diamond $D$
• Boundary $\rho =0$ is null hypersurface
• Unruh temperature: $T_U=\frac{a}{2\pi }$ ($a$ is acceleration)

Connection to Causal Diamond:

• Rindler horizon = boundary of causal diamond
• Unruh radiation = thermal effect on boundary
• Modular Hamiltonian $K_D$ = Rindler boost generator

graph TB
    subgraph "Rindler Wedge"
        HORIZON["Rindler Horizon<br/>ρ = 0"] --> WEDGE["Observable Wedge<br/>(Causal Diamond)"]
        WEDGE --> OBSERVER["Accelerating Observer"]
    end

    HORIZON -.->|"Equivalent"| BOUNDARY["Causal Diamond Boundary<br/>E⁺ ∪ E⁻"]
    OBSERVER -.->|"Measures"| UNRUH["Unruh Temperature<br/>T_U = a/(2π)"]

    style HORIZON fill:#ffe1e1
    style WEDGE fill:#fff4e1

🧮 Boundary Localization of Modular Hamiltonian

Core Theorem (Null-Modular Double Cover)

Theorem (detailed in next article): For causal diamond $D(p,q)$, its modular Hamiltonian $K_D$ is completely localized on null boundaries $E^{+}\cup E^{-}$:

$$K_D=2\pi \sum _{\sigma =\pm }{\int }_{E^{\sigma }}{g}_{\sigma }(\lambda ,x_{\perp })T_{\sigma \sigma }(\lambda ,x_{\perp })\mathrm{d}\lambda {\mathrm{d}}^{d-2}{x}_{\perp }$$

where:

• $\lambda$: Affine parameter along null geodesic
• $x_{\perp }$: Transverse coordinates
• $g_{\sigma }$: Modulation function
• $T_{\sigma \sigma }$: Component of stress tensor in null direction

graph TB
    DIAMOND["Causal Diamond D(p,q)"] --> BOUNDARY["Boundary E⁺ ∪ E⁻"]
    BOUNDARY --> MOD["Modular Hamiltonian<br/>K_D"]

    MOD --> EPLUS["E⁺ Contribution<br/>∫ g₊ T₊₊"]
    MOD --> EMINUS["E⁻ Contribution<br/>∫ g₋ T₋₋"]

    EPLUS -.->|"Sum"| TOTAL["Total Modular Hamiltonian"]
    EMINUS -.->|"Sum"| TOTAL

    style DIAMOND fill:#e1f5ff
    style BOUNDARY fill:#fff4e1
    style TOTAL fill:#e1ffe1

Physical Meaning

This formula reveals:

1. Boundary Completeness: Bulk operator $K_D$ completely determined by boundary data $T_{\sigma \sigma }{|}_{E^{\pm }}$
2. Null Localization: Only energy-momentum in null direction $T_{\sigma \sigma }$ contributes
3. Modulation Function: $g_{\sigma }(\lambda ,x_{\perp })$ encodes geometric information (similar to Radon transform)

Connection to Boundary Theory (Chapter 6):

• GHY boundary term on null boundaries: $S_{\mathrm{GHY}}{|}_{\text{null}}\sim \int (\theta +\kappa )$
• Brown-York energy: $E_{\mathrm{BY}}\sim \int K$ (extrinsic curvature)
• Modular Hamiltonian: $K_D\sim \int gT$

Unification of Three:

$$\text{Modular Hamiltonian}⟷\text{GHY Boundary Term}⟷\text{Brown-York Energy}$$

🔗 Connection to Unified Time Chapter

In Unified Time chapter (Chapter 5), we learned:

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

How Do Causal Diamonds Generate Time Scale?

Answer: Through boundary expansion $\theta$!

For null boundary $E^{+}$, define expansion:

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

Physical meaning: “Divergence rate” of null geodesic bundle.

