Causal Structure: Overview

“Causality is not a relation, but a structure; not additional, but essential.”

🎯 Core Ideas of This Chapter

In GLS theory, causality is never a simple “cause → effect” relation, but a triply unified mathematical structure:

graph TB
    CAUSALITY["Causal Structure"] --> THREE["Triple Equivalence"]

    THREE --> ORDER["Geometric Partial Order<br/>∃p≺q"]
    THREE --> TIME["Time Scale Monotonicity<br/>τ(q)>τ(p)"]
    THREE --> ENTROPY["Generalized Entropy Monotonicity<br/>S_gen↑"]

    ORDER -.->|"Equivalent"| TIME
    TIME -.->|"Equivalent"| ENTROPY
    ENTROPY -.->|"Equivalent"| ORDER

    CAUSALITY --> DIAMOND["Basic Unit: Small Causal Diamond"]
    DIAMOND --> NULL["Null Boundary<br/>Null surfaces"]
    NULL --> MODULAR["Null-Modular Double Cover"]

    MODULAR --> MARKOV["Markov Property<br/>Memoryless Propagation"]

    style CAUSALITY fill:#fff4e1,stroke:#ff6b6b,stroke-width:4px
    style THREE fill:#e1f5ff,stroke:#0066cc,stroke-width:3px
    style DIAMOND fill:#e1ffe1,stroke:#00cc00,stroke-width:3px

Core Insight:

$$\boxed{\text{Causality}=\text{Partial Order}=\text{Time}=\text{Entropy}}$$

These are not three different things, but three projections of the same structure!

📚 Chapter Content Map

This chapter consists of 10 articles, revealing the complete picture of causal structure:

Article 1: What is Causality?

Core Question: What exactly is causality?

Three Equivalent Definitions:

1. Geometric Definition: $q\in J^{+}(p)$ (inside light cone)
2. Time Definition: $\tau (q)>\tau (p)$ (time scale increasing)
3. Entropy Definition: $S_{\mathrm{gen}}$ monotonically non-decreasing along path

Stunning Theorem: These three definitions are completely equivalent!

graph LR
    GEO["Geometric Causality<br/>q∈J⁺(p)"] -->|"Implies"| TIME["Time Monotonicity<br/>τ(q)>τ(p)"]
    TIME -->|"Implies"| ENT["Entropy Monotonicity<br/>S_gen↑"]
    ENT -->|"Implies"| GEO

    style GEO fill:#ffe1e1
    style TIME fill:#e1f5ff
    style ENT fill:#e1ffe1

Article 2: Geometry of Causal Diamond

Core Object: Small causal diamond

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

Why Important:

• Is smallest causally complete region
• IGVP variation defined here
• Generalized entropy extremized here
• Null-Modular double cover unfolds here

Structure:

graph TB
    DIAMOND["Causal Diamond D(p,q)"] --> BOUNDARY["Boundary Structure"]

    BOUNDARY --> FUTURE["Future Null Hypersurface<br/>N⁺"]
    BOUNDARY --> PAST["Past Null Hypersurface<br/>N⁻"]
    BOUNDARY --> CORNER["Corner Points<br/>p,q"]

    FUTURE --> AFFINE1["Affine Parameter λ⁺"]
    PAST --> AFFINE2["Affine Parameter λ⁻"]

    AFFINE1 --> MODULAR["Modular Hamiltonian K_D"]
    AFFINE2 --> MODULAR

    style DIAMOND fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style MODULAR fill:#e1f5ff,stroke:#0066cc,stroke-width:3px

Article 3: Partial Order Structure

Core Concept: Partial order $(M,\prec )$

Three Properties:

1. Reflexivity: $p⪯p$
2. Antisymmetry: $p⪯q$ and $q⪯p$ ⇒ $p=q$
3. Transitivity: $p⪯q$ and $q⪯r$ ⇒ $p⪯r$

Physical Realizations:

• Relativity: Light cone structure $J^{+}(p)$
• Quantum Field Theory: Operator commutativity (microcausality)
• Causal Set Theory: Discrete partially ordered set $(C,\prec )$

Gluing Problem: How do local partial orders of multiple observers combine into global partial order?

