Causal Geometrization: Spacetime as Minimal Lossless Compression

“Spacetime geometry is not given a priori, but is the optimal encoding of causal constraints.”

🎯 Core Ideas

In the previous seven articles, we explored various aspects of causal structure. Now, a deeper question emerges:

Why is spacetime curved? Why does curvature exist?

Traditional general relativity answer: Matter and energy cause spacetime to curve (Einstein equation).

GLS Theory New Perspective:

$$\boxed{\text{Spacetime Geometry = Minimal Lossless Compression of Causal Constraints}}$$

Analogy: Imagine drawing a complex traffic network on paper:

• Flat Space: All roads can be drawn on a plane without crossing conflicts
• Curved Space: Constraints between roads are too complex, must bend the paper to accommodate

Curvature is not “extra stuff”, but accounting for correlations between causal constraints that cannot be eliminated!

📖 Problem Formulation

Confusion from Traditional Perspective

In general relativity, spacetime metric $g_{\mu \nu }$ simultaneously plays two roles:

1. Causal Role: Determines light cone structure, determines “which events can affect which events”
2. Metric Role: Determines spacetime lengths, areas, volumes

But numerous theorems (Hawking-King-McCarthy, 1976, etc.) show:

$$\text{Causal Structure}\stackrel{\text{Strong Causality}}{\to }\text{Conformal Class}[g]$$

Intuition: Just from “who can affect whom” we can recover most information about the metric!

New Questions from Information Theory

Since causal structure can recover conformal class, we can ask:

1. Minimal Encoding: How much information is needed to record causal structure?
2. Redundancy: Does curvature correspond to “incompressible redundancy”?
3. Variational Principle: Can geometry be derived from “minimum description length” principle?

This is the core question of causal geometrization!

🧩 Three-Step Reconstruction from Causality to Geometry

Step 1: Causal Partial Order → Topology

Input: Causal relation $(M,\prec )$ on event set

Tool: Alexandrov topology

Define Double Cone Open Sets: For $p\ll q$ (strictly causally before),

$$A(p,q):=I^{+}(p)\cap I^{-}(q)$$

where $I^{+}(p)$ is the strict timelike future of $p$, $I^{-}(q)$ is the strict timelike past of $q$.

graph TB
    subgraph "Generation of Alexandrov Topology"
        P["Event p"] --> FUTURE["Future Light Cone I⁺(p)"]
        Q["Event q"] --> PAST["Past Light Cone I⁻(q)"]
        FUTURE -.->|"Intersection"| DIAMOND["Double Cone A(p,q)"]
        PAST -.->|"Intersection"| DIAMOND
    end

    DIAMOND --> TOPOLOGY["Alexandrov Topology<br/>(Generated by all double cones)"]

    style DIAMOND fill:#fff4e1

Core Theorem: Under strong causality conditions,

$$\text{Alexandrov Topology}=\text{Manifold Topology}$$

Physical Meaning: Can reconstruct spacetime topology structure just from “who can affect whom”!

Step 2: Causal Structure + Time Orientation → Conformal Class

Input: Causal partial order + global time direction choice

Output: Conformal class of metric $[g]=\{{\Omega }^{2}g:\Omega >0\}$

Key Observation: Conformally equivalent metrics have the same light cone structure

$$g\sim g⟺\text{Light Cone}(g)=\text{Light Cone}(g)$$

Causal Homeomorphism Theorem:

Let $(M,g)$ and $(M,g)$ be strongly causal spacetimes. If there exists a causal homeomorphism $\Phi :M\to \tilde{M}$ (i.e., bijection preserving causal relations), then $\Phi$ is a conformal homeomorphism:

$${\Phi }^{\ast }\tilde{g}={\Omega }^{2}g$$

Analogy:

• Causal structure is like the temporal order of plot in a script
• Conformal class is like the layout of scenes on stage
• Same plot order → Same stage layout (allowing overall scaling)

Step 3: Causality + Volume Scale → Complete Metric

Problem: Conformal class only determines “shape”, not “scale”

Solution: Introduce volume measure $\mu$

Postulate: Given Borel measure $\mu$, compatible with volume form of some representative metric $g$:

$$\mu (A)={\int }_A\rho {\mathrm{dVol}}_g$$

Intuition: $\mu$ tells us “event density” or “volume scale”

Reconstruction Theorem: By comparing volumes of different Alexandrov sets $A(p,q)$, we can invert the conformal factor $\Omega$, thus recovering metric $g$.

