What is Causality?

“Causality is not a relation, but a structure.”

🎯 Core of This Article

In GLS theory, causality is no longer a simple naive concept of “cause leads to effect”, but a trinitarian mathematical structure:

$$\boxed{\text{Causal Structure}=\text{Geometric Partial Order}=\text{Time Monotonicity}=\text{Entropy Increase}}$$

These three seemingly different concepts are actually projections of the same object from different perspectives!

This article will reveal this profound equivalence.

🌊 Analogy: Causality in a River

Imagine a river:

graph LR
    SOURCE["Source"] -->|"Water Flow"| MID["Midstream"]
    MID -->|"Water Flow"| SEA["Estuary"]

    SOURCE -.->|"Geometry"| GEO["Upstream Spatially Higher Than Downstream"]
    SOURCE -.->|"Time"| TIME["Water Flows from Early to Late"]
    SOURCE -.->|"Entropy"| ENT["Disorder Increases"]

    style SOURCE fill:#e1f5ff
    style SEA fill:#ffe1e1

In this analogy:

• Geometric Causality: Upstream is in “past light cone” of downstream (higher gravitational potential)
• Time Causality: Time monotonically increases along river
• Entropy Causality: Entropy continuously increases during water flow (from ordered to chaotic)

Key Insight: These three descriptions are equivalent! You can use any one to define “causality”, and the other two automatically hold.

📐 Definition One: Geometric Causality (Partial Order Structure)

Classical Definition

In spacetime $(M,g_{\mu \nu })$, causal relation is defined as a partial order $\prec$:

$$p\prec q⟺q\in J^{+}(p)$$

where:

• $J^{+}(p)$: Causal future of point $p$
• $J^{-}(q)$: Causal past of point $q$

graph TB
    subgraph "Minkowski Spacetime"
        P["p"] --> LIGHT["Light Cone"]
        LIGHT --> Q["q ∈ J⁺(p)"]
        LIGHT --> R["r ∈ J⁺(p)"]
    end

    P -.->|"Causal Relation"| REL["p ≺ q"]

    style P fill:#e1f5ff
    style Q fill:#ffe1e1
    style LIGHT fill:#f0f0f0

Partial Order Axioms

Causal partial order $\prec$ must satisfy:

1. Reflexivity: $p\prec p$

   • Physical meaning: Any event is in its own causal cone

2. Transitivity: $p\prec q\wedge q\prec r\Rightarrow p\prec r$

   • Physical meaning: Causal chains can propagate

3. Antisymmetry: $p\prec q\wedge q\prec p\Rightarrow p=q$

   • Physical meaning: No causal loops (unless same event)

Light Cone Structure

Geometric causality is completely determined by light cone structure:

$$J^{+}(p)=\{q\in M\mid \exists \gamma :p\to q,{\dot{\gamma }}^{\mu }{\dot{\gamma }}_{\mu }\le 0\}$$

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

Intuitive Understanding:

• Inside light cone: Can transmit information via “signals” (light speed or sub-light speed)
• Outside light cone: Causally unreachable, cannot transmit information

⏰ Definition Two: Time Causality (Time Monotonicity)

Time Function

If there exists a time function $\tau :M\to \mathbb{R}$ such that:

$$p\prec q⟺\tau (q)>\tau (p)$$

then spacetime is said to have time orientation.

graph LR
    subgraph "Time Function τ"
        T0["τ = 0<br/>(Spacelike Hypersurface Σ₀)"] -->|"Evolution"| T1["τ = 1<br/>(Spacelike Hypersurface Σ₁)"]
        T1 -->|"Evolution"| T2["τ = 2<br/>(Spacelike Hypersurface Σ₂)"]
    end

    T0 -.->|"Causal Monotonicity"| MONO["τ(q) > τ(p)<br/>⟺ p ≺ q"]

    style T0 fill:#e1f5ff
    style T2 fill:#ffe1e1

Time Scale and Causality

Recall core result from Unified Time chapter (Chapter 5):

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

This unified time scale $\kappa (\omega )$ is completely determined by causal structure!

Key Theorem (Existence of Time Function):

For globally hyperbolic spacetime, there exists a smooth time function $\tau$ such that each constant-time hypersurface $\{\tau =t\}$ is a Cauchy hypersurface.