Čech Consistency Condition:

$$\boxed{\text{Local Partial Orders Must Agree on Overlap Regions}}$$

Article 4: Null-Modular Double Cover

Core Construction:

For causal diamond $D$, its null boundary decomposes as:

$${\tilde{E}}_D=E^{+}\bigsqcup E^{-}$$

where $E^{+}$ is future leaf, $E^{-}$ is past leaf.

Modular Hamiltonian Localization:

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

Deep Meaning:

• Modular flow completely localized on null boundary
• Time evolution generated by null boundary energy flux
• Bulk dynamics is projection of boundary data

graph TB
    DIAMOND["Causal Diamond D"] --> COVER["Null-Modular Double Cover"]

    COVER --> FUT["E⁺<br/>Future Leaf"]
    COVER --> PAST["E⁻<br/>Past Leaf"]

    FUT --> FLUX1["Energy Flux T₊₊"]
    PAST --> FLUX2["Energy Flux T₋₋"]

    FLUX1 --> KD["Modular Hamiltonian K_D"]
    FLUX2 --> KD

    KD --> MODULAR["Modular Flow σᵗ"]
    MODULAR --> TIME["Intrinsic Time"]

    style COVER fill:#e1f5ff,stroke:#0066cc,stroke-width:3px
    style KD fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px

Article 5: Markov Property and Information Propagation

Core Theorem (Casini-Teste-Torroba):

Information propagation on causal diamond chains satisfies Markov property:

$$\boxed{I(A:C|B)=0}$$

Physical Meaning:

• Information propagation memoryless
• Intermediate state $B$ completely screens $A$ and $C$
• Causal chain is first-order Markov process

Inclusion-Exclusion Formula:

$$K_{{\cup }_j{D}_j}=\sum _{k\ge 1}(-1{)}^{k-1}\sum _{j_1<\cdots <j_k}{K}_{D_{j_1}\cap \cdots \cap D_{j_k}}$$

Modular Hamiltonian satisfies perfect inclusion-exclusion structure!

Article 6: Observer Consensus Geometry

Core Question: How do multiple observers reach consensus on same cosmic causal network?

Observer Formalization:

$$O_i=(C_i,{\prec }_i,{\Lambda }_i,{\mathcal{A}}_i,{\omega }_i,{\mathcal{M}}_i,U_i,u_i,\{{\mathcal{C}}_{ij}\})$$

Contains: Geometric domain, partial order, resolution, algebra, state, model, update, utility, communication.

Three Types of Consensus:

graph TB
    CONSENSUS["Observer Consensus"] --> THREE["Three Levels"]

    THREE --> CAUSAL["Causal Consensus<br/>Local Partial Order Gluing"]
    THREE --> STATE["State Consensus<br/>Relative Entropy Convergence"]
    THREE --> MODEL["Model Consensus<br/>Unique True Model"]

    CAUSAL --> CECH["Čech Consistency"]
    CECH --> GLOBAL["Global Partial Order (M,≺)"]

    STATE --> LYAP["Lyapunov Function<br/>Φ = Σλᵢ D(ωᵢ‖ω*)"]
    LYAP --> CONVERGE["ωᵢ → ω_cons"]

    MODEL --> KL["KL Divergence<br/>D(P_{M*}‖P_M)>0"]
    KL --> UNIQUE["∩Mᵢ = {M*}"]

    style CONSENSUS fill:#fff4e1,stroke:#ff6b6b,stroke-width:4px
    style THREE fill:#e1f5ff,stroke:#0066cc,stroke-width:3px

State Consensus Theorem: Under conditions of strongly connected communication graph, primitive weight matrix, and existence of common fixed point ${\omega }_{\ast }$,

$$\underset{t\to \infty }{\lim }{\omega }_i^{(t)}={\omega }_{\ast },\forall i$$

Article 7: Causal Structure Mid-Summary

First summary of previous 6 articles, establishing initial connections between causality-time-entropy-boundary.