Three-Step Summary

graph LR
    CAUSAL["Causal Partial Order<br/>(M, ≺)"] -->|"Step 1<br/>Alexandrov Topology"| TOPO["Topology"]
    TOPO -->|"Step 2<br/>Light Cone Reconstruction"| CONFORMAL["Conformal Class [g]"]
    CONFORMAL -->|"Step 3<br/>Volume Scale"| METRIC["Complete Metric g"]

    VOLUME["Volume Measure μ"] -.->|"Additional Info"| METRIC

    style CAUSAL fill:#e1f5ff
    style METRIC fill:#fff4e1

Key Insight:

$$\boxed{(M,g_{\mu \nu })⟷(M,\prec ,\mu )}$$

Right side data is more “primitive”, more like “compressed encoding”!

🗜️ Causal Reachability Graph and Description Complexity

From Continuous to Discrete

Discretize spacetime into finite event set:

• Vertices: Events $p_1,p_2,\dots ,p_N$
• Directed Edges: Causal relations $p_i\prec p_j$

Obtain causal reachability graph $\mathcal{G}=(V,E)$ (directed acyclic graph)

Example (Discrete Sampling of Minkowski Spacetime):

graph TB
    subgraph "Causal Reachability Graph"
        P1["p₁"] --> P3["p₃"]
        P1 --> P4["p₄"]
        P2["p₂"] --> P4
        P2 --> P5["p₅"]
        P3 --> P6["p₆"]
        P4 --> P6
        P5 --> P6
    end

    style P1 fill:#e1f5ff
    style P6 fill:#ffe1e1

Description Complexity

Definition: Description complexity $\mathcal{C}(\mathcal{G})$ is the minimum information (bits) needed to precisely record graph $\mathcal{G}$

Encoding Methods:

1. Adjacency Matrix: $N\times N$ matrix, requires $O(N^2)$ bits
2. Adjacency List: Only record existing edges, requires $O(|E|)$ bits
3. Hierarchical Decomposition: Exploit hierarchical nature of causal structure, possibly more optimal

Continuum Limit: In continuous limit,

$$\mathcal{C}(\text{Reach}(g)):=\underset{ϵ\to 0}{\lim }{\mathcal{C}}_{ϵ}({\mathcal{G}}_{ϵ})$$

where ${\mathcal{G}}_{ϵ}$ is discrete approximation at resolution $ϵ$.

High Symmetry = Low Complexity

Minkowski Spacetime:

• Causal structure has Poincaré symmetry
• Highly regular → Description complexity extremely low
• Can encode entire structure with few parameters (translations, rotations)

Curved Spacetime (e.g., FRW Universe):

• Reduced symmetry (only spatial rotational symmetry)
• Need more information to describe causal structure
• Description complexity higher

Analogy:

• Flat spacetime = Perfect checkerboard (repeating pattern, high compression ratio)
• Curved spacetime = Irregular jigsaw puzzle (must record shape of each piece, low compression ratio)

🌀 Curvature as Redundancy Density

Meaning of Flatness

Locally Flat: Near any point $p$, can choose coordinates such that metric approximates Minkowski:

$$g_{\mu \nu }(x)\approx {\eta }_{\mu \nu }+O(|x-p{|}^2)$$

Globally Flat: Exists global inertial frame, entire spacetime metric is ${\eta }_{\mu \nu }$

Key Difference: Can local constraints be globally compatibly patched?