Physical Meaning:

• Cauchy hypersurface: “Complete time slice” of spacetime
• Existence of time function $⟺$ Spacetime has well-defined causal structure

Proper Time and Time Causality

For timelike curve $\gamma$, proper time ${\tau }_{\mathrm{proper}}$ monotonically increases along curve:

$${\tau }_{\mathrm{proper}}[\gamma ]={\int }_{\gamma }\sqrt{-g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }}\mathrm{d}\lambda$$

This gives another definition of time causality:

$$p\prec q⟺\exists \gamma :p\to q,{\tau }_{\mathrm{proper}}[\gamma ]>0$$

📈 Definition Three: Entropy Causality (Generalized Entropy Monotonicity)

Generalized Entropy

In GLS theory, generalized entropy is defined as:

$$S_{\mathrm{gen}}[\Sigma ]=\frac{A(\partial \Sigma )}{4G}+S_{\mathrm{matter}}[\Sigma ]$$

where:

• $A(\partial \Sigma )$: Boundary area
• $S_{\mathrm{matter}}$: Matter entropy

Entropy Causality Principle

Core Idea: Entropy monotonically increases along causal direction.

If ${\Sigma }_1\prec {\Sigma }_2$ (i.e., ${\Sigma }_2$ is in causal future of ${\Sigma }_1$), then:

$$S_{\mathrm{gen}}[{\Sigma }_2]\ge S_{\mathrm{gen}}[{\Sigma }_1]$$

graph TB
    SIGMA1["Σ₁<br/>S_gen = S₁"] -->|"Causal Evolution"| SIGMA2["Σ₂<br/>S_gen = S₂ ≥ S₁"]
    SIGMA2 -->|"Causal Evolution"| SIGMA3["Σ₃<br/>S_gen = S₃ ≥ S₂"]

    SIGMA1 -.->|"Entropy Increase"| LAW["Generalized Second Law"]

    style SIGMA1 fill:#e1f5ff
    style SIGMA3 fill:#ffe1e1

QNEC and Entropy Causality

Quantum Null Energy Condition (QNEC) provides differential form of entropy causality:

$$⟨T_{kk}⟩\ge \frac{\hslash }{2\pi A^{\prime }}{S}^{\prime \prime }$$

where:

• $T_{kk}$: Component of stress tensor in null direction
• $A^{\prime }$: First derivative of area
• $S^{\prime \prime }$: Second derivative of von Neumann entropy

Physical Meaning:

• QNEC unifies geometry ($A^{\prime }$), matter ($T_{kk}$), entropy ($S^{\prime \prime }$)
• Convexity of entropy $⟹$ Energy condition $⟹$ Stability of causal structure

🔄 Trinity: Equivalence Proof

Now we prove equivalence of three definitions.

Step One: Geometry $\Rightarrow$ Time

Theorem (Geroch 1970): For globally hyperbolic spacetime $(M,g_{\mu \nu })$, there exists smooth time function $\tau :M\to \mathbb{R}$ such that:

$$p\prec q⟺\tau (q)>\tau (p)$$

Proof Outline:

1. Choose arbitrary Cauchy hypersurface ${\Sigma }_0$
2. Define $\tau (p)$ as proper time from ${\Sigma }_0$ to $p$
3. By global hyperbolicity, this function is well-defined and monotonic

Step Two: Time $\Rightarrow$ Entropy

Theorem (Generalized Second Law): If $\tau (q)>\tau (p)$, then:

$$S_{\mathrm{gen}}[{\Sigma }_q]\ge S_{\mathrm{gen}}[{\Sigma }_p]$$

where ${\Sigma }_p$, ${\Sigma }_q$ are constant-time hypersurfaces $\{\tau =\mathrm{const}\}$.

Proof Basis:

• Quantum Focusing Theorem
• Wall’s proof of Generalized Second Law (2011)
• QNEC as differential form

Step Three: Entropy $\Rightarrow$ Geometry

Theorem (Reverse Implication): If for all Cauchy hypersurfaces ${\Sigma }_1\prec {\Sigma }_2$ we have $S_{\mathrm{gen}}[{\Sigma }_2]\ge S_{\mathrm{gen}}[{\Sigma }_1]$, then causal structure of spacetime is determined by Einstein equations:

$$G_{\mu \nu }=8\pi G{T}_{\mu \nu }$$

Proof Outline (from IGVP framework):