Article 8: Causal Geometrization—Spacetime as Minimal Lossless Compression

Core Idea: Spacetime geometry = minimal lossless compression of causal constraints

Three-Step Reconstruction Theorem:

1. Causal Partial Order → Topology: Alexandrov topology uniquely determined by causal structure
2. Causality + Time Orientation → Conformal Class: Light cone structure reconstructs conformal geometry
3. Causality + Volume Scale → Complete Metric: $(M,g)⟺(M,\prec ,\mu )$

Description Complexity-Curvature Functional:

$$\mathcal{F}[g]=\mathcal{C}(\text{Reach}(g))+\lambda {\int }_M|\text{Riem}(g){|}^2{\mathrm{dVol}}_g$$

Causal Interpretation of Curvature: Curvature = incompressible correlation redundancy density between causal constraints

graph TB
    CAUSAL["Causal Partial Order<br/>(M,≺)"] -->|"Alexandrov Topology"| TOPO["Topology"]
    TOPO -->|"Light Cone Reconstruction"| CONFORMAL["Conformal Class [g]"]
    CONFORMAL -->|"Volume Scale μ"| METRIC["Complete Metric g"]

    METRIC -.->|"Compression Perspective"| COMPRESS["Description Complexity 𝒞"]
    COMPRESS -->|"Redundancy"| CURV["Curvature Riem"]

    style CAUSAL fill:#e1f5ff
    style METRIC fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px

Article 9: Error Geometry and Causal Robustness

Core Idea: Error = geometric boundary, robustness = geometric invariance

Confidence Ellipsoid (Credible Region):

$${\mathcal{R}}_n(\alpha )=\left\{\theta :n(\theta -{\hat{\theta }}_n{)}^{\top }I({\hat{\theta }}_n)(\theta -{\hat{\theta }}_n)\le {\chi }_{d,1-\alpha }^2\right\}$$

Geometric Robustness Criterion: Causal conclusions should be based on intersection of credible region and identifiable set

$${\mathcal{C}}_n(\alpha ):=\{\psi (\theta ):\theta \in {\mathcal{R}}_n(\alpha )\cap {\mathcal{I}}_n\}$$

Multi-Experiment Consensus Region:

$${\mathcal{R}}_{\text{cons}}:=\bigcap _{k=1}^K{\mathcal{R}}_k({\alpha }_k)$$

graph TB
    ESTIMATE["Point Estimate θ̂"] --> ELLIPSE["Confidence Ellipsoid ℛₙ(α)"]
    IDENT["Identifiable Set ℐₙ"] --> INTER["Intersection ℛₙ∩ℐₙ"]
    ELLIPSE --> INTER

    INTER --> ROBUST["Robust Causal Conclusion"]

    MULTI["Multiple Experiments"] --> CONS["Consensus Region ℛ_cons"]
    CONS -->|"Non-Empty"| AGREEMENT["Results Agree"]
    CONS -->|"Empty"| CONFLICT["Significant Conflict"]

    style INTER fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style ROBUST fill:#e1ffe1

Article 10: Unified Theorem Complete Proof

Core Theorem: Within semiclassical-holographic window, three are completely equivalent:

$$\boxed{\begin{array}{rl}& \text{Geometric Causality: }q\in J^{+}(p)\\ & \text{Time Monotonicity: }\tau (q)>\tau (p)\\ & \text{Entropy Arrow: }{S}_{\text{gen}}[{\Sigma }_q]\ge S_{\text{gen}}[{\Sigma }_p]\end{array}}$$

