What is Causality?

“Everything has a cause. But what does ‘cause leads to effect’ really mean? This question is far deeper than you imagine.”

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Starting from Dominoes

Imagine a row of dominoes:

graph LR
    D1["🀅"] -->|knock down| D2["🀅"]
    D2 -->|knock down| D3["🀅"]
    D3 -->|knock down| D4["🀅"]
    D4 -->|knock down| D5["🀅"]

    style D1 fill:#ff6b6b,color:#fff

You knock down the first domino, it falls, hits the second, the second falls, hits the third…

We say: The fall of the first domino “causes” the fall of the second domino.

🤔 But What Does “Cause” Mean?

Let’s analyze:

1. Temporal Order: First domino falls before, second falls after
2. Spatial Contact: First domino must touch the second
3. Regularity: Every experiment, same result (if first falls, second must fall)
4. Counterfactual Condition: If first didn’t fall, second wouldn’t fall

💡 Key Question: Are these four conditions enough? Can we give “causality” a strict definition?

Hume’s Challenge: Is Causality an Illusion?

18th-century philosopher David Hume proposed a shocking view:

We never “see” causality, we only see events occurring in sequence.

Example: Billiard ball collision

graph LR
    A["White Ball Moving"] --> B["White Ball Collides with Red Ball"]
    B --> C["Red Ball Starts Moving"]

    style B fill:#ffe66d,stroke:#f59f00,stroke-width:2px

What do you see?

1. White ball is moving
2. White ball touches red ball
3. Red ball starts moving

But you don’t see “causing” itself. You only infer: White ball “causes” red ball’s motion.

⚠️ Hume’s View: Causality is not an objective thing, but a habit of our mind. We’re used to calling events that “always occur in sequence” as “causal relationships.”

This raises a profound question: Is causality real, or an interpretation we impose on the world?

Physics’ Causality: Light Cone Structure

Physics has a stricter definition of causality, centered on the light cone.

🌟 What is a Light Cone?

In relativity, any event has a “light cone” that divides three regions:

graph TB
    subgraph "Time (Future)"
        Future["Future Light Cone<br/>Events You Can Affect"]
    end

    Event["Event: Here and Now<br/>(You, Now)"]

    subgraph "Time (Past)"
        Past["Past Light Cone<br/>Events That Can Affect You"]
    end

    subgraph "Elsewhere"
        Elsewhere["Spacelike Separation<br/>Cannot Affect You,<br/>You Cannot Affect"]
    end

    Past -->|light-speed signal| Event
    Event -->|light-speed signal| Future
    Event -.no causality.-> Elsewhere

    style Event fill:#ff6b6b,color:#fff,stroke-width:3px
    style Future fill:#a8e6cf
    style Past fill:#ffd3b6
    style Elsewhere fill:#e0e0e0

Three Regions:

1. Past Light Cone: All events that can affect “you now”

   • Must be in your past
   • Close enough that light can travel from there to here

2. Future Light Cone: All events that “you now” can affect

   • Must be in your future
   • Close enough that you can affect it with light-speed signals

3. Spacelike Separation: Events with no causal relationship to “you now”

   • Too far, light can’t reach in time
   • Cannot affect you, you cannot affect it

💡 Physics’ Definition of Causality: Event A can “cause” event B if and only if B is in A’s future light cone.

📏 Physical Limit of Causality: Speed of Light

Key Constraint: No information or influence can propagate faster than light!

This gives strict constraints on causality:

$$\text{If }A\text{ causes }B\Rightarrow (x_B-x_A{)}^2-c^2(t_B-t_A{)}^2\le 0$$

Translation:

• $x_B-x_A$ = Spatial distance
• $t_B-t_A$ = Time interval
• If $c(t_B-t_A)\ge |x_B-x_A|$ (enough time for light to travel), then causality is possible

Example: Sun Explosion

graph LR
    A["Sun Explodes<br/>t = 0"] -->|8 minutes| B["You See Explosion on Earth<br/>t = 8 minutes"]
    A -.cannot affect.-> C["Earth 1 Second Later<br/>t = 1 second"]

    style A fill:#ff6b6b,color:#fff
    style B fill:#a8e6cf
    style C fill:#e0e0e0

• Sun is 150 million km from Earth, light takes 8 minutes
• 1 second after sun explodes, you on Earth don’t know yet (light hasn’t arrived)
• After 8 minutes, you can be affected

During those 8 minutes, the sun explosion and you on Earth have no causal relationship (spacelike separation).