1. Extremal variation of generalized entropy: $\delta S_{\mathrm{gen}}=0$
2. Derive Einstein equations as first-order condition
3. QNEC/QFC as second-order condition, ensuring causal structure stability

Complete Cycle:

$$\text{Geometric Partial Order}\underset{\text{Geroch}}{\overset{\text{Time Function}}{\to }}\text{Time Monotonicity}\underset{\text{Wall}}{\overset{\text{Generalized Second Law}}{\to }}\text{Entropy Increase}\underset{\text{IGVP}}{\overset{\text{Variational Principle}}{\to }}\text{Einstein Equations}$$

graph TB
    GEO["Geometric Causality<br/>p ≺ q ⟺ q ∈ J⁺(p)"] -->|"Geroch Theorem<br/>Time Function Exists"| TIME["Time Causality<br/>τ(q) > τ(p)"]
    TIME -->|"Wall Theorem<br/>Generalized Second Law"| ENT["Entropy Causality<br/>S_gen↑"]
    ENT -->|"IGVP Variation<br/>Einstein Equations"| GEO

    GEO -.->|"Three Equivalent"| EQUIV["Same Structure<br/>Three Projections"]
    TIME -.->|"Three Equivalent"| EQUIV
    ENT -.->|"Three Equivalent"| EQUIV

    style GEO fill:#e1f5ff
    style TIME fill:#fff4e1
    style ENT fill:#ffe1e1
    style EQUIV fill:#e1ffe1

🌍 Physical Examples

Example 1: Minkowski Spacetime

Geometric Causality: $$p\prec q⟺(t_q-t_p{)}^2\ge ({\mathbf{x}}_q-{\mathbf{x}}_p{)}^2$$

Time Causality: $$\tau (p)=t_p(\text{proper time of inertial observer})$$

Entropy Causality: $$S_{\mathrm{gen}}[{\Sigma }_t]=S_{\mathrm{matter}}[{\Sigma }_t](\text{no gravitational entropy})$$

All three perfectly consistent!

Example 2: Schwarzschild Black Hole

Geometric Causality:

• Horizon $r=2M$ is null hypersurface
• Interior: All timelike curves terminate at singularity $r=0$

Time Causality:

• Kruskal time $T$ monotonically increases crossing horizon
• Schwarzschild time $t$ diverges at horizon (coordinate singularity)

Entropy Causality:

• Bekenstein-Hawking entropy: $S_{\mathrm{BH}}=\frac{A}{4G}=4\pi M^2$
• Horizon area theorem (Hawking 1971): $\delta A\ge 0$

Example 3: Cosmology (FLRW Metric)

Geometric Causality:

• Particle horizon: $d_{\mathrm{hor}}=a(t){\int }_0^t\frac{\mathrm{d}{t}^{\prime }}{a(t^{\prime })}$
• Cosmological redshift defines causal structure

Time Causality:

• Cosmic proper time $t$
• Conformal time $\eta =\int \frac{\mathrm{d}t}{a(t)}$

Entropy Causality:

• Cosmological horizon entropy: $S_{\mathrm{hor}}\sim A_{\mathrm{hor}}/4G$
• Gibbons-Hawking temperature: $T_{\mathrm{GH}}\sim H$

🔗 Connection to Unified Time Chapter

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

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

Now we see: This unified time scale is completely determined by causal structure!

graph TB
    CAUSALITY["Causal Structure<br/>(M, ≺)"] --> TIMEFUNC["Time Function<br/>τ: M → ℝ"]
    TIMEFUNC --> KAPPA["Unified Scale<br/>κ(ω)"]

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

    KAPPA -.->|"All Equivalent"| UNIFIED["Same Time"]

    style CAUSALITY fill:#e1f5ff
    style KAPPA fill:#fff4e1
    style UNIFIED fill:#e1ffe1

Profound Insight:

• Causal structure $⟹$ Time function exists
• Time function $⟹$ Unified time scale
• Time scale $⟹$ All physical times (scattering, spectral shift, modular flow, geometry)

Therefore: All time concepts in physics originate from causal structure!