Unified Time Scale Equivalence Class:

$$[\tau ]=\{{\tau }_{\text{scatt}},{\tau }_{\text{mod}},{\tau }_{\text{geom}}\}/\sim$$

Equivalence of IGVP and Einstein Equation:

$$\delta S_{\text{gen}}=0⟺G_{\mu \nu }+\Lambda g_{\mu \nu }=8\pi G{T}_{\mu \nu }$$

Markov Property and Inclusion-Exclusion Formula:

$$K_{{\cup }_j{D}_j}=\sum _{k\ge 1}(-1{)}^{k-1}\sum _{j_1<\cdots <j_k}{K}_{D_{j_1}\cap \cdots \cap D_{j_k}}$$

graph TB
    subgraph "Unification Theorem Core"
        CAUSAL["Causal Partial Order"] <-->|"Theorem 2"| TIME["Time Scale"]
        TIME <-->|"Theorem 2"| ENTROPY["Entropy Arrow"]
        ENTROPY <-->|"Theorem 2"| CAUSAL
    end

    subgraph "Time Scale Unification"
        SCATT["Scattering Time"] <-->|"Theorem 1"| MOD["Modular Flow Time"]
        MOD <-->|"Theorem 1"| GEOM["Geometric Time"]
    end

    subgraph "Variational Principle"
        IGVP["δS_gen=0"] <-->|"Theorem 3"| EINSTEIN["Einstein Equation"]
    end

    TIME -.-> MOD
    ENTROPY -.-> IGVP

    style CAUSAL fill:#fff4e1,stroke:#ff6b6b,stroke-width:4px
    style TIME fill:#e1f5ff
    style ENTROPY fill:#e1ffe1

Complete Picture:

Reviewing the unified picture of entire causal structure:

$$\boxed{\text{Partial Order}⟺\text{Time Scale}⟺\text{Generalized Entropy}⟺\text{Einstein Equation}}$$

Most Profound Insight:

Causality is not an additional structure, but intrinsic structure unifying time, geometry, and entropy, proven through rigorous mathematical theorems.

🔗 Connections with Other Chapters

Looking Back: Boundary Theory Chapter (Chapter 6)

In Boundary Theory, we saw:

• Form of GHY boundary term on null boundary
• Brown-York energy as boundary time generator
• Trinity of boundary observers

Now Deepened:

• Null boundary is not additional, but essential component of causal diamond
• Null-Modular double cover gives geometric realization of modular flow
• Observer consensus geometry reveals how causality is reconstructed by multiple observers

Looking Back: Unified Time Chapter (Chapter 5)

In Unified Time chapter, we proved:

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

Now Deepened:

• This unified scale defines direction of causal partial order
• Time scale monotonicity equivalent to causal partial order
• Unified time is coordinate of causal structure

Looking Forward: Topological Constraints Chapter (Chapter 8)

Causal structure provides foundation for topological constraints:

Causal Topology: Topological structure induced by causal relations

Alexandrov Topology: Topology with causal diamonds as open set basis

Topological Anomaly: ${\mathbb{Z}}_2$ sector $[K]\in H^2(Y,\partial Y;{\mathbb{Z}}_2)$

Undecidability: Some causal structure problems are inherently undecidable

💡 Learning Roadmap

graph TB
    START["Start Causal Structure"] --> WHAT["01-What is Causality"]
    WHAT --> DIAMOND["02-Causal Diamond"]
    DIAMOND --> ORDER["03-Partial Order Structure"]
    ORDER --> NULL["04-Null-Modular Double Cover"]
    NULL --> MARKOV["05-Markov Property"]
    MARKOV --> OBSERVER["06-Observer Consensus"]
    OBSERVER --> SUM1["07-Mid-Summary"]

    SUM1 --> COMPRESS["08-Causal Geometrization"]
    COMPRESS --> ERROR["09-Error Geometry"]
    ERROR --> PROOF["10-Unified Theorem Proof"]

