02. Detailed Explanation of Three Components: Events, Geometry, Measure

Introduction: Foundation Tripod of Universe

In tenfold structure, first three components constitute most fundamental “foundation” of universe:

1. Event and Causality Layer $U_{\text{evt}}$: Defines “what happens” and “who influences whom”
2. Geometry and Spacetime Layer $U_{\text{geo}}$: Defines “where it happens” and “distance/angle”
3. Measure and Probability Layer $U_{\text{meas}}$: Defines “how probable” and “how to integrate”

Relationship among these three is similar to:

• Script (Event Causality): Defines plot development order
• Stage (Geometric Spacetime): Provides physical space for performance
• Lighting (Measure Probability): Determines “weight” audience sees each scene

Without compatibility and alignment of these three, universe cannot be consistently defined.

Part I: Event and Causality Layer $U_{\text{evt}}$

1.1 Intuitive Picture: Domino Network

Imagine huge domino network:

• Each domino = an event
• Path of dominoes falling = causal chain
• Cannot fall backwards = causality irreversible
• Can branch = one cause produces multiple effects
• Can converge = multiple causes jointly lead to one effect

Global structure of this network is $U_{\text{evt}}$.

graph TD
    A["Event x1<br/>(Cause)"] --> B["Event x2<br/>(Direct Effect)"]
    A --> C["Event x3<br/>(Direct Effect)"]
    B --> D["Event x4<br/>(Common Effect)"]
    C --> D
    D --> E["Event x5<br/>(Final Effect)"]

    style A fill:#ff9999
    style E fill:#99ff99
    style D fill:#ffcc99

1.2 Strict Mathematical Definition

Definition 1.1 (Event Causality Layer): $$U_{\text{evt}}=(X,⪯,\mathcal{C})$$

where:

(1) Event Set $X$:

• Each element $x\in X$ represents an indivisible event
• Can be: particle collision, observation behavior, information transmission
• Does not contain “continuous processes” (those decomposed into multiple events)

(2) Causal Partial Order $⪯$:

• $x⪯y$: Read as “$x$ may causally influence $y$”
• Reflexivity: $x⪯x$ (event can influence itself)
• Transitivity: $x⪯y\wedge y⪯z\Rightarrow x⪯z$
• Antisymmetry: $x⪯y\wedge y⪯x\Rightarrow x=y$ (no causal closed loops)

Key Constraint: $$x\prec y\Rightarrow \exists \text{ causal path }x\to \cdots \to y$$

(3) Family of Causal Fragments $\mathcal{C}$: $$\mathcal{C}=\{C_{\alpha }\subseteq X\mid C_{\alpha }\text{ is causally closed}\}$$

Each $C_{\alpha }$ satisfies:

• Downward closed: $x\in C_{\alpha }\wedge y⪯x\Rightarrow y\in C_{\alpha }$
• Finitely generated: Exists finite set $F$ such that $C_{\alpha }={\bigcup }_{x\in F}\text{past}(x)$

Physical Meaning: $C_{\alpha }$ represents “all events observer $\alpha$ can know so far”.

1.3 Core Properties and Physical Interpretation

Property 1.1 (Global Causal Consistency):

Exists causal time function $T_{\text{cau}}:X\to \mathbb{R}$ such that: $$x\prec y\Rightarrow T_{\text{cau}}(x)<T_{\text{cau}}(y)$$

Physical Meaning: Entire universe has global “plot development order”, no time paradoxes like “grandson kills grandfather”.

Property 1.2 (Causal Diamond Boundedness):

For any $x,y\in X$, causal diamond: $$◊(x,y):=\{z\in X\mid x⪯z⪯y\}$$ is either empty set, or finite set or compact set.

Physical Meaning: “Intermediate events” between any two events are not infinitely many, information transmission is discrete or local.

Property 1.3 (Existence of Lightlike Hypersurfaces):

Exists Cauchy hypersurface family $\{{\Sigma }_t{\}}_{t\in \mathbb{R}}$ such that: $$\forall x\in X,\exists !t:x\in {\Sigma }_t$$ and: $$t_1<t_2\Rightarrow \forall x\in {\Sigma }_{t_1},y\in {\Sigma }_{t_2}:x⪯y$$

Physical Meaning: Can reconstruct entire causal structure using “layers of time slices”, similar to frame-by-frame playback of animation.