🔗 Connection to Boundary Theory Chapter

In Boundary Theory chapter (Chapter 6), we learned:

• GHY boundary term: $S_{\mathrm{GHY}}=\frac{\epsilon }{8\pi G}{\int }_{\partial M}\sqrt{|h|}K{\mathrm{d}}^{3}x$
• Brown-York quasi-local energy: $E_{\mathrm{BY}}=\int \sqrt{\sigma }{u}_a{u}_b{T}_{\mathrm{BY}}^{ab}{\mathrm{d}}^{2}x$

Connection Between Causality and Boundary:

In subsequent articles of this chapter, we will see:

• Boundary of causal diamond is null hypersurfaces
• Modular Hamiltonian completely localized on these null boundaries
• Form of GHY boundary term on null boundaries: $(\theta +\kappa )$ structure

graph LR
    DIAMOND["Causal Diamond<br/>D(p,q)"] --> BOUNDARY["Null Boundaries<br/>E⁺ ∪ E⁻"]
    BOUNDARY --> GHY["GHY Term<br/>θ + κ"]
    BOUNDARY --> MOD["Modular Hamiltonian<br/>K_D"]

    GHY -.->|"Same Object"| MOD

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

💡 Key Points Summary

1. Trinitarian Definition

Causality has three equivalent definitions:

• Geometry: $p\prec q\iff q\in J^{+}(p)$
• Time: $p\prec q\iff \tau (q)>\tau (p)$
• Entropy: $p\prec q\Rightarrow S_{\mathrm{gen}}[{\Sigma }_q]\ge S_{\mathrm{gen}}[{\Sigma }_p]$

2. Mathematical Structure

Causal structure is partial order $(M,\prec )$, satisfying:

• Reflexivity, transitivity, antisymmetry
• Completely determined by light cone structure $J^{\pm }(p)$
• Equivalent to existence of time function (globally hyperbolic spacetime)

3. Physical Meaning

Causal structure determines:

• Possibility of information transmission (inside vs outside light cone)
• Direction of time evolution (time arrow)
• Necessity of entropy increase (generalized second law)

4. Unified Framework

Causal structure is core of GLS theory:

• Unification of geometry ↔ time ↔ entropy
• Source of unified time scale
• Foundation of boundary theory (null boundaries)

🤔 Thought Questions

Question 1: Why Are Causal Loops Forbidden?

Hint: If $p\prec q\prec p$, what does this mean for time function $\tau$?

Answer: This means $\tau (p)<\tau (q)<\tau (p)$, contradiction! Therefore globally hyperbolic spacetime cannot have causal loops (closed timelike curves, CTC).

Question 2: Why Does Schwarzschild Time Diverge at Horizon?

Hint: Is this real singularity or coordinate choice?

Answer: This is coordinate singularity, not real singularity. Can be removed using Kruskal coordinates. Causal structure itself is smooth at horizon.

Question 3: How Is Causal Equivalence Embodied in AdS/CFT?

Hint: How does boundary CFT time correspond to causal structure of bulk AdS?

Answer: Conformal time of boundary CFT directly corresponds to radial coordinate of AdS bulk, and causal structure of AdS is completely determined by boundary conformal structure (detailed in next chapter).

Question 4: How Is Causality Defined in Quantum Field Theory?

Hint: Recall commutator $[\varphi (x),\varphi (y)]$.

Answer: If $x-y$ is spacelike separated, then $[\varphi (x),\varphi (y)]=0$ (microcausality). This is equivalent to no superluminal signal transmission.

📖 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:

• Triple equivalence of causal structure
• Causal origin of time scale identity
• Small causal diamonds and modular Hamiltonian
• IGVP framework and Einstein equations

Important Theorem:

“Partial order structure $\prec$, time scale monotonicity, generalized entropy increase are equivalent, together defining causal structure.”

Supporting Source Theory

Document: docs/euler-gls-causal/observer-properties-consensus-geometry-causal-network.md

Key Content:

• Observer’s definition of causal structure
• Čech-type consistency conditions
• Global gluing of local partial orders

🎯 Next Steps

We’ve understood what causality is (trinitarian definition), next article will delve into geometric structure of causal diamonds:

Next Article: 02-causal-diamond_en.md - Geometry and Topology of Small Causal Lozenge

There, we will see:

• Complete geometry of causal diamond $D(p,q)=J^{+}(p)\cap J^{-}(q)$
• Structure of null boundaries $E^{+}$ and $E^{-}$
• Relationship between boundary area and bulk volume
• Why causal diamonds are “atoms” of GLS theory

Back: Causal Structure Chapter Overview