    NULL -.->|"Deep Dive"| TECH["Technical Appendix"]
    PROOF -.->|"Rigorous Proof"| TECH

    style START fill:#e1f5ff
    style PROOF fill:#e1ffe1,stroke:#ff6b6b,stroke-width:4px
    style NULL fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px

Recommended Reading Order

Quick Path (grasp core):

1. 01-What is Causality (triple equivalence)
2. 04-Null-Modular Double Cover (core construction)
3. 08-Causal Geometrization (compression perspective)
4. 10-Unified Theorem Proof (complete picture)

Deep Learning (complete understanding): Read 01-10 in order, with source theory documents

Technical Research (rigorous derivation): Focus on:

• 03-Partial Order Structure gluing theorem
• 04-Null-Modular localization proof
• 08-Description complexity-curvature functional
• 09-Fisher information geometry
• 10-Complete proof of unified theorem (Axioms G/S/M/B/E/T)

🎓 Core Conclusions Preview

After completing this chapter, you will understand:

1. Triple Equivalence Theorem of Causality

Theorem: The following three propositions are equivalent:

1. Geometric Causality: $q\in J^{+}(p)$
2. Time Monotonicity: Exists $\tau \in [\tau ]$ such that $\tau (q)>\tau (p)$
3. Entropy Monotonicity: $S_{\mathrm{gen}}$ monotonically non-decreasing along path

Proof Idea:

• $(1)\Rightarrow (2)$: Time function existence
• $(2)\Rightarrow (3)$: QNEC + generalized entropy variation
• $(3)\Rightarrow (1)$: Entropy monotonicity excludes closed causal curves

2. Null-Modular Double Cover Theorem

Theorem: Modular Hamiltonian of causal diamond $D$ can be completely localized on null boundary:

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

where $E^{\sigma }$ are two null boundary leaves of causal diamond.

Physical Meaning:

• Modular flow doesn’t need bulk, only needs boundary
• Time evolution completely determined by null boundary energy flux
• Information-theoretic foundation of holographic principle

3. Markov Property Theorem

Theorem (Casini-Teste-Torroba): Causal diamond chains on null planes satisfy:

1. Inclusion-Exclusion Formula:

   $$K_{{\cup }_j{D}_j}=\sum _{k\ge 1}(-1{)}^{k-1}\sum _{j_1<\cdots <j_k}{K}_{D_{j_1}\cap \cdots \cap D_{j_k}}$$

2. Markov Property: For nested regions $A\subset B\subset C$,

   $$I(A:C|B)=0$$

Physical Meaning: Information propagation on causal chains is memoryless first-order Markov process.

4. Observer Consensus Theorem

Theorem (Causal Consensus): Local partial order family $\{(C_i,{\prec }_i)\}$ can be glued into global partial order $(M,\prec )$ if and only if:

1. Covering: ${\bigcup }_i{C}_i=M$
2. Finite Overlap: Each point covered by finitely many $C_i$
3. Čech Consistency: Local partial orders agree on all overlap regions

Theorem (State Consensus): Under conditions of strongly connected communication graph, primitive weight matrix, and existence of common fixed point, state iteration converges to unique consensus:

$$\underset{t\to \infty }{\lim }{\omega }_i^{(t)}={\omega }_{\ast },\forall i$$

Lyapunov Function:

$${\Phi }^{(t)}=\sum _i{\lambda }_{i}D({\omega }_i^{(t)}\Vert {\omega }_{\ast })$$

Strictly monotonically decreasing!

🤔 Thinking Questions (Chapter Preview)

Question 1: Why is Causality Equivalent to Time?

Hint: Recall relation between stable causality and time function.

Answer in: 01-What is Causality, 03-Partial Order Structure

Question 2: Why Can Modular Hamiltonian Be Completely Localized on Null Boundary?