1.4 Examples and Counterexamples

Example 1 (Causal Structure of Minkowski Spacetime):

In special relativity: $$X={\mathbb{R}}^4,x⪯y\iff (y-x)\in \overline{V^{+}}$$ where $\overline{V^{+}}$ is closed future light cone: $$\overline{V^{+}}=\{(t,\mathbf{x})\mid t\ge 0,t^2\ge |\mathbf{x}{|}^2\}$$

Causal fragment $C$ corresponds to “past horizon of an observer”: $$C=\{x\in {\mathbb{R}}^4\mid x⪯x_{\text{obs}}\}$$

Counterexample 1 (Gödel Spacetime):

In rotating universe model constructed by Gödel in 1949, exists closed timelike curves (CTC): $$\exists x_1,\dots ,x_n:x_1\prec x_2\prec \cdots \prec x_n\prec x_1$$

This violates Property 1.1, therefore does not satisfy definition of $U_{\text{evt}}$. GLS theory excludes such pathological spacetimes.

Counterexample 2 (Quantum Causal Uncertainty):

Some quantum gravity models allow “superposition of causal orders”: $$|\psi ⟩=\frac{1}{\sqrt{2}}(|x\prec y⟩+|y\prec x⟩)$$

This also does not satisfy antisymmetry of partial order. But can be compatible through probability measure on causal fragments, see Part III.

1.5 Analogy Summary: City Traffic Network

Imagine $U_{\text{evt}}$ as one-way traffic network of a city:

• Intersections = events
• One-way streets = causal relations (can only drive from $x$ to $y$, cannot reverse)
• No circular one-way streets = no causal closed loops
• Reachable areas = causal fragments (all intersections reachable from some intersection)

Topological structure of this network determines “how information flows in universe”.

Part II: Geometry and Spacetime Layer $U_{\text{geo}}$

2.1 Intuitive Picture: Light Cones on Rubber Membrane

Imagine a stretched rubber membrane:

• Shape of membrane = spacetime geometry
• Light cones on membrane = causal structure
• Curvature of membrane = gravitational effects
• Tilting of light cones = distribution of matter-energy

Core requirement of $U_{\text{geo}}$: Geometry and causality must align—orientation of light cones must match causal partial order.

graph TD
    A["Manifold M<br/>(Rubber Membrane)"] --> B["Lorentz Metric g<br/>(Distance/Angle)"]
    B --> C["Light Cone Structure<br/>(Causal Cone)"]
    C --> D["Alignment Map Φ_evt<br/>(Membrane↔Domino Net)"]
    D --> E["Causal Consistency Check<br/>(Light Cone=Causality)"]

    style A fill:#e6f3ff
    style C fill:#fff4e6
    style E fill:#e6ffe6

2.2 Strict Mathematical Definition

Definition 2.1 (Geometry and Spacetime Layer): $$U_{\text{geo}}=(M,g,{\Phi }_{\text{evt}},{\Phi }_{\text{cau}})$$

where:

(1) Spacetime Manifold $M$:

• Four-dimensional smooth manifold (usually assume $M\cong {\mathbb{R}}^4$ topologically)
• Orientable, Hausdorff, paracompact
• Each point $p\in M$ represents a spacetime coordinate

(2) Lorentz Metric $g$: $$g:TM\times TM\to \mathbb{R}$$ satisfying:

• Signature $(-,+,+,+)$: one time direction, three space directions
• Non-degenerate: For any $v\ne 0$, exists $w$ such that $g(v,w)\ne 0$
• Smooth dependence on $p\in M$

Lightlike vector: $v\in T_{p}M$ satisfying $g(v,v)=0$ and $v\ne 0$

(3) Event Embedding Map ${\Phi }_{\text{evt}}$: $${\Phi }_{\text{evt}}:X\to M$$ satisfying:

• Injective: Different events correspond to different spacetime points
• Locality: ${\Phi }_{\text{evt}}(X)$ dense or full coverage in $M$

(4) Causal Alignment Map ${\Phi }_{\text{cau}}$: $${\Phi }_{\text{cau}}:(X,⪯)\to (M,{⪯}_g)$$ where ${⪯}_g$ is metric causality: $$p{⪯}_{g}q\iff \exists \text{ non-spacelike curve }\gamma :p\to q$$

Core Constraint: $$\boxed{x⪯y\iff {\Phi }_{\text{evt}}(x){⪯}_g{\Phi }_{\text{evt}}(y)}$$

Physical Meaning: Causal partial order (dominoes) and light cone structure (rubber membrane) completely consistent.

2.3 Core Properties and Physical Interpretation

Property 2.1 (Solution of Einstein Equation):

Metric $g$ must satisfy (in classical approximation): $$G_{ab}+\Lambda g_{ab}=8\pi G⟨T_{ab}⟩$$ where:

• $G_{ab}=R_{ab}-\frac{1}{2}R{g}_{ab}$: Einstein tensor
• $\Lambda$: Cosmological constant
• $⟨T_{ab}⟩$: Expectation value of energy-momentum tensor (quantum corrections)

Physical Meaning: Spacetime geometry determined by matter-energy distribution—“matter tells spacetime how to curve”.

Property 2.2 (Global Hyperbolicity):

Exists Cauchy hypersurface $\Sigma \subset M$ such that: $$\forall p\in M,\exists !\text{ timelike curve }\gamma \text{ passing through }\Sigma \text{ reaching }p$$

Physical Meaning: Can uniquely determine entire universe evolution from “initial data at one moment” (determinism).

Property 2.3 (Time Orientation):

Exists continuous timelike vector field $T^a$ such that: $$\forall p\in M:g(T,T)<0$$

Physical Meaning: Globally defines “time forward” direction, excludes regions with “time arrow reversal”.

2.4 Examples and Non-Trivial Structures

Example 2 (Schwarzschild Black Hole Spacetime):

Metric: $$d{s}^2=-\left(1-\frac{2GM}{r}\right)d{t}^2+{\left(1-\frac{2GM}{r}\right)}^{-1}d{r}^2+r^{2}d{\Omega }^2$$

Key features:

• Horizon $r=2GM$: Light cones “completely tilted”, collapsing inward
• Singularity $r=0$: Curvature diverges, theory breaks down

Causal structure: $$\begin{gathered} r>2GM:\text{timelike observers can escape} \\ r<2GM:\text{all future paths point to singularity} \end{gathered}$$

Example 3 (FLRW Expanding Universe):

Metric: $$d{s}^2=-d{t}^2+a(t{)}^2\left(\frac{d{r}^2}{1-k{r}^2}+r^{2}d{\Omega }^2\right)$$

where $a(t)$ is scale factor, satisfying Friedmann equation: $${\left(\frac{\dot{a}}{a}\right)}^2=\frac{8\pi G}{3}\rho -\frac{k}{a^2}+\frac{\Lambda }{3}$$

Causal structure features:

• Particle horizon: ${\chi }_p={\int }_0^t\frac{d{t}^{\prime }}{a(t^{\prime })}$ (finite means horizon exists)
• Event horizon: ${\chi }_e={\int }_t^{\infty }\frac{d{t}^{\prime }}{a(t^{\prime })}$ (finite means accelerated expansion)

2.5 Analogy Summary: City 3D Map

Imagine $U_{\text{geo}}$ as 3D map of a city:

• Map surface = spacetime manifold $M$
• Contour lines = time slices
• Slope = gravitational potential
• Traffic flow direction = light cone direction
• Forbidden zones = horizons or singularities

“Traffic flow direction” on map must completely match “one-way network” of Part I.

Part III: Measure and Probability Layer $U_{\text{meas}}$

3.1 Intuitive Picture: Spotlight on Stage

Imagine a stage play:

• Stage = spacetime manifold $M$
• Script = causal structure $⪯$
• Spotlight = probability measure $\mu$

Spotlight determines weight audience “sees” each scene:

• Bright regions = high probability events
• Shadow regions = low probability events
• Complete darkness = zero measure sets

But movement of spotlight must obey:

• Continuity: Cannot suddenly jump
• Normalization: Total brightness conserved
• Causal compatibility: Cannot illuminate “causally unreachable” regions

graph TD
    A["Probability Space (Ω, F, P)"] --> B["Measure μ on Spacetime M"]
    B --> C["Normalized on Cauchy Surface Σ"]
    C --> D["Induced by Quantum State ρ"]
    D --> E["Path Integral Weight"]
    E --> F["Compatible with Causal Structure"]

    style A fill:#ffe6f0
    style C fill:#e6f0ff
    style F fill:#f0ffe6

3.2 Strict Mathematical Definition

Definition 3.1 (Measure and Probability Layer): $$U_{\text{meas}}=(\Omega ,\mathcal{F},\mathbb{P},{\mu }_M,\{{\rho }_{\Sigma }{\}}_{\Sigma })$$

where:

(1) Probability Space $(\Omega ,\mathcal{F},\mathbb{P})$:

• $\Omega$: Sample space (all possible “universe histories”)
• $\mathcal{F}$: $\sigma$-algebra (set of observable events)
• $\mathbb{P}:\mathcal{F}\to [0,1]$: Probability measure

Satisfying Kolmogorov axioms: $$\mathbb{P}(\Omega )=1,\mathbb{P}\left(\bigcup _{i=1}^{\infty }{A}_i\right)=\sum _{i=1}^{\infty }\mathbb{P}(A_i)(\text{disjoint})$$

(2) Measure on Spacetime ${\mu }_M$:

Borel measure on $M$, satisfying: $${\mu }_M(M)<\infty \text{or}{\mu }_M\text{ is }\sigma \text{-finite}$$

Relation to metric: $$d{\mu }_M=\sqrt{-g}{d}^{4}x$$ where $g=\det (g_{ab})$ is metric determinant.

Physical Meaning: ${\mu }_M$ defines “volume element”—weight for integrating physical quantities in spacetime.

(3) Family of Quantum States on Cauchy Surfaces $\{{\rho }_{\Sigma }{\}}_{\Sigma }$:

For each Cauchy hypersurface $\Sigma$, define density matrix: $${\rho }_{\Sigma }:{\mathcal{H}}_{\Sigma }\otimes {\mathcal{H}}_{\Sigma }^{†}\to \mathbb{C}$$ satisfying:

• Hermiticity: ${\rho }_{\Sigma }^{†}={\rho }_{\Sigma }$
• Positive semidefiniteness: $⟨\psi |{\rho }_{\Sigma }|\psi ⟩\ge 0$
• Normalization: $\text{tr}({\rho }_{\Sigma })=1$

Physical Meaning: ${\rho }_{\Sigma }$ completely encodes “quantum state at moment $\Sigma$”, including entanglement and mixed states.

(4) Compatibility Condition:

$$\boxed{{\int }_{\Sigma }\text{tr}({\rho }_{\Sigma })d\sigma =1}$$

where $d\sigma$ is induced volume element of Cauchy surface: $$d\sigma =\sqrt{h}{d}^{3}x,h=\det (h_{ij})$$ $h_{ij}$ is induced metric on $\Sigma$.

Time evolution compatibility: $${\rho }_{{\Sigma }_2}=\mathcal{U}({\Sigma }_1\to {\Sigma }_2){\rho }_{{\Sigma }_1}{\mathcal{U}}^{†}({\Sigma }_1\to {\Sigma }_2)$$ where $\mathcal{U}$ is unitary evolution operator.

3.3 Core Properties and Physical Interpretation

Property 3.1 (Born Rule):

Probability of observing event $A\in \mathcal{F}$: $$\mathbb{P}(A|\Sigma )=\text{tr}({\rho }_{\Sigma }{\hat{P}}_A)$$ where ${\hat{P}}_A$ is projection operator.

Physical Meaning: Probability of quantum measurement determined by density matrix and observation operator—fundamental postulate of quantum mechanics.

Property 3.2 (Path Integral Representation):

Evolution amplitude from ${\Sigma }_1$ to ${\Sigma }_2$: $$\mathcal{U}({\Sigma }_1\to {\Sigma }_2)={\int }_{\varphi ({\Sigma }_1)\to \varphi ({\Sigma }_2)}\mathcal{D}\varphi e^{iS[\varphi ]/\hslash }$$ where:

• $\varphi$: Field configuration
• $S[\varphi ]$: Action
• $\mathcal{D}\varphi$: Path integral measure (needs regularization)

Physical Meaning: Quantum state evolves through “superposition of all possible paths”, each path weighted by $e^{iS/\hslash }$.

Property 3.3 (Entanglement Entropy and Geometry):

For subregion $A$ of Cauchy surface $\Sigma$, entanglement entropy: $$S_{\text{ent}}(A)=-\text{tr}({\rho }_A\log {\rho }_A)$$ where ${\rho }_A={\text{tr}}_{\bar{A}}({\rho }_{\Sigma })$ is reduced density matrix.

Ryu-Takayanagi Formula (result in AdS/CFT): $$S_{\text{ent}}(A)=\frac{\text{Area}({\gamma }_A)}{4G\hslash }$$ where ${\gamma }_A$ is minimal surface of $A$ in bulk.

Physical Meaning: Entanglement entropy directly relates to spacetime geometry—“geometry is measure of entanglement”.

3.4 Examples: Vacuum States of Quantum Field Theory

Example 4 (Minkowski Vacuum):

In flat spacetime, vacuum state of scalar field: $${\rho }_{\text{Mink}}=|0⟩⟨0|$$ where $|0⟩$ is Poincaré-invariant vacuum.

Key properties:

• Pure state: ${\rho }^2=\rho$
• Translation invariant: ${\hat{P}}^{\mu }\rho {\hat{P}}^{\mu †}=\rho$
• Zero entanglement entropy: $S_{\text{ent}}(A)=0$ (for spatial region $A$)

Example 5 (Unruh Temperature and Rindler Horizon):

Minkowski vacuum as seen by accelerated observer (acceleration $a$) is thermal state: $${\rho }_{\text{Rindler}}=\frac{1}{Z}{e}^{-\beta {\hat{H}}_{\text{Rindler}}}$$ where: $$\beta =\frac{2\pi }{a},T_{\text{Unruh}}=\frac{a}{2\pi k_B}\approx 4\times 10^{-23}\text{K}\cdot \left(\frac{a}{1{\text{m/s}}^2}\right)$$

Physical Meaning: Vacuum state depends on observer’s motion state—“relativistic effect” in quantum field theory.