GLS Theory’s Causality: Partial Order = Entropy Monotonicity

GLS unified theory proposes a third understanding of causality, attempting to unify the above views:

Causal Relationship = Partial Order Relationship = Monotonicity of Entropy

📊 What is Partial Order?

“Partial order” is a mathematical concept describing “ordering.”

Example: Family Tree

graph TB
    A["Grandfather"] --> B["Father"]
    A --> C["Uncle"]
    B --> D["You"]
    B --> E["Brother"]
    C --> F["Cousin"]

    style D fill:#ffe66d,stroke:#f59f00,stroke-width:2px

Properties of Partial Order:

1. Reflexivity: You ≤ You (obvious, but mathematically needed)
2. Antisymmetry: If A ≤ B and B ≤ A, then A = B
3. Transitivity: If A ≤ B and B ≤ C, then A ≤ C

In the family tree:

• Grandfather < Father < You (transitivity)
• You and cousin cannot compare (this is “partial” order, not all elements can be compared)

🔗 Causality = Partial Order

GLS theory proposes: Causal relationships in spacetime might be mathematically equivalent to partial order relationships of events!

$$A\prec B\iff \text{A is in B’s causal past}$$

Symbol $\prec$ reads as “before.”

Properties:

1. Reflexive: $A\prec A$ (event can affect itself)
2. Antisymmetric: If $A\prec B$ and $B\prec A$, then $A=B$ (no closed causal loops)
3. Transitive: If $A\prec B$ and $B\prec C$, then $A\prec C$ (causality is transitive)

📈 Causality = Entropy Monotonicity

A core theoretical inference: Causal order might be equivalent to monotonicity of entropy!

$$A\prec B\iff S(A)\le S(B)$$

where $S$ is generalized entropy (will be detailed later).

💡 Key Insight: Saying “A is before B” is equivalent to saying “A’s entropy is not greater than B’s entropy”!

Why?

Because entropy always increases (or stays constant), so:

• If $S(A)<S(B)$, then A must be before B
• If $S(A)=S(B)$, A and B may be simultaneous, or in a reversible process
• If $S(A)>S(B)$, then A cannot be before B (violates second law of thermodynamics)

graph LR
    A["Event A<br/>Entropy = 100"] -->|time| B["Event B<br/>Entropy = 150"]
    B -->|time| C["Event C<br/>Entropy = 200"]

    A -.cannot return to.-> C

    style A fill:#a8e6cf
    style B fill:#ffd3b6
    style C fill:#ffaaa5

Small Causal Diamonds: Minimal Units of Causality

GLS theory introduces a core concept: Small Causal Diamonds (causal diamond) or Causal Rhombus.

💎 What is a Causal Diamond?

Imagine two events $p$ and $q$ in spacetime, where $q$ is in $p$’s future.

Causal diamond is:

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

Translation:

• $J^{+}(p)$ = Future of p (all events p can affect)
• $J^{-}(q)$ = Past of q (all events that can affect q)
• $D(p,q)$ = Intersection (all events affected by p and can affect q)

graph TB
    subgraph "Causal Diamond D(p,q)"
        q["Event q<br/>Future Vertex"]
        middle["Diamond Interior<br/>Affected by p, Can Affect q"]
        p["Event p<br/>Past Vertex"]
    end

    p -->|causal influence| middle
    middle -->|causal influence| q

    style p fill:#ffd3b6
    style q fill:#a8e6cf
    style middle fill:#ffe66d,stroke:#f59f00,stroke-width:2px

Why Called “Diamond”?

Drawn in two-dimensional spacetime, it looks like a diamond:

      q (future)
     /│\
    / │ \
   /  │  \
  /   │   \
 /____|____\
      │
      p (past)

🔬 Importance of Small Causal Diamonds

In GLS theory, small causal diamonds are fundamental building blocks of spacetime, like LEGO bricks:

1. Local Causality: Causal relationships within diamond are clear
2. Generalized Entropy: Entropy can be defined on diamond
3. Field Equations Emerge: Einstein equations can be derived from entropy extremum on small diamond

💡 Analogy: If spacetime is a building, small causal diamonds are the bricks. Understanding brick properties helps understand the whole building.

