Core Insight 5: Entropy Modeled as Arrow

“GLS theory proposes: Why does time have direction? Possibly because entropy is increasing.”

🎯 Core Idea

In previous insights, we saw:

• Time is geometry
• Causality is partial order
• Boundary is reality
• Scattering is evolution

But there’s a deeper question: Why does time have direction?

Physical laws (Newton, Maxwell, Schrödinger, Einstein) are mostly time-reversal symmetric—play the tape backwards, they still hold.

So where does the “past→future” arrow come from?

GLS theory’s answer:

In GLS framework: Arrow of time ⟺ Arrow of entropy ⟺ Arrow of causality

And these three, under unified time scale, might be mathematically equivalent to the same object!

🏠 Starting from Room Chaos: What is Entropy?

Room “Entropy”

Imagine your room:

Tidy state:

• Books on shelves, alphabetically sorted
• Clothes in wardrobe, neatly organized
• Desk clean, stationery in place

Chaotic state:

• Books scattered on floor
• Clothes piled up
• Desk in complete disarray

graph LR
    O["Tidy<br/>Low Entropy<br/>S_low"] --> |"Spontaneous"| D["Chaotic<br/>High Entropy<br/>S_high"]
    D -.-> |"Requires Work"| O

    style O fill:#e1f5ff
    style D fill:#ffe1e1

Observation:

• Room spontaneously goes from tidy to chaotic (entropy increase)
• Restoring order from chaos requires work (you have to clean up)
• Time arrow aligns with direction of entropy increase

Boltzmann Formula: Definition of Entropy

Ludwig Boltzmann gave the precise definition of entropy in 1877:

$$\boxed{S=k_B\ln \Omega }$$

where:

• $S$: Entropy
• $k_B$: Boltzmann constant
• $\Omega$: Number of microstates

Physical meaning:

Entropy measures “how many microscopic arrangements correspond to the same macroscopic state.”

For example:

• Tidy room: ${\Omega }_{\text{tidy}}\approx 10^{10}$ (only a few tidy ways)
• Chaotic room: ${\Omega }_{\text{chaotic}}\approx 10^{100}$ (countless chaotic ways)

So:

$$S_{\text{chaotic}}=k_B\ln (10^{100})\gg S_{\text{tidy}}=k_B\ln (10^{10})$$

📈 Second Law of Thermodynamics: Entropy Always Increases

Statements of Second Law

Clausius statement (1850):

“Heat cannot spontaneously flow from cold object to hot object.”

Kelvin statement (1851):

“It is impossible to extract heat from a single heat source and completely convert it to useful work without other effects.”

Statistical mechanics statement (Boltzmann):

“Entropy of isolated system never decreases.”

$$\boxed{\Delta S\ge 0}\text{(Isolated System)}$$

GLS insight:

Under unified time scale framework, the second law might not be an independent “law,” but a necessary consequence of causal structure!

🌌 Generalized Entropy: Geometry + Quantum

Inspiration from Bekenstein-Hawking Entropy

We already saw in “Boundary is Reality” that black hole entropy is proportional to area:

$$S_{\text{BH}}=\frac{A}{4G\hslash }$$

This suggests: Entropy might not just be “number of microstates,” but has geometric meaning!

Definition of Generalized Entropy

In quantum gravity, generalized entropy contains two parts:

$$\boxed{S_{\text{gen}}(\Sigma )=\underset{\text{Geometric Entropy (Area)}}{\underbrace{\frac{A(\Sigma )}{4G\hslash }}}+\underset{\text{Quantum Field Entropy (von Neumann)}}{\underbrace{S_{\text{out}}(\Sigma )}}}$$

where:

• $\Sigma$: A spatial hypersurface (Cauchy slice)
• $A(\Sigma )$: Area of hypersurface boundary
• $S_{\text{out}}(\Sigma )$: von Neumann entropy of quantum fields outside boundary

graph TB
    subgraph "Spatial Hypersurface Σ"
        B["Boundary ∂Σ<br/>Area A"]
    end

    subgraph "Outside Region"
        O["Quantum Fields<br/>Entropy S_out"]
    end

    S["Generalized Entropy<br/>S_gen = A/4Gℏ + S_out"]

    B --> S
    O --> S

    style B fill:#e1f5ff
    style O fill:#ffe1e1
    style S fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px

Physical meaning:

1. Geometric part $A/(4G\hslash )$: From degrees of freedom of gravity/spacetime
2. Quantum part $S_{\text{out}}$: From degrees of freedom of matter fields
3. Total entropy: Sum of both is complete entropy

Key insight:

In GLS theory, extremum/monotonicity conditions of generalized entropy are derived to directly yield Einstein’s field equation!