Causal Interpretation of Curvature

Consider three events $p,q,r$ forming a “causal triangle”:

graph LR
    P["p"] -->|"Causal Path 1"| Q["q"]
    Q -->|"Causal Path 2"| R["r"]
    P -->|"Causal Path 3"| R

    style P fill:#e1f5ff
    style R fill:#ffe1e1

Flat Spacetime: “Total causal delay” of paths 1+2 completely matches path 3

Curved Spacetime: Exists closure error (similar to non-closure of parallel transport)

$$\text{Closure Error}\sim {\int }_{△}{R}_{\mu \nu \rho \sigma }\mathrm{d}{S}^{\mu \nu }$$

Definition:

$$\boxed{\text{Curvature = Incompressible Correlation Redundancy Density Between Causal Constraints}}$$

Analogy:

Imagine building triangular network with rigid sticks:

• Plane: All triangles can tile flat, no internal stress
• Surface: Triangles have internal stress, must bend to patch

Curvature is this “internal stress density”!

Mathematical Form

Riemann curvature tensor:

$$R^{\rho }{}_{\sigma \mu \nu }={\partial }_{\mu }{\Gamma }_{\nu \sigma }^{\rho }-{\partial }_{\nu }{\Gamma }_{\mu \sigma }^{\rho }+{\Gamma }_{\mu \lambda }^{\rho }{\Gamma }_{\nu \sigma }^{\lambda }-{\Gamma }_{\nu \lambda }^{\rho }{\Gamma }_{\mu \sigma }^{\lambda }$$

Physical Meaning (GLS Interpretation):

• ${\Gamma }_{\mu \sigma }^{\rho }$: Local causal constraints (connection)
• $R^{\rho }{}_{\sigma \mu \nu }$: Inconsistency when combining local constraints along different paths

⚖️ Description Length-Curvature Variational Principle

Functional Construction

Given causal structure class ${\mathcal{C}}_{\text{caus}}$ and volume scale, define:

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

Meaning of Two Terms:

1. $\mathcal{C}(\text{Reach}(g))$: Description complexity

   • Minimum bits needed to record causal reachability structure
   • High symmetry → Low complexity
   • Encourages “concise causal structure”

2. $\int |\text{Riem}(g){|}^2{\mathrm{dVol}}_g$: Curvature penalty term

   • Penalizes high curvature
   • Encourages “local constraints globally compatible”
   • Corresponds to “as flat as possible”

Parameter $\lambda$: Trade-off between description conciseness and geometric flatness

Variational Principle

Physical Selection: Actual spacetime geometry is minimizer of $\mathcal{F}[g]$

$$\delta \mathcal{F}[g]=0$$

Special Case: If causal structure is fixed ($\mathcal{C}$ constant), reduces to:

$$\delta {\int }_M|\text{Riem}(g){|}^2{\mathrm{dVol}}_g=0$$

This corresponds to critical points of $L^2$-curvature flow!

Relation to Einstein-Hilbert Action

Einstein-Hilbert action:

$$S_{\text{EH}}[g]=\frac{1}{16\pi G}{\int }_{M}R{\mathrm{dVol}}_g+S_{\text{matter}}$$

Connection Conjecture: Under appropriate coarse-graining,

$$\mathcal{C}(\text{Reach}(g))+\lambda \int |\text{Riem}{|}^2⟷S_{\text{EH}}[g]$$

Evidence:

• Description complexity term $\leftrightarrow$ Ricci scalar $R$ (related to Euler characteristic)
• Curvature penalty term $\leftrightarrow$ Higher-order gravitational corrections (e.g., $R^2$ gravity)

This is the information-theoretic interpretation of emergent gravity!

🔬 Concrete Examples

Example 1: Minkowski Spacetime

Causal Structure:

$$p\prec q⟺{\eta }_{\mu \nu }(q-p{)}^{\mu }(q-p{)}^{\nu }<0\text{ and }{t}_q>t_p$$

Symmetry: Poincaré group $ISO(3,1)$

Description Complexity: $\mathcal{C}\approx 10$ parameters (4 translations + 6 rotations)

Curvature: $R_{\mu \nu \rho \sigma }\equiv 0$

Functional Value:

$$\mathcal{F}[\eta ]={\mathcal{C}}_{\text{min}}+0={\mathcal{C}}_{\text{min}}$$

Conclusion: Minkowski spacetime achieves absolute minimum (zero curvature + highest symmetry)!

Example 2: FRW Universe

Metric:

$$g=-\mathrm{d}{t}^2+a(t{)}^2{\gamma }_{ij}\mathrm{d}{x}^i\mathrm{d}{x}^j$$

where ${\gamma }_{ij}$ is constant curvature three-dimensional spatial metric.

Symmetry: Spatial isotropy $SO(3)$ (time direction symmetry broken)

Description Complexity: Need to record complete functional form of $a(t)$ (infinitely many parameters)

$$\mathcal{C}(\text{FRW})\gg \mathcal{C}(\text{Minkowski})$$

Curvature: Non-zero (spatial curvature $k\ne 0$ or cosmological curvature)

$$\int |\text{Riem}{|}^2\mathrm{dVol}>0$$

Functional Value:

$$\mathcal{F}[\text{FRW}]={\mathcal{C}}_{\text{high}}+\lambda \cdot (\text{positive})$$

Explanation: Cosmological horizon, particle horizon, etc. causal boundaries → Complex causal structure → High description complexity + High curvature

Example 3: Black Hole Spacetime (Schwarzschild)

Metric (exterior region):

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

Causal Structure Features:

• Horizon $r=2GM$ where causal structure undergoes qualitative change
• Interior region $r<2GM$ has time and radial roles swapped
• Singularity $r=0$ is causal boundary

Description Complexity:

• Highly symmetric (spherical symmetry $SO(3)$)
• But horizon and singularity cause complex causal topology

$$\mathcal{C}(\text{Schwarzschild})\sim \text{moderate}$$

Curvature:

• Riemann tensor non-zero (tidal forces)
• Curvature diverges at singularity

Variational Interpretation: Black hole is solution achieving local minimum of $\mathcal{F}[g]$ under given mass constraint (information-theoretic version of no-hair theorem)!

🌉 Connection with Quantum Field Theory

Microcausality

In quantum field theory, local observable algebras $\mathcal{A}(\mathcal{O})$ satisfy:

Microcausality Axiom: If ${\mathcal{O}}_1,{\mathcal{O}}_2$ are spacelike separated, then

$$[A,B]=0,\forall A\in \mathcal{A}({\mathcal{O}}_1),B\in \mathcal{A}({\mathcal{O}}_2)$$

GLS Interpretation: Microcausality completely determined by causal structure!

$$\text{Causal Structure}⟶\text{Commutativity of Observable Algebras}$$

Relative Entropy and Causal Cone

Given two states $\omega ,{\omega }^{\prime }$ restricted to region $\mathcal{O}$, define relative entropy:

$$S(\omega |{\omega }^{\prime })=⟨\log {\Delta }_{\omega ,{\omega }^{\prime }}{⟩}_{\omega }$$

Monotonicity Theorem (Araki, 1976): If $\mathcal{O}\subset {\mathcal{O}}^{\prime }$ (causal inclusion), then

$$S(\omega {|}_{{\mathcal{O}}^{\prime }}|{\omega }^{\prime }{|}_{{\mathcal{O}}^{\prime }})\ge S(\omega {|}_{\mathcal{O}}|{\omega }^{\prime }{|}_{\mathcal{O}})$$

Physical Meaning: When extending observable domain along causal flow, distinguishability non-decreasing

Connection with Description Complexity:

$$\mathcal{C}(\text{Reach}(g))\sim S_{\text{rel}}\sim \text{"Information Needed to Distinguish Causal Structures"}$$

Fisher Information and Causal Metric

Under appropriate smoothness conditions, second-order differential of relative entropy gives Fisher information metric:

$$g_{\mu \nu }^{\text{Fisher}}=\frac{{\partial }^{2}S(\omega |{\omega }^{\prime })}{\partial {\theta }^{\mu }\partial {\theta }^{\nu }}{|}_{\theta =0}$$

GLS Insight: Inside causal cone, Fisher information metric adds meaning of “distinguishability rate” to spacetime geometry

$$\text{Spacetime Geometry}⟷\text{Information Geometry (under causal constraints)}$$

🔍 Deep Understanding

Why Can’t Curvature Be Arbitrarily Eliminated?

Topological Obstacle:

Some spacetimes have non-trivial topology (e.g., $S^3\times \mathbb{R}$), cannot be globally flattened

Gauss-Bonnet Theorem (2D generalization to 4D):

$${\int }_{M}R\mathrm{dVol}=4\pi \chi (M)$$

where $\chi (M)$ is Euler characteristic (topological invariant)

Conclusion: Non-trivial topology $\Rightarrow$ Curvature cannot be identically zero

Relation Between Description Complexity and Entropy

Shannon Entropy:

$$H(X)=-\sum _i{p}_i\log p_i$$

Kolmogorov Complexity:

$$K(s)=\min \{\ell (\text{program}):\text{program outputs }s\}$$

Connection:

$$\mathcal{C}(\mathcal{G})\approx K(\text{encoding}(\mathcal{G}))$$

In statistical sense, $\mathcal{C}\sim H$ (coding theorem)

GLS Unification:

$$\begin{array}{rl}\text{Thermodynamic Entropy}& ⟷\text{Shannon Entropy}\\ \text{Generalized Entropy}& ⟷\text{Description Complexity}\\ \text{Time Arrow}& ⟷\text{Compression Efficiency Unidirectionality}\end{array}$$

Why Choose $|\text{Riem}{|}^2$ Instead of $|R|$?

Ricci Scalar $R$:

$$R=g^{\mu \nu }{R}_{\mu \nu }$$

is “trace” (average) of curvature

Riemann Tensor Norm $|\text{Riem}{|}^2$:

$$||\text{Riem}{|}^2=R_{\mu \nu \rho \sigma }{R}^{\mu \nu \rho \sigma }$$

contains complete tidal information

Physical Difference:

• $R$ related to matter density (Einstein equation: $R\sim T$)
• $|\text{Riem}{|}^2$ related to incompatibility of causal constraints (Weyl curvature)

Variational Principle Choice:

$$\int |\text{Riem}{|}^2⟷\text{Bach Equation (Fourth-Order Gravity)}$$

$$\int R⟷\text{Einstein Equation (Second-Order Gravity)}$$

The two describe spacetime at different levels!

🌟 Core Formula Summary

Three Steps of Causal Reconstruction

$$\boxed{\begin{array}{rl}& \text{Step 1:}(M,\prec )\stackrel{\text{Alexandrov}}{\to }\text{Topology}\\ & \text{Step 2:}(M,\prec ,\text{Time Orientation})\stackrel{\text{Light Cone}}{\to }[g]\\ & \text{Step 3:}(M,\prec ,\mu )\stackrel{\text{Volume Scale}}{\to }g\end{array}}$$

Equivalent Encoding

$$\boxed{(M,g_{\mu \nu })⟺(M,\prec ,\mu )}$$

Description Complexity-Curvature Functional

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

Variational Principle

$$\boxed{\delta \mathcal{F}[g]=0⟺\text{Physical Spacetime Geometry}}$$

Causal Interpretation of Curvature

$$\boxed{\text{Curvature}=\text{Incompressible Correlation Redundancy Density Between Causal Constraints}}$$

💭 Thinking Questions

Question 1: Why Does Flat Spacetime Have Lowest Description Complexity?

Hint: Consider relation between symmetry and compression

Answer:

Flat spacetime (Minkowski) has highest symmetry (Poincaré group):

$$ISO(3,1):10\text{ parameters (4 translations + 6 rotations/boosts)}$$

High symmetry means high regularity → high compression ratio → low description complexity

Analogy:

• Completely repeating pattern: Only need to record “unit + repetition rule”
• Random pattern: Must record every pixel

Question 2: Does Black Hole Entropy Correspond to Description Complexity of Causal Structure?