3.5 Compatibility with Causal Structure

Constraint 3.1 (Causal Measure Support):

For causal fragment $C\in \mathcal{C}$, define support set: $$\text{supp}({\mu }_C)=\{p\in M\mid \forall U\ni p:{\mu }_C(U)>0\}$$

Requirement: $$\text{supp}({\mu }_C)\subseteq {\Phi }_{\text{evt}}(C)$$

Physical Meaning: Support of probability measure must be within causally reachable region—cannot assign non-zero probability to causally unreachable events.

Constraint 3.2 (Quantum Causal Order):

For events $x,y\in X$: $$x⪯y\Rightarrow [{\hat{O}}_x(t_x),{\hat{O}}_y(t_y)]=0\text{when}{t}_x<t_y$$

where ${\hat{O}}_x,{\hat{O}}_y$ are corresponding observation operators.

Physical Meaning: Causally unrelated observation operators commute—microcausality of quantum field theory.

3.6 Analogy Summary: City Heat Map

Imagine $U_{\text{meas}}$ as real-time traffic heat map of a city:

• Color depth = probability density
• Heat regions = high probability event concentration areas
• Cold regions = low probability or zero measure sets
• Heat flow = quantum state evolution

Heat map must satisfy:

• Total “heat” conserved (normalization)
• Heat can only propagate along “one-way streets” (causal compatibility)
• Heat cannot appear in “traffic control zones” (support constraint)

Part IV: Deep Unification of the Three

4.1 Unified Time Scale: Red Thread Penetrating Three Layers

In three components, each has definition of “time”:

(1) Causal Time $T_{\text{cau}}$: $$x\prec y\Rightarrow T_{\text{cau}}(x)<T_{\text{cau}}(y)$$

(2) Geometric Proper Time ${\tau }_{\text{geo}}$: $${\tau }_{\text{geo}}={\int }_{\gamma }\sqrt{-g_{ab}{\dot{x}}^a{\dot{x}}^b}d\lambda$$ integral along timelike curve $\gamma$.

(3) Measure Time $t_{\text{meas}}$:

“Foliation time” defined through Cauchy surface family $\{{\Sigma }_t\}$: $$t_{\text{meas}}:M\to \mathbb{R},p\in {\Sigma }_t\Rightarrow t_{\text{meas}}(p)=t$$

Core Proposition (Time Scale Unification):

Three time definitions are affinely equivalent: $$\boxed{T_{\text{cau}}\sim {\tau }_{\text{geo}}\sim t_{\text{meas}}}$$

i.e., exist affine transformations: $${\tau }_{\text{geo}}=\alpha T_{\text{cau}}+\beta ,t_{\text{meas}}={\alpha }^{\prime }{T}_{\text{cau}}+{\beta }^{\prime }$$

Physical Meaning: Universe has unique time flow direction, only “unit conversion” differs in different perspectives.

graph TD
    A["Causal Time T_cau"] --> D["Affine Equivalence Class [τ]"]
    B["Geometric Proper Time τ_geo"] --> D
    C["Measure Foliation Time t_meas"] --> D
    D --> E["Unified Time Scale"]

    style D fill:#ffcccc
    style E fill:#ccffcc

4.2 Compatibility Conditions: Triangle Identities

Condition 4.1 (Causal-Geometric Alignment): $$x⪯y\iff {\Phi }_{\text{evt}}(x){⪯}_g{\Phi }_{\text{evt}}(y)$$

Condition 4.2 (Geometric-Measure Alignment): $${\int }_{M}f(p)d{\mu }_M={\int }_{M}f(p)\sqrt{-g}{d}^{4}x$$ for all integrable functions $f$.

Condition 4.3 (Measure-Causal Alignment): $$\text{supp}({\mu }_C)\subseteq {\Phi }_{\text{evt}}(C),\forall C\in \mathcal{C}$$

These three conditions form closed triangle constraint:

graph LR
    A["U_evt<br/>(Causality)"] -- "Φ_evt, Φ_cau" --> B["U_geo<br/>(Geometry)"]
    B -- "√-g Induced Measure" --> C["U_meas<br/>(Measure)"]
    C -- "Support⊆Causal Fragment" --> A

    style A fill:#ffe6e6
    style B fill:#e6f0ff
    style C fill:#f0ffe6

Lemma 4.1 (Necessary and Sufficient Condition for Triangle Closure):

Three components compatible $\iff$ exists global Cauchy hypersurface family $\{{\Sigma }_t\}$ such that: $$\forall t:{\rho }_{{\Sigma }_t}\text{ uniquely determined by }g{|}_{{\Sigma }_t}\text{ and }{T}_{\text{cau}}{|}_{{\Sigma }_t}$$

Physical Meaning: Quantum state, spacetime geometry, causal structure are trinity, cannot be independently specified.

4.3 Core Theorem: Uniqueness of Foundation Triplet

Theorem 4.1 (Moduli Space of Foundation Triplet):

Fix topology $M\cong {\mathbb{R}}^4$ and global causal structure type (e.g., “globally hyperbolic”), then triplets satisfying all compatibility conditions: $$(U_{\text{evt}},U_{\text{geo}},U_{\text{meas}})$$ constitute a finite-dimensional moduli space ${\mathcal{M}}_{\text{base}}$.

Dimension Estimate: $$\dim {\mathcal{M}}_{\text{base}}\le \dim {\mathcal{M}}_{\text{geo}}+\dim {\mathcal{M}}_{\text{QFT}}$$ where:

• $\dim {\mathcal{M}}_{\text{geo}}\sim \infty$ (degrees of freedom of metric)
• $\dim {\mathcal{M}}_{\text{QFT}}=\text{finite}$ (boundary data of fields)

But causal constraints and IGVP (see Component 7) greatly compress degrees of freedom.

Corollary 4.1 (No Free Lunch Principle):

Cannot simultaneously arbitrarily specify:

1. Arbitrary causal structure
2. Arbitrary spacetime geometry
3. Arbitrary quantum state

At most specify two of the three, third uniquely determined by compatibility conditions.

4.4 Practical Calculation Example

Problem: Given flat causal structure (Minkowski) and scalar field vacuum state, calculate induced spacetime metric.

Solution:

(1) Causal Structure: $$U_{\text{evt}}=({\mathbb{R}}^4,{⪯}_{\text{Mink}})$$ where ${⪯}_{\text{Mink}}$ is standard light cone causality.

(2) Quantum State: $${\rho }_{{\Sigma }_0}=|0⟩⟨0|\text{(Poincareˊ-invariant vacuum)}$$

(3) Reverse Derive Metric:

Since vacuum state satisfies: $$⟨0|T_{ab}|0⟩=0$$

Einstein equation gives: $$G_{ab}+\Lambda g_{ab}=0$$

When $\Lambda =0$, solution uniquely determined as: $$g={\eta }_{ab}=\text{diag}(-1,1,1,1)$$

i.e., Minkowski metric.

Conclusion: Flat causality + translation-invariant vacuum $\Rightarrow$ flat spacetime (self-consistent).

Part V: Physical Picture and Philosophical Meaning

5.1 “Trinity” Foundation of Universe

Traditional physics treats spacetime, causality, quantum state as three independent levels:

• General relativity handles spacetime geometry
• Quantum field theory handles quantum state evolution
• Causal structure as “background constraint”

But GLS theory reveals: the three are three perspectives of same reality:

Core Insight: These three perspectives locked into one whole through compatibility conditions—changing any one, other two must adjust accordingly.

5.2 Emergence of Classical Limit

In limit $\hslash \to 0$ and $G\to 0$:

(1) Causal Structure Degenerates to Determinism: $$U_{\text{evt}}\to \text{unique time flow}(\text{no quantum superposition})$$

(2) Spacetime Geometry Degenerates to Newtonian Absolute Spacetime: $$U_{\text{geo}}\to ({\mathbb{R}}^3\times \mathbb{R},d{t}^2-d{x}^2)(\text{spacetime separated})$$

(3) Measure Degenerates to Classical Probability: $$U_{\text{meas}}\to \delta (\varphi -{\varphi }_{\text{classical}})(\text{state vector collapses to point})$$

Physical Meaning: Classical physics is special simplified case of quantum gravity theory, not fundamental level.

5.3 Role of Observer

Note: In first three components, no explicit mention of observer. But:

• Causal fragments $C\in \mathcal{C}$ implicitly contain “events some perspective can know”
• Quantum states on Cauchy surfaces ${\rho }_{\Sigma }$ implicitly contain “measurement configuration at some moment”

This foreshadows subsequent introduction of $U_{\text{obs}}$ (Observer Network Layer): $$U_{\text{obs}}=\{({\mathcal{O}}_{\alpha },C_{\alpha },{\rho }_{\alpha }){\}}_{\alpha \in \mathcal{A}}$$

Each observer ${\mathcal{O}}_{\alpha }$ possesses:

• Causal fragment $C_{\alpha }$: Events they can know
• Reduced state ${\rho }_{\alpha }={\text{tr}}_{\bar{\alpha }}({\rho }_{\text{global}})$: Quantum state they see

Key Question: How do $(C_{\alpha },{\rho }_{\alpha })$ of different observers reach consensus? This requires strict definition of Component 8 $U_{\text{obs}}$.

5.4 Information Geometric Perspective

Can unify three components into information geometry framework:

(1) Causal Structure = Topology of information transmission

(2) Spacetime Geometry = Capacity of information encoding (holographic principle)

(3) Probability Measure = Statistics of information distribution (maximum entropy principle)

Unified Formula: $${\mathcal{I}}_{\text{total}}=\underset{\text{causal entropy}}{\underbrace{S_{\text{topo}}}}+\underset{\text{geometric entropy}}{\underbrace{S_{\text{geom}}}}+\underset{\text{probabilistic entropy}}{\underbrace{S_{\text{prob}}}}$$

where:

• $S_{\text{topo}}\sim \log |\text{number of causal paths}|$
• $S_{\text{geom}}=\frac{A}{4G\hslash }$ (holographic entropy)
• $S_{\text{prob}}=-\text{tr}(\rho \log \rho )$ (von Neumann entropy)

Conjecture: ${\mathcal{I}}_{\text{total}}$ conserved in physical evolution (generalized second law).

5.5 Analogy Summary: Symphony Trio

Imagine universe as a symphony:

• Causal Structure = Melody Line (order of notes)
• Spacetime Geometry = Pitch and Intervals (distance and harmonics)
• Probability Measure = Volume and Dynamics (loudness contrast)

Three must perfectly harmonize:

• Melody cannot conflict with harmony (causal-geometric alignment)
• Volume cannot appear in silent zones (measure-causal alignment)
• Pitch must support melody (geometric-causal alignment)

And “full score” of entire symphony is complete universe definition $\mathfrak{U}$.

Part VI: Advanced Topics and Open Problems

6.1 Quantum Gravity Corrections

Near Planck scale ${\ell }_P\sim 10^{-35}\text{m}$, three components may need corrections:

(1) Fuzzification of Causal Structure: $$U_{\text{evt}}\to U_{\text{evt}}^{\text{fuzzy}}=(X,{⪯}_{ϵ},{\mathcal{C}}_{ϵ})$$ where ${⪯}_{ϵ}$ allows causal uncertainty of $ϵ\sim {\ell }_P$.

(2) Non-Commutativization of Spacetime: $$[x^{\mu },x^{\nu }]=i{\theta }^{\mu \nu },\theta \sim {\ell }_P^2$$

(3) Non-Commutativization of Measure: $${\mu }_M\to \text{non-commutative measure}(\text{spectral triple})$$

Challenge: How to maintain compatibility of the three? May need to categorify entire structure (see $U_{\text{cat}}$).

6.2 Topological Phase Transitions and Spacetime Emergence

Some quantum gravity models allow topology changes: $$M_{\text{before}}\ncong M_{\text{after}}$$

Examples:

• Black hole formation/evaporation
• Universe creation/annihilation
• Wheeler’s “spacetime foam”

Problem: At topological phase transition point, how to define $U_{\text{geo}}$? May need: $$U_{\text{geo}}\to \text{stratified manifold}\text{or}\text{orbifold}$$

GLS Scheme: Use continuity of $U_{\text{evt}}$ as “topologically neutral” anchor: $$\text{causal structure continuous}\Rightarrow \text{spacetime topology can jump}$$

6.3 Observer Dependence and Relational Quantum Mechanics

Rovelli et al. propose: Physical quantities are always relational—no “God’s eye view” absolute state.

In GLS framework: $${\rho }_{\Sigma }\to {\rho }_{\Sigma }^{\text{obs}}=f(\mathcal{O},{\rho }_{\text{global}})$$

Different observers ${\mathcal{O}}_{\alpha },{\mathcal{O}}_{\beta }$ may have: $${\rho }_{\Sigma }^{\alpha }\ne {\rho }_{\Sigma }^{\beta }$$

But must satisfy Wigner Friendship Constraint (consistency): $${\text{tr}}_{\bar{\alpha }}({\rho }_{\Sigma }^{\alpha })={\text{tr}}_{\bar{\alpha }}({\rho }_{\Sigma }^{\beta })$$

See Component 8 $U_{\text{obs}}$ for details.