Key Connection:

$$\kappa (\omega )⟷\theta +{\kappa }_{\mathrm{surf}}$$

where ${\kappa }_{\mathrm{surf}}$ is surface gravity.

graph LR
    DIAMOND["Causal Diamond D"] --> BOUNDARY["Null Boundary E⁺"]
    BOUNDARY --> EXPANSION["Expansion θ"]
    EXPANSION --> TIMESCALE["Time Scale κ(ω)"]

    TIMESCALE --> PHASE["Scattering Phase φ'/π"]
    TIMESCALE --> SPECTRAL["Spectral Shift ρ_rel"]
    TIMESCALE --> WIGNER["Wigner Delay tr Q/(2π)"]

    style DIAMOND fill:#e1f5ff
    style TIMESCALE fill:#fff4e1

Profound Insight:

• Geometry of causal diamond (expansion $\theta$)
• Determines time scale $\kappa (\omega )$
• Which determines all physical times (scattering, spectral, modular flow)

This is concrete mechanism of Causality → Time!

🔗 Causal Diamonds Are “Atoms” of GLS Theory

Why Called “Atoms”?

1. Smallest Causal Units

   • Causal diamonds are smallest causally complete regions
   • Any observable must be defined within some causal diamond

2. Boundary Determines Bulk

   • Boundary data $(E^{+},E^{-},T_{\sigma \sigma })$ completely determines bulk
   • This is manifestation of holographic principle

3. Composability

   • Multiple causal diamonds can construct larger regions through gluing
   • Partial order structure induced by local causal diamonds (detailed in next article)

4. Quantum Entanglement Units

   • Causal diamond boundaries define entanglement wedges
   • Modular Hamiltonian $K_D$ generates entanglement evolution of boundary states

graph TB
    ATOM["Causal Diamond = Spacetime Atom"] --> MIN["Smallest Causal Unit"]
    ATOM --> HOLO["Boundary Determines Bulk"]
    ATOM --> COMP["Composability"]
    ATOM --> ENT["Quantum Entanglement Unit"]

    MIN --> OBS["Observable Localization"]
    HOLO --> BOUNDARY["Boundary Data Completeness"]
    COMP --> GLUE["Partial Order Gluing"]
    ENT --> MODULAR["Modular Flow Evolution"]

    style ATOM fill:#fff4e1
    style HOLO fill:#e1f5ff

Connection to AdS/CFT

In AdS/CFT correspondence:

• Bulk AdS: Continuous superposition of causal diamonds
• Boundary CFT: Conformal field theory on causal diamond boundaries
• Entanglement Wedge Reconstruction: Causal diamond boundaries reconstruct bulk

RT Formula (Ryu-Takayanagi): $$S(A)=\frac{\text{Area}({\gamma }_A)}{4G}$$

where ${\gamma }_A$ is extremal surface, essentially generalization of causal diamond boundary!

💡 Key Points Summary

1. Causal Diamond Definition

$$D(p,q)=J^{+}(p)\cap J^{-}(q),p\prec q$$

• All events simultaneously in future of $p$ and past of $q$
• Compact causally complete region

2. Null Boundary Structure

$$\partial D=E^{+}(p,q)\cup E^{-}(p,q)$$

• $E^{+}$: Forward light cone
• $E^{-}$: Backward light cone
• Both are null hypersurfaces

3. Holographic Scaling

$$\frac{V(D)}{A(\partial D)}\sim \tau \Rightarrow \text{Boundary Degrees of Freedom}\sim \text{Bulk Degrees of Freedom}$$

4. Boundary Localization of Modular Hamiltonian

$$K_D=2\pi \sum _{\sigma =\pm }{\int }_{E^{\sigma }}{g}_{\sigma }{T}_{\sigma \sigma }\mathrm{d}\lambda {\mathrm{d}}^{d-2}{x}_{\perp }$$

• Bulk operator completely determined by boundary
• Energy-momentum in null direction contributes

5. “Atoms” of Spacetime

• Smallest causal units
• Boundary determines bulk
• Composable into larger structures
• Basic units of quantum entanglement

🤔 Thought Questions

Question 1: Why Must Boundaries of Causal Diamonds Be Null?