Hint: Recall Bekenstein-Hawking entropy formula

Answer:

Bekenstein-Hawking entropy:

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

GLS Interpretation:

1. Black hole horizon is causal boundary (interior and exterior causal structures completely different)
2. Horizon area $A$ encodes number of interior causal microstates
3. Therefore $S_{\text{BH}}\sim \log (\text{number of interior causal configurations})$

Connection with Description Complexity:

$$S_{\text{BH}}\sim \mathcal{C}(\text{interior causal structure})\sim \frac{A}{4G\hslash }$$

Conclusion: Black hole entropy is perfect verification of causal geometric compression theory!

Question 3: How Is Description Complexity Modified in Quantum Gravity?

Hint: Consider discreteness at Planck scale

Answer:

In full quantum gravity, spacetime is discretized at Planck scale ${\ell }_P$:

Causal Set Theory:

$$(M,g)⟶(C,\prec )$$

where $C$ is discrete partially ordered set, satisfying:

• Local Finiteness: $\{r:p\prec r\prec q\}$ is finite
• Poisson Sprinkling: Event density $\sim {\ell }_P^{-4}$

Description Complexity:

$${\mathcal{C}}_{\text{quantum}}(C)={\log }_2(\text{number of partial orders satisfying constraints})$$

Holographic Principle Prediction:

$${\mathcal{C}}_{\text{quantum}}\le \frac{A(\partial V)}{4G\hslash }$$

where $A(\partial V)$ is boundary area of region $V$!

🔗 Connections with Previous Chapters

With Unified Time Chapter (Chapter 5)

Null boundary of causal diamond → Expansion $\theta$ → Time scale $\kappa (\omega )$

$$\kappa (\omega )⟷\theta +{\kappa }_{\text{surf}}$$

Compression Interpretation: Time scale is optimal projection of causal structure in time direction

With Boundary Theory Chapter (Chapter 6)

Boundary triplet $(\partial M,{\mathcal{A}}_{\partial },{\omega }_{\partial })$ encodes complete information

$$\text{Bulk Information}\le \text{Boundary Information}$$

Compression Interpretation: Boundary is lossless compressed representation of bulk causal structure

With Causal Structure Chapter (Previous 7 Articles)

Null-Modular Double Cover Theorem:

$$K_D=2\pi \sum _{\sigma =\pm }{\int }_{E^{\sigma }}{g}_{\sigma }{T}_{\sigma \sigma }$$

Compression Interpretation: Modulation function $g_{\sigma }$ of modular Hamiltonian $K_D$ encodes geometric (curvature) information

🎯 Core Insights of This Chapter

1. Spacetime Geometry = Minimal Lossless Compression of Causal Constraints

   $$(M,g)⟷(M,\prec ,\mu )$$

2. Curvature = Incompressible Causal Redundancy Density

   Flat ↔ All local constraints globally compatible Curved ↔ Exists “closure error”

3. Variational Principle = Information Optimization

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

4. Information-Theoretic Foundation of Emergent Gravity

   Einstein-Hilbert action ↔ Description complexity-curvature functional

📚 Further Reading

Classical References:

• Malament (1977): “The Class of Continuous Timelike Curves Determines the Topology”
• Hawking & Ellis (1973): “The Large Scale Structure of Space-Time” (Chapter 6)

Modern Developments:

• Bombelli et al. (1987): “Space-Time as a Causal Set” (foundation of causal set theory)
• Sorkin (2005): “Causal Sets: Discrete Gravity” (review)

Information-Theoretic Perspective:

• Bousso (2002): “The Holographic Principle” (holographic principle)
• Lloyd (2002): “Computational Capacity of the Universe”

GLS Theory Source Documents:

• causal-structure-geometrization-spacetime-minimal-lossless-compression.md (source of this chapter)

Next Chapter Preview: 09-Error Geometry and Causal Robustness

We will explore: When causal structure has small perturbations (measurement errors, quantum fluctuations), how does spacetime geometry remain robust?

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