🔗 Unification of Triple Arrows

Now we can reveal one of the core insights of GLS theory:

Theoretical Inference 2 (Equivalent Characterizations of Causal Partial Order)

In GLS framework, for any two events $p,q$, the following are mathematically equivalent:

1. Geometric Causality: $q\in J^{+}(p)$
2. Time Monotonicity: $\tau (p)\le \tau (q)$
3. Entropy Monotonicity: $S_{\text{gen}}(p)\le S_{\text{gen}}(q)$

$$\boxed{p\prec q⟺\tau (p)\le \tau (q)⟺S_{\text{gen}}(p)\le S_{\text{gen}}(q)}$$

This means:

• Causal arrow (past→future)
• Time arrow (clock advances)
• Entropy arrow (chaos increases)

Might be three manifestations of the same arrow!

graph TB
    U["Unified Arrow<br/>Directionality of Universe"] --> C["Causal Arrow<br/>p ≺ q"]
    U --> T["Time Arrow<br/>τ(p) < τ(q)"]
    U --> S["Entropy Arrow<br/>S(p) ≤ S(q)"]

    C <--> |"Equivalent"| T
    T <--> |"Equivalent"| S
    S <--> |"Equivalent"| C

    style U fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style C fill:#e1f5ff
    style T fill:#e1ffe1
    style S fill:#ffe1e1

🎓 IGVP: From Entropy to Einstein Equation

This is one of the most amazing achievements of GLS theory: Attempting to derive gravitational field equation using variational principle of entropy!

Information Geometric Variational Principle (IGVP)

Core idea:

On small causal diamond ${\mathcal{D}}_{\ell }(p)$ near each spacetime point $p$, require:

(First-order condition): Under fixed volume constraint $\delta V=0$, generalized entropy takes extremum:

$$\delta S_{\text{gen}}=0$$

(Second-order condition): Relative entropy non-negative:

$${\delta }^2{S}_{\text{rel}}\ge 0$$

Steps to Derive Einstein Equation

Step 1: Calculate variation of generalized entropy:

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

Step 2: Use first law of external entropy (from modular theory):

$$\delta S_{\text{out}}=\frac{\delta Q}{T}=\frac{2\pi }{\hslash }{\int }_{\mathcal{H}}\lambda T_{kk}d\lambda dA$$

where $T_{kk}=T_{ab}{k}^a{k}^b$ is stress-energy tensor along null direction.

Step 3: Using Raychaudhuri equation, area variation relates to curvature:

$$\delta A\sim -\int R_{kk}(\text{light ray measure})$$

Step 4: Setting $\delta S_{\text{gen}}=0$, we get:

$$\frac{\delta A}{4G\hslash }+\frac{2\pi }{\hslash }\int \lambda T_{kk}d\lambda dA=0$$

Step 5: Through Radon-type closure (converting integral condition to pointwise condition), we get:

$$\boxed{R_{kk}=8\pi G{T}_{kk}}$$

Step 6: Holding for all null directions $k^a$, upgrading to tensor equation:

$$\boxed{G_{ab}+\Lambda g_{ab}=8\pi G{T}_{ab}}$$

This formally yields Einstein’s field equation!

graph TB
    I["IGVP<br/>δS_gen = 0"] --> |"First-Order Condition"| A["Area Variation<br/>δA/4Gℏ"]
    I --> |"First-Order Condition"| Q["External Entropy<br/>δS_out = δQ/T"]

    A --> |"Raychaudhuri"| R["Curvature Term<br/>R_kk"]
    Q --> |"Modular Theory"| T["Matter Term<br/>T_kk"]

    R --> E["Einstein Equation<br/>G_ab + Λg_ab = 8πGT_ab"]
    T --> E

    style I fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style E fill:#e1ffe1

Profound meaning:

GLS theory argues: Gravitational field equation might not be a fundamental axiom, but a necessary consequence of entropy extremum condition!

🌊 QNEC: Quantum Null Energy Condition

Dilemma of Classical Energy Conditions

In classical general relativity, we often assume energy conditions, for example:

Null Energy Condition (NEC):

$$T_{ab}{k}^a{k}^b\ge 0(\text{for all null vectors }{k}^a)$$

But in quantum field theory, this condition can be violated! (e.g., Casimir effect)

Quantum Null Energy Condition (QNEC)

GLS theory (and related work) discovered a deeper condition:

QNEC (Quantum Null Energy Condition):

$$\boxed{⟨T_{kk}(x){⟩}_{\psi }\ge \frac{\hslash }{2\pi }\frac{d^2{S}_{\text{out}}}{d{\lambda }^2}(x)}$$

where:

• $⟨T_{kk}{⟩}_{\psi }$: Quantum expectation value of null stress-energy tensor
• $d^2{S}_{\text{out}}/d{\lambda }^2$: Second derivative of entropy along null geodesic
• $\lambda$: Affine parameter of null geodesic

Physical meaning:

Lower bound of energy density might be determined by rate of entropy change!