Hint: Consider boundary $\partial J^{+}(p)$ of causal future $J^{+}(p)$, what curves generate it?

Answer: $\partial J^{+}(p)$ is generated by null geodesics from $p$. Any timelike curve is in interior of $J^{+}(p)$, any spacelike separated point is not in $J^{+}(p)$. Therefore boundary is exactly null hypersurface.

Question 2: Why Does Modular Hamiltonian Only Depend on $T_{\sigma \sigma }$, Not All Components of $T_{\mu \nu }$?

Hint: Recall special properties of null hypersurfaces, and geometric meaning of modular flow.

Answer: Normal vector ${\ell }^{\mu }$ of null hypersurface is also tangent vector ($\ell \cdot \ell =0$). Modular flow corresponds to “boost” along null direction, so only invariant $T_{\ell \ell }:=T_{\mu \nu }{\ell }^{\mu }{\ell }^{\nu }$ contributes. Other components correspond to transverse or mixed directions, don’t participate in modular flow.

Question 3: In Minkowski Spacetime, How to Explicitly Write Modulation Function $g_{\sigma }(\lambda ,x_{\perp })$?

Hint: Consider relationship between Rindler coordinates and Minkowski coordinates.

Answer: For standard Rindler wedge, modulation function is: $$g(\rho ,\eta ,x_{\perp })=\frac{2\pi }{\beta }\rho$$ where $\beta =2\pi /a$ (inverse Unruh temperature), $\rho$ is Rindler radial coordinate. This corresponds to linear modulation (similar to triangular weight).

Question 4: Is Black Hole Horizon Boundary of Causal Diamond?

Hint: Consider “limiting” causal diamond, where one endpoint tends to horizon.

Answer: Yes! Black hole horizon can be seen as boundary of series of causal diamonds in limit. More precisely, horizon is Killing horizon, corresponding to boundary of fixed point set of modular flow. Therefore horizon has deep connection with causal diamond boundaries.

📖 Source Theory References

Content of this article mainly from following source theories:

Core Source Theory

Document: docs/euler-gls-causal/unified-theory-causal-structure-time-scale-partial-order-generalized-entropy.md

Key Content:

• Definition of causal diamond $D(p,q)=J^{+}(p)\cap J^{-}(q)$
• Structure of null boundaries $E^{+}\cup E^{-}$
• Boundary localization formula of modular Hamiltonian
• Null-Modular double cover theorem

Important Formula:

$$K_D=2\pi \sum _{\sigma =\pm }{\int }_{E^{\sigma }}{g}_{\sigma }(\lambda ,x_{\perp })T_{\sigma \sigma }(\lambda ,x_{\perp })\mathrm{d}\lambda {\mathrm{d}}^{d-2}{x}_{\perp }$$

Supporting Theories

Boundary Theory Chapter (Chapter 6):

• Form of GHY boundary term on null boundaries
• Geometric meaning of Brown-York energy
• Boundary triple $(\partial M,{\mathcal{A}}_{\partial },{\omega }_{\partial })$

Unified Time Chapter (Chapter 5):

• Definition of unified time scale $\kappa (\omega )$
• Connection between expansion $\theta$ and time scale
• Unification of scattering phase, spectral shift, Wigner delay

🎯 Next Steps

We’ve understood geometric structure of causal diamonds, next article will explore how to glue local causal diamonds into global partial order structure:

Next Article: 03-partial-order_en.md - Causal Network from Local to Global

There, we will see:

• Axiomatic definition of partial order $(M,\prec )$
• Čech-type consistency conditions
• How local causal diamonds glue into global spacetime
• Mathematical foundation of observer consensus

Back: Causal Structure Chapter Overview

Previous: 01-what-is-causality_en.md