Hint: Think about relation between Bisognano-Wichmann theorem and null generators.

Answer in: 04-Null-Modular Double Cover

Question 3: Why Does Information Propagation Satisfy Markov Property?

Hint: What does strong subadditivity saturation mean?

Answer in: 05-Markov Property

Question 4: What Happens if Local Partial Orders Are Inconsistent?

Hint: Imagine three observers forming causal loop.

Answer in: 06-Observer Consensus (three-node loop example in appendix)

📖 Notation Conventions

This chapter uses the following core symbols:

Geometric Symbols

• $(M,g)$: Spacetime manifold and metric
• $J^{+}(p)$: Future light cone of point $p$
• $J^{-}(q)$: Past light cone of point $q$
• $D(p,q)=J^{+}(p)\cap J^{-}(q)$: Causal diamond

Partial Order Symbols

• $(M,\prec )$: Partially ordered set
• $p\prec q$: $p$ in causal past of $q$
• $p⪯q$: $p\prec q$ or $p=q$

Causal Diamond Boundary

• ${\mathcal{N}}^{+}$: Future null hypersurface
• ${\mathcal{N}}^{-}$: Past null hypersurface
• $E^{+},E^{-}$: Two leaves of Null-Modular double cover
• $\lambda$: Affine parameter of null geodesic

Modular Flow and Algebra

• $\mathcal{A}(D)$: Local algebra of causal diamond $D$
• $\omega$: State
• $K_{\omega }$: Modular Hamiltonian
• ${\sigma }_t^{\omega }$: Modular flow
• $T_{\sigma \sigma }$: Stress-energy component along null direction

Observer Symbols

• $O_i$: $i$-th observer
• $C_i$: Causal domain of observer $i$
• ${\prec }_i$: Local partial order
• ${\mathcal{A}}_i$: Observable algebra
• ${\omega }_i$: Local state

Consensus Symbols

• ${\mathcal{A}}_{\mathrm{com}}={\bigcap }_i{\mathcal{A}}_i$: Common algebra
• ${\omega }_{\ast }$: Consensus state
• $D({\omega }_i\Vert {\omega }_{\ast })$: Relative entropy (Umegaki)
• $W=(w_{ij})$: Communication weight matrix

🔍 Unique Contributions of This Chapter

Compared to traditional causality theory, this chapter:

1. Unifies Three Perspectives

Traditional:

• Geometry: Light cones and partial orders
• Algebra: Microcausality and commutativity
• Information: Entropy and time arrow

Discussed separately, not connected.

This Chapter: Unifies three as triple equivalence of causality, revealing they are different projections of same structure.

2. Emphasizes Null-Modular Double Cover

Traditional: Null boundary is technical boundary condition.

This Chapter: Null-Modular double cover is essential structure of causal diamond, modular flow completely localized here.

3. Introduces Observer Consensus Geometry

Traditional: Causal structure is objectively given.

This Chapter: How do multiple observers reconstruct global causal network from local partial orders? Čech consistency, state consensus, model consensus.

4. Connects Markov Property and Causality

Traditional: Markov property is probabilistic concept.

This Chapter: Markov property is essential attribute of causal chains, strictly characterized by inclusion-exclusion formula.

🌟 Why Is This Chapter Important?

Causal Structure chapter is the hub of GLS theory, because:

Theoretical Level

• Reveals four-in-one of causality, time, geometry, entropy
• Provides gluing framework from local to global
• Gives Markov structure of information propagation

Application Level

• Quantum gravity: Foundation of causal set theory
• Holographic principle: How boundary encodes bulk causality
• Quantum computation: Causal networks and information processing

Philosophical Level

• What is the essence of causality?
• How do multiple observers share the same causal world?
• How does information propagate on causal network?

Ready?

Let’s begin this causal journey from partial order to Markov, from local to global, from observer to consensus!

Next Article: 01-What is Causality - Triple equivalent definitions of causality

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