This condition:

1. Recovers NEC in classical limit
2. Allows local negative energy in quantum case
3. Has been rigorously proven in many CFTs
4. Is manifestation of IGVP second-order condition

graph LR
    C["Classical NEC<br/>T_kk ≥ 0"] --> |"Quantum Correction"| Q["Quantum QNEC<br/>T_kk ≥ ℏ/(2π) d²S/dλ²"]
    Q --> |"Entropy Monotonicity"| S["Generalized Entropy Monotonicity<br/>∂_s S_gen ≥ 0"]

    style C fill:#e1f5ff
    style Q fill:#fff4e1
    style S fill:#ffe1e1

🔄 Entropy Increase = Information Increase?

Shannon Information Entropy

In information theory, Claude Shannon defined information entropy:

$$S=-\sum _i{p}_i\ln p_i$$

where $p_i$ is probability of event $i$.

Physical meaning:

• Entropy = Uncertainty = Missing information
• Larger entropy, less we know about system

Information Never Lost?

Quantum mechanical unitary evolution guarantees information conservation (von Neumann entropy unchanged).

But second law of thermodynamics says entropy increases (information lost?).

Contradiction?

GLS explanation:

1. Microscopic level: Unitary evolution, information conserved
2. Macroscopic level: After coarse-graining, accessible information decreases
3. Generalized entropy: Contains geometry + quantum, total entropy monotonically increases
4. Boundary perspective: Information might not be lost, just transferred to boundary!

Inspiration from black hole information paradox:

• Information falling into black hole encoded on horizon area
• Hawking radiation carries information
• Generalized entropy $S_{\text{gen}}=A/(4G\hslash )+S_{\text{rad}}$ monotonically increases

🔗 Connections to Other Core Ideas

• Time is Geometry: Time scale $\tau$ is equivalent to entropy monotonicity
• Causality is Partial Order: Partial order relation $p\prec q\iff S(p)\le S(q)$
• Boundary is Reality: Geometric part of entropy $A/(4G\hslash )$ is boundary area
• Scattering is Evolution: Scattering process satisfies unitarity (entropy reversible), entropy increases after coarse-graining

🎓 Further Reading

To understand more technical details, you can read:

• Theory document: igvp-einstein-complete.md
• Causality and entropy: unified-theory-causal-structure-time-scale-partial-order-generalized-entropy.md
• Previous: 04-scattering-is-evolution_en.md - Scattering is Evolution
• Next: 06-unity-of-five_en.md - Five into One (Detailed)
• Summary: 07-core-summary_en.md - Core Ideas Summary

🤔 Questions for Reflection

1. Why can an egg spontaneously break, but broken eggs don’t spontaneously restore?
2. In Boltzmann formula $S=k_B\ln \Omega$, why use logarithm instead of directly using $\Omega$?
3. If microscopic laws are time-reversible, why is macroscopic irreversible? What role does coarse-graining play?
4. In the two parts of generalized entropy (geometry + quantum), which dominates under what circumstances?
5. QNEC allows local negative energy, does this contradict the intuition of “energy non-negative”?
6. What is the core problem of black hole information paradox? How does GLS theory solve it?

📝 Key Formulas Review

$$\boxed{S=k_B\ln \Omega }\text{(Boltzmann Formula)}$$

$$\boxed{S_{\text{gen}}(\Sigma )=\frac{A(\Sigma )}{4G\hslash }+S_{\text{out}}(\Sigma )}\text{(Generalized Entropy)}$$

$$\boxed{p\prec q\iff \tau (p)\le \tau (q)\iff S_{\text{gen}}(p)\le S_{\text{gen}}(q)}\text{(Triple Equivalence)}$$

$$\boxed{\delta S_{\text{gen}}=0\Rightarrow G_{ab}+\Lambda g_{ab}=8\pi G{T}_{ab}}\text{(IGVP)}$$

$$\boxed{⟨T_{kk}⟩\ge \frac{\hslash }{2\pi }\frac{d^2{S}_{\text{out}}}{d{\lambda }^2}}\text{(QNEC)}$$

Next Step: We have understood five core insights. Next, in “Five into One,” we will see how they perfectly combine into one through the Unified Time Scale Identity!