6.4 Cosmological Boundary Conditions

In FLRW universe, how to determine initial condition ${\rho }_{{\Sigma }_0}$? Possible schemes:

(1) No-Boundary Hypothesis (Hartle-Hawking): $${\rho }_{{\Sigma }_0}=\text{path integral, no boundary}$$

(2) Tunneling Boundary (Vilenkin): $${\rho }_{{\Sigma }_0}\sim e^{-S_E[\text{Euclidean instanton}]}$$

(3) Maximum Entropy Principle: $${\rho }_{{\Sigma }_0}=\arg \underset{\rho }{\max }S(\rho )\text{subject to physical constraints}$$

GLS theory provides criterion through IGVP of $U_{\text{ent}}$: $$\delta S_{\text{gen}}=0\Rightarrow \text{select unique initial state}$$

Part VII: Learning Path and Practical Suggestions

7.1 Steps for Deep Understanding of Three Components

Stage 1: Familiarize with basic concepts (1-2 weeks)

• Causal partial order and DAG
• Lorentz manifolds and light cones
• Probability measures and density matrices

Stage 2: Derive compatibility conditions (2-3 weeks)

• Prove necessary and sufficient conditions for causal-geometric alignment
• Calculate measure normalization of path integral
• Verify Ryu-Takayanagi formula (simple cases)

Stage 3: Study classical examples (3-4 weeks)

• Three components of Minkowski spacetime
• Causal structure of Schwarzschild black hole
• Measure analysis of Unruh effect

Stage 4: Explore frontier problems (long-term)

• Causal dynamics in quantum gravity
• Spacetime topological phase transitions
• Observer-dependent quantum states

7.2 Recommended References

Classical Textbooks:

1. Wald, General Relativity (spacetime geometry)
2. Haag, Local Quantum Physics (algebraic QFT)
3. Naber, Topology, Geometry and Gauge Fields (mathematical tools)

Modern Progress:

1. Sorkin, Causal Sets (causal structure)
2. Van Raamsdonk, Building up spacetime with quantum entanglement (geometry-quantum connection)
3. Bousso, The Holographic Principle (entropy and geometry)

GLS Specific:

1. Chapter 1 of this tutorial (foundations of causal dynamics)
2. Chapter 5 of this tutorial (scattering matrix and time)
3. Chapter 7 of this tutorial (generalized entropy and gravity)

7.3 Common Misconceptions and Avoidance Guide

Misconception 1: “Causal structure is accessory of spacetime”

• Correction: In GLS theory, causality and geometry are equal foundation layers, no hierarchy.

Misconception 2: “Quantum states can arbitrarily superpose”

• Correction: Superposition must respect causal constraints (microcausality) and geometric constraints (energy conditions).

Misconception 3: “Observer does not affect ontology”

• Correction: ${\rho }_{\Sigma }$ in $U_{\text{meas}}$ already implicitly contains observation configuration, subsequent $U_{\text{obs}}$ will explicitly introduce.

Summary and Outlook

Core Points Review

1. Event Causality Layer $U_{\text{evt}}$: Defines partial order structure of “what happens”
2. Geometry Spacetime Layer $U_{\text{geo}}$: Defines Lorentz manifold of “where it happens”
3. Measure Probability Layer $U_{\text{meas}}$: Defines family of quantum states of “how probable”

Three locked into one through compatibility conditions, forming “foundation” of universe.

Connections with Subsequent Components

• $U_{\text{QFT}}$: Define field operators on $(M,g,{\rho }_{\Sigma })$
• $U_{\text{scat}}$: Extract scattering matrix from $U_{\text{QFT}}$, penetrate back to causal time
• $U_{\text{ent}}$: Calculate generalized entropy from ${\rho }_{\Sigma }$, reverse derive $g$ (IGVP)
• $U_{\text{obs}}$: Assign $\mathcal{C}$ and $\rho$ to specific observers

First three components are static framework, later components introduce dynamic evolution and multi-perspective observers.

Philosophical Implication

Universe is not simple combination of “matter + spacetime + laws”, but trinity of causality, geometry, probability:

• Change causality, geometry and probability must follow
• Change geometry, causality and probability must adjust
• Change probability, causality and geometry must adapt

This global self-consistency may be deep reason for “why universe is comprehensible”.

Next Article Preview:

• 03. Quantum Field Theory, Scattering, Modular Flow: Trio of Dynamics
  • $U_{\text{QFT}}$: How to define field operators in curved spacetime?
  • $U_{\text{scat}}$: How does scattering matrix encode all dynamics?
  • $U_{\text{mod}}$: How does modular flow define “thermodynamic time”?