Triple Equivalence of Causality

One of GLS theory’s core propositions:

graph TD
    Causal["Causal Relationship"] --> Order["Partial Order Structure<br/>A ≺ B"]
    Causal --> Entropy["Entropy Monotonicity<br/>S(A) ≤ S(B)"]
    Causal --> Time["Time Function<br/>t(A) ≤ t(B)"]

    Order -.equivalent.-> Entropy
    Entropy -.equivalent.-> Time
    Time -.equivalent.-> Order

    style Causal fill:#ff6b6b,stroke:#c92a2a,stroke-width:3px,color:#fff

Three Formulations, One Essence:

1. Geometric Formulation: There exists a time function $t:M\to \mathbb{R}$ such that $A\prec B\iff t(A)\le t(B)$
2. Partial Order Formulation: Causal relationships satisfy reflexivity, antisymmetry, transitivity
3. Thermodynamic Formulation: Generalized entropy monotonically increases along causal direction

Why Are They Equivalent?

Because they all describe different aspects of the same spacetime structure:

• Time function = “Ordering” events
• Partial order = Mathematical language of “ordering”
• Entropy monotonicity = Physical content of “ordering”

Causality and Free Will

A philosophical question: If everything has causes, do we have free will?

🤖 Determinism vs Free Will

Determinism:

• Given current state, future is completely determined
• Like billiard table: knowing all ball positions and velocities, can predict future
• Classical physics is deterministic

Quantum Uncertainty:

• Quantum mechanics introduces true randomness
• Even knowing current state, future has multiple possibilities
• But this is just “random,” not “free choice”

GLS Theory’s Perspective:

Causality is not “force,” but “constraint”:

• Allowed: A before B, A can affect B
• Forbidden: A after B, A cannot affect B
• Free: Under causal constraints, system has multiple possible evolution paths

graph TD
    A["Now"] --> B1["Possible Future 1"]
    A --> B2["Possible Future 2"]
    A --> B3["Possible Future 3"]
    B1 --> C["Final Outcome"]
    B2 --> C
    B3 --> C

    A -.cannot reach.-> Past["Past"]

    style A fill:#ffe66d,stroke:#f59f00,stroke-width:2px
    style Past fill:#e0e0e0

💡 Analogy: Causality is like a road network. You can’t walk through walls (causal limits), but you can choose which road to take (free will).

Anti-Causality and Time Travel

⏰ Is Time Travel Possible?

If you could go back and kill your grandfather, you wouldn’t be born, so you couldn’t go back… This is the famous Grandfather Paradox.

GLS Theory’s Inference: Causal structure mathematically forbids closed timelike curves (CTCs).

In spacetimes satisfying stable causality, no closed causal loops exist:

$$A\prec B\prec C\prec \cdots \prec A\text{Impossible!}$$

⚠️ Why? What happens to entropy if causal loops exist?

If $A\prec B\prec A$, then:

$$S(A)\le S(B)\le S(A)\Rightarrow S(A)=S(B)$$

But this means entropy is completely constant, violating all physical processes except reversible ones.

The universe rejects time travel, not because of technical limitations, but because of the fundamental self-consistency of causal-entropy structure.

Summary: Multiple Faces of Causality

🎯 Key Points

1. Causality is not absolute: Light speed limits the range of causal influence
2. Causality has structure: Causal relationships satisfy mathematical properties of partial order
3. Causality equals entropy: Saying “A before B” equals “A’s entropy ≤ B’s entropy”
4. Small Causal Diamonds: Fundamental causal units of spacetime
5. Triple Equivalence of Causality: Geometry(time function) = Partial Order(≺) = Thermodynamics(entropy increase)

💡 Most Profound Insight

GLS theory proposes: Causality might not be a “mysterious force” between things, but a necessary consequence of spacetime geometry and entropy structure.

Just as “straight line” is not fundamental (it’s a geodesic), “causality” is not fundamental—it’s a manifestation of deeper partial order-entropy-time unified structure.

What’s Next

We’ve understood time and causality. Next question:

• What is boundary? Why say “boundary is reality”?
• Does the physical world really exist in “volume”? Or is everything encoded on the “surface”?
• What is the holographic principle?

Answers to these questions are in the next article:

Next: What is Boundary? →

Remember: Causality is not magic, but geometry. Understanding the partial order structure of causality, you’ve taken the second step in understanding the universe.

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