How Information Becomes Spacetime: When Relative Entropy Meets Einstein

Einstein’s Unfulfilled Dream

In 1955, Einstein was still calculating on his deathbed. His lifelong dream was to find a unified field theory—one formula explaining gravity, electromagnetic force, and everything.

He failed.

But 70 years later, physicists discovered: Einstein was actually very close. He was only one step away—treating information as more fundamental than spacetime.

What we’re discussing today is this stunning discovery:

Spacetime is not the stage, but the actor. It’s not background, but the geometric expression of information relationships.

The gravitational equation (Einstein’s greatest achievement) and electromagnetic equations (Maxwell’s masterpiece) can be derived from the same information principle.

This story begins with a seemingly boring mathematical concept: relative entropy.

Act I: Relative Entropy—The “Distance” of Information

Imagine two friends, Xiao Ming and Xiao Hong, describing the same soccer match.

• Xiao Ming says: “Team A has 90% chance to win, Team B 10%”
• Xiao Hong says: “Team A has 60% chance to win, Team B 40%”

How “different” are their predictions?

Intuitively, you might calculate the difference: $|90\%-60\%|=30\%$. But this isn’t precise enough—because probability isn’t a simple number, but distribution of information.

Information theory has a better measure: relative entropy (also called KL divergence), denoted $D(P\Vert Q)$.

$$D(P\Vert Q)=\sum _{i}P(i)\log \frac{P(i)}{Q(i)}$$

It measures: If the true distribution is $P$, but you mistakenly think it’s $Q$, how surprised will you be.

• If $P$ and $Q$ are identical, $D(P\Vert Q)=0$ (no surprise)
• If $P$ and $Q$ differ greatly, $D(P\Vert Q)$ is large (very surprised)

Relative entropy isn’t a true “distance” (because it’s asymmetric: $D(P\Vert Q)\ne D(Q\Vert P)$), but it’s the core measure in information geometry.

Why Is Relative Entropy Important?

Because it measures reduction of uncertainty—that is, acquisition of information.

When you go from “don’t know” (uniform distribution $Q$) to “know” (true distribution $P$), the information you gain is $D(P\Vert Q)$.

Relative entropy = Surprise = Information = Growth of knowledge

Now, something magical happens.

Act II: Fisher Metric—The “Geometry” of Information

Suppose you have a family of probability distributions, each labeled by parameter $\theta$ (like mean and variance of a normal distribution).

Now you change the parameter: $\theta \to \theta +d\theta$.

Question: How much information difference does this change bring?

Answer: Expand $D(P_{\theta }\Vert P_{\theta +d\theta })$ in Taylor series as $d\theta \to 0$, you get:

$$D(P_{\theta }\Vert P_{\theta +d\theta })\approx \frac{1}{2}{g}_{ij}(\theta )d{\theta }^{i}d{\theta }^j$$

Here $g_{ij}$ is called the Fisher Information Metric:

$$g_{ij}=\mathbb{E}\left[\frac{\partial \log P}{\partial {\theta }^i}\frac{\partial \log P}{\partial {\theta }^j}\right]$$

It has an amazing property: It’s a metric—like a ruler measuring distance.

But this time it’s not measuring spatial distance, but distance in information space.

Chentsov’s Theorem: Uniqueness

In 1982, mathematician Chentsov proved a profound theorem:

Under certain natural symmetry conditions, Fisher metric is the unique metric on information space.

In other words: If you want to define “distance” on probability distribution space, and this distance must be invariant under coordinate transformations (like physical laws don’t depend on coordinate choice), then you can only use Fisher metric.

The geometry of information is unique.

This is like saying: If you want to measure distance in 3D space, you can only use the Pythagorean theorem (Euclidean metric) or its generalization (Riemannian metric).

Fisher metric is the “Pythagorean theorem” of information space.

Act III: From Information to Spacetime

Now, the crucial leap comes.

Physicists asked a bold question:

What if spacetime is also an information geometric space?

Specifically:

• Each point $x$ in spacetime corresponds to a quantum state $\rho (x)$ (local information distribution)
• Spacetime distance corresponds to relative entropy of information
• Spacetime curvature corresponds to changes in information distribution

This sounds crazy, but mathematically it works perfectly.

Key Insight: Second-Order Expansion of Relative Entropy

Consider quantum states $\rho (x)$ and $\rho (x+dx)$ at two adjacent points $x$ and $x+dx$.

Their relative entropy is:

$$D(\rho (x+dx)\Vert \rho (x))\approx g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$$

Here $g_{\mu \nu }$ is the spacetime metric—the protagonist in Einstein’s equation!

So, spacetime metric = information metric = Fisher metric.

In one sentence:

The geometry of spacetime is the geometry of information distribution.

Act IV: IGVP—One Principle Rules Everything

Now we’re ready to reveal the ultimate secret.

Physicists defined something called the Information Geometry Variational Principle (IGVP):

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

Here $S_{\text{gen}}$ is generalized entropy, including:

• Gravitational entropy (horizon area / 4G, Bekenstein-Hawking)
• Matter entropy (von Neumann entropy of quantum fields)
• Boundary terms (boundary contributions from information flow)

The principle is simple: Nature chooses configurations that extremize generalized entropy.

Like soap bubbles choosing minimal surface area, nature chooses optimal information distribution.

The Miracle Happens

When you vary $S_{\text{gen}}$ (find extremum), you simultaneously get:

1. Einstein Field Equation

$$G_{\mu \nu }+\Lambda g_{\mu \nu }=8\pi G⟨T_{\mu \nu }⟩$$

This is the equation of gravity—telling you how spacetime curves.

2. Yang-Mills Equation

$$D_{\mu }{F}^{\mu \nu ,a}=J^{\nu ,a}$$

This is the equation of gauge fields—describing electromagnetic, strong, and weak nuclear forces.

One variational principle, two equations, ruling all fundamental forces of the universe.

Act V: Gravity Is Not a Force, But Geometry of Information

Let’s digest what this means.

Traditional View: Gravity Is a Force

Newton said: Gravity is attraction between masses. Einstein said: Gravity is manifestation of spacetime curvature.

But both assume: Gravity is fundamental, spacetime is given background.

New View: Gravity Is Extremum of Information Entropy

IGVP says: Gravity is not fundamental, but emerges from information principles.

Spacetime curves because information seeks optimal distribution.

Analogy:

• Soap bubbles aren’t “shaped by some force” into spheres, but because surface tension minimizes surface area
• Water doesn’t flow downhill “driven by some force,” but because potential energy minimizes
• Spacetime doesn’t curve “by gravity,” but because information entropy reaches extremum

Gravity is a byproduct of information geometry.

Why Does Mass Curve Spacetime?

Traditional answer: “Mass just curves spacetime, it’s a law of nature.”

New answer: Mass carries information (quantum states), and information changes local information geometry, this geometric change is spacetime curvature.

In formula:

$$\text{mass}\to \text{quantum state}\to \text{information distribution}\to \text{Fisher metric}\to \text{spacetime curvature}$$

Mass curves spacetime because mass is a form of information.

Act VI: Electromagnetic Force Is Also Grammar of Information

Even more astonishing, IGVP not only gives gravity, but also electromagnetic force and other gauge forces.

What Are Gauge Fields?

In the quantum world, particles don’t just have position and velocity, but also internal degrees of freedom—like electron’s “phase,” quark’s “color”.

These internal degrees of freedom form an internal space (called fiber), attached to each spacetime point.

Gauge fields are “connections” describing how this internal space changes with spacetime.

Why Do We Need Gauge Fields?

Because of locality of information.

If you measure the phase of the same particle at two different places, you need a “dictionary” to translate—this “dictionary” is the gauge field.

Analogy:

• In Beijing, you say “苹果” is apple
• In New York, you say “apple” is 苹果
• You need a translation (gauge field) to keep information consistent

Gauge fields are translation rules for information between different local “frames”.

How Does IGVP Give Gauge Field Equations?

When you require information consistency between different local frames (this is gauge invariance), and extremize generalized entropy, you naturally get Yang-Mills equations.

Electromagnetic, strong, weak nuclear forces all emerge from information maintaining causal consistency.

Act VII: One Principle, the Entire Universe

Let’s stand high and survey the whole picture.

The “Operating System” of the Universe

If the universe is a computer, IGVP is its operating system:

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

“Let generalized entropy reach extremum”

From this one principle emerge:

• Geometry of spacetime (metric $g_{\mu \nu }$)
• Dynamics of gravity (Einstein’s equation)
• Interactions of matter (Yang-Mills equations)
• Evolution of quantum fields (Dirac equation, Schrödinger equation…)

Just as all number theory can be derived from “addition and multiplication of integers,” all fundamental physics can be derived from IGVP.

Why Entropy?

Because entropy is a measure of information.

• High entropy = large uncertainty = little information
• Low entropy = small uncertainty = much information

Nature extremizing entropy is optimizing information distribution.

But note: Not simple “entropy increase” (that’s the second law of thermodynamics), but extremum of generalized entropy—can increase, can decrease, key is balance.

Like “market equilibrium” in economics—not everyone profits, not everyone loses, but overall reaches some optimal allocation.

The universe is performing “market equilibrium” of information.

Act VIII: Where Does Spacetime Come From?

Now we can answer an ultimate question: Where does spacetime come from?

Traditional Cosmology

Big Bang theory says: 13.8 billion years ago, spacetime was born from a singularity, then expanded.

But this doesn’t answer: Why is there spacetime? Why not nothing?

Information Geometry’s Answer

Spacetime doesn’t “emerge,” but “emerges.”

Once there is information (quantum states, probability distributions, uncertainty…), there must be:

• Relative entropy (difference of information)
• Fisher metric (geometry of information)
• Extremum principle (systems tend toward optimal configuration)

Then, IGVP automatically gives:

• Metric $g_{\mu \nu }$ (geometry of spacetime)
• Einstein’s equation (dynamics of spacetime)

So, wherever there is information, there is spacetime.

Spacetime is not fundamental, information is.

Then, Where Does Information Come From?

This is the next question—perhaps the ultimate question.

Possible answers:

1. Information is eternal (Platonism)—mathematical structures exist by themselves, don’t need “where from”
2. Information is self-referential (bootstrapping)—information defines itself, like “this sentence is true”
3. Information is existence (pan-information theory)—“existence” is “distinguished by information,” they’re synonymous

We don’t know the answer yet. But at least we know: Spacetime is not the starting point of the answer, but emergence of information.

Real Examples: Seeing Information Become Spacetime

Example 1: Black Hole Entropy

Black holes have entropy (Bekenstein-Hawking entropy):

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

where $A$ is horizon area.

This entropy is information geometric entropy—measuring difference of information inside and outside the black hole.

And black hole temperature (Hawking radiation):

$$T_{\text{H}}=\frac{\hslash c^3}{8\pi GM{k}_B}$$

Exactly satisfies the first law of thermodynamics:

$$dM=T_{\text{H}}d{S}_{\text{BH}}-\frac{\kappa }{8\pi G}dA$$

Black hole thermodynamics is information geometric thermodynamics.

Example 2: Accelerated Expansion of the Universe

We observe the universe accelerating expansion (1998 Nobel Prize).

Traditional explanation: Mysterious “dark energy” drives it.

IGVP explanation: Cosmological constant $\Lambda$ is vacuum’s information geometric curvature.

Vacuum isn’t “empty,” but has quantum fluctuations (virtual particles constantly created and annihilated). These fluctuations carry information, and information has geometry.

This geometry’s “background curvature” is $\Lambda$.

Dark energy is not “energy,” but vacuum curvature of information geometry.

Example 3: Gravitational Waves

In 2015, LIGO first detected gravitational waves (2017 Nobel Prize).

What are gravitational waves? Traditional answer: Spacetime ripples.

IGVP answer: Perturbations of information geometry propagating.

Two black holes merge, changing local information distribution, this change propagates outward at light speed—that’s gravitational waves.

Gravitational waves are information waves.

Philosophical Reflection: Do We Live in Mathematics?

Let’s pause and think about what this means.

Plato’s Cave

Plato said: The world we see is shadows, the real world is “ideas” (mathematical forms).

IGVP seems to say: Plato was half right.

• Spacetime is indeed “shadows”—projections of information geometry
• But “ideas” aren’t mysterious existence transcending physics, but information relationships themselves

Mathematics doesn’t “describe” physics, but defines physics.

The Role of Observers

Deeper question: Without observers, is there still information?

Quantum mechanics says: Without measurement, particles have no definite state.

Information geometry says: Without observers, there’s no relative entropy, no Fisher metric, no spacetime.

Spacetime needs observers to emerge.

This doesn’t mean “your brain created the universe” (that’s solipsism), but: Observers and universe co-emerge—you can’t have one without the other.

Self-Referential Universe

Finally, we arrive at a self-referential cycle:

• Information defines spacetime
• Spacetime carries observers
• Observers measure information
• Measurement defines information

This is a self-consistent closed loop, no beginning, no end.

The universe wasn’t created, but self-defined.

Like mathematical axiom systems—you can’t ask “where does 1+1=2 come from,” because it’s definition.

The universe is its own definition.

Take-Home Thoughts

Next time when you:

• See an apple fall (gravity)
• Feel magnet attraction (electromagnetic force)
• Think about time passing (causality)

Remember: These aren’t fundamental forces or background, but emergence of information seeking optimal distribution.

Gravity doesn’t “pull,” but information geometry tells mass “here is optimal configuration”.

Time doesn’t “flow,” but irreversible evolution direction of information.

You don’t “exist” in spacetime, but spacetime emerges because of your observation.

You’re not in the universe, you are the way the universe sees itself.

Next: “Particles Are Network Resonances: Why Electrons Are Not Little Balls”

We will see that all particles are standing waves in information scattering networks—not “things,” but “patterns”.