Time From Dimensions

Part I: Without the Math

TL;DR: Time emerges from correlations through a dimensional aperture. When the aperture contracts—whether from gravity, velocity, or neural synchronization—time dilates. This isn't three separate phenomena. It's one phenomenon in three regimes.

This is Part 3 of a three-part series. Part 1 introduced the projection framework for quantum mechanics. Part 2 applied it to quantum gravity.

Dimension glossary (used throughout this series):

• D (Hilbert/state dimension): size of the full state space
• d (spacetime dimension): 3+1 in our classical description
• k (aperture): effective degrees of freedom accessible to a given observer+interface

Now we ask: what about time?

Disambiguation: In this post, "time" can mean (i) proper time measured by a physical clock, (ii) coordinate time used to describe a system from afar, or (iii) subjective time in experience. The first two are established physics with precise definitions. The third is an analogy—a structural claim about shared geometry, not an identity.

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The pattern

In the previous post, we noticed something:

Black holes:

• External observer's accessible description becomes surface-confined (k→2, area scaling)
• External observer sees infalling objects freeze in time
• Infalling observer maintains full 3D access (k=3)
• Infalling observer experiences normal time flow

High velocity:

• Accessible variation along travel direction is compressed (effective k reduction)
• Time dilates proportionally
• Co-moving observer maintains full access
• Co-moving observer experiences normal time

Nothing "loses a dimension" ontologically. The claim is that the accessible channel for correlations narrows, reducing effective degrees of freedom.

The pattern: when k deviates from its optimal value, time slows.

This isn't a coincidence. It's a relationship.

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Time from correlations

In the Wheeler-DeWitt formulation of quantum gravity, the universal wavefunction satisfies:

$\hat{H}|\Psi\rangle = 0$

The universe is "frozen." There's no time parameter in the fundamental equation.

So where does our experience of time come from?

From correlations. When a subsystem S is entangled with a "clock" subsystem C, the correlations between them look like time evolution from S's perspective. Time is not fundamental—it emerges from the pattern of correlations between subsystems.

But here's the key: you can only correlate with degrees of freedom you have access to.

The dimensional aperture k determines how many degrees of freedom are accessible. When k drops, you lose access to degrees of freedom. Fewer accessible degrees of freedom → fewer possible correlations → slower time.

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The unified picture

The pattern: k=3 is the classical aperture. That's not a hypothesis—it's the dimensionality of space. What's new is the claim that deviations from k=3 in either direction disrupt correlation rate, and that this is what time dilation is.

Time = rate of correlation accumulation through the dimensional aperture

Squeeze the aperture → reduce correlation rate → dilate time

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Why this makes sense

Think about what "time passing" actually means operationally:

• Recording memories
• Updating states based on interactions
• Accumulating correlations with your environment

All of these require access to degrees of freedom. If your dimensional aperture contracts, you can interact with fewer degrees of freedom per unit of coordinate time. From your perspective, less happens. Time slows.

This is why:

• The external observer sees the infalling astronaut freeze—they've lost access to the degrees of freedom that would let them see further evolution
• The astronaut experiences normal time—their aperture is unchanged, they're still correlating with their local environment
• You lose the record of time during deep sleep—your neural dimensionality collapses, correlations aren't being formed
• Flow states distort time—narrowed attention means fewer peripheral correlations

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The aperture is observer-relative

A crucial point: k is not a property of spacetime alone. It's a property of the observer-system relationship.

The black hole horizon is k=2 for the external observer and k=3 for the infalling observer. Neither is wrong. The aperture depends on your embedding in the system.

This is black hole complementarity—and it generalizes. Your dimensional aperture is determined by:

• Your trajectory through spacetime (velocity, gravitational field)
• Your measurement apparatus (what degrees of freedom you couple to)
• Your neural state (attention, synchronization, neurochemistry)

Different observers can have different k for the same underlying physics.

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The minimum k for complexity

There's another piece to this puzzle: the minimal embedding dimension.

Any dynamical system has a minimum dimensionality $k_{min}$ needed to faithfully represent its behavior. Below $k_{min}$, trajectories cross that shouldn't—the dynamics become ill-defined. This is Takens' embedding theorem applied to state space.

So k has a floor. Below it, complexity itself cannot exist.

This connects to time in a striking way:

• $k < k_{min}$: Dynamics are undefined. No correlations possible. No time.
• $k = k_{min}$: Minimal complexity. Time exists but is constrained.
• $k = 3$ (optimal): Full classical dynamics. Maximal correlation rate. Time flows.
• $k > 3$: Too many degrees of freedom to integrate. Time fragments.

The implication: classical observers operate at $k = 3$ because that's what "classical" means in our universe.

The constraints are asymmetric:

• Below k=3: Complexity is impossible. You can't embed the dynamics needed for memory, prediction, or self-reference. Hard floor.
• Above k=3: Complexity might be possible, but it's thermodynamically expensive. Integration costs scale with dimensionality. Soft ceiling.

k=3 is the minimum aperture that supports observers—and probably the thermodynamically optimal one. Not anthropic hand-waving; a constraint.

The question isn't whether k=3 is special—it is. The question is whether this aperture framework correctly describes why deviations produce time effects.

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What this doesn't explain

Let me be clear about limits:

We don't have a derivation. The k-time relationship is a pattern, not (yet) a theorem. I'm claiming the correlation is real and meaningful, not that I've derived it from first principles.

The exponent is unknown. Is time dilation proportional to k? To k²? To log(k)? The functional form needs to be worked out.

Neural "time" is slippery. Subjective time distortion isn't the same as physical time dilation. The analogy may be structural rather than identical.

This is speculative. The relativistic cases (black holes, velocity) are established physics with established math. The claim that they're instances of a more general dimensional principle is the new part.

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Part II: With the Math

GR/SR: time dilation as a metric factor

In general relativity, proper time τ relates to coordinate time t via the metric:

$d\tau^2 = -g_{\mu\nu}dx^\mu dx^\nu$

For a stationary observer in Schwarzschild spacetime:

$\frac{d\tau}{dt} = \sqrt{1 - \frac{2GM}{rc^2}}$

At the horizon (r = 2GM/c²), this goes to zero. Time freezes.

In special relativity, the Lorentz factor:

$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$

gives both length contraction (L = L₀/γ) and time dilation (Δt = γΔt₀).

The question is: can we express these as dimensional aperture effects?

Aperture: participation ratio k_eff

Define the effective dimensionality k_eff via a participation ratio over accessible degrees of freedom:

$k_{eff} = \frac{(\sum_i \lambda_i)^2}{\sum_i \lambda_i^2}$

where λᵢ are eigenvalues of:

• Spacetime: spatial metric eigenvalues (or measurement operator tied to observer coupling)
• Neural: covariance matrix of activity
• Information geometry: Fisher metric eigenvalues

The conjecture:

$\frac{d\tau}{dt} \propto \left(\frac{k_{eff}}{k_{max}}\right)^\alpha$

for some exponent α.

Connecting to the metric (heuristic)

The spacetime volume element is √(-g)d⁴x, where g is the metric determinant.

In Schwarzschild:

$\sqrt{-g} = r^2 \sin\theta \sqrt{1 - \frac{2GM}{rc^2}}$

The radial factor √(1 - 2GM/rc²) appears in both the volume element and the time dilation factor.

Conjecture (heuristic): the same factor that shrinks local volume elements also shrinks the accessible correlation bandwidth. When √(-g) → 0, the "room for events" vanishes—and time freezes.

This isn't claiming √(-g) is the aperture (that conflates coordinates with physics). It's noting that the mathematical structure of time dilation mirrors the structure of volume contraction. Both might be downstream of the same underlying constraint on accessible degrees of freedom.

For Lorentz contraction, the spatial volume contracts by 1/γ in the direction of motion. The accessible variation in that direction decreases—and time dilates by the same factor γ.

Toy example: k_eff tracks the Lorentz factor

This doesn't derive SR—but it shows the aperture proxy tracks the same contraction.

Consider a 3D spatial metric at rest: $g_{ij} = \text{diag}(1, 1, 1)$. The eigenvalues are $\lambda = (1, 1, 1)$.

Participation ratio: $k_{eff} = \frac{(1+1+1)^2}{1^2+1^2+1^2} = \frac{9}{3} = 3$

Now boost to velocity $v$ along the x-axis. In the moving frame, the x-direction contracts by $1/\gamma$. The effective metric eigenvalues become $\lambda = (1/\gamma^2, 1, 1)$.

$k_{eff}(v) = \frac{(1/\gamma^2 + 1 + 1)^2}{1/\gamma^4 + 1 + 1}$

At $v = 0$: $k_{eff} = 3$ (full access).

At $v \to c$: $\gamma \to \infty$, so $1/\gamma^2 \to 0$, and:

$k_{eff} \to \frac{(0 + 1 + 1)^2}{0 + 1 + 1} = \frac{4}{2} = 2$

The aperture drops from 3 to 2 as velocity approaches c—and time dilation goes to infinity. The k_eff proxy is monotone with $\gamma$.

This is not a derivation. It's a sanity check: if you model "accessible dimensions" via participation ratio of metric eigenvalues, the contraction factor tracks the relativistic one.

Neural dimensional time (speculative)

For neural dynamics, define:

$D_{eff} = \frac{(\text{tr } C)^2}{\text{tr } C^2}$

where C is the covariance matrix of neural activity.

Hypothesis:

$\frac{d\tau_{subjective}}{dt} \propto D_{eff}^\beta$

for some exponent β.

Testable predictions:

• During slow-wave sleep (low D_eff), subjective time should be minimal
• During psychedelic states (high D_eff), subjective time should stretch or fragment
• During focused attention (reduced D_eff), time distortion should occur

Info geometry: Fisher volume and Cramér-Rao

The Fisher information metric on a statistical manifold defines a natural notion of "distance" between states:

$ds^2 = g_{ij}(\theta)d\theta^i d\theta^j$

The volume element √(det g) determines how much "space" is available for states to differ.

The Cramér-Rao bound sets a fundamental limit: the precision with which you can distinguish state $\theta$ from state $\theta + d\theta$ is bounded by the Fisher information. If the geometry of state space contracts (lower Fisher volume), the "speed limit" for distinguishing adjacent states drops. This isn't a practical limitation—it's a mathematical necessity.

The mechanism becomes inevitable: if you can't tell states apart quickly, you can't evolve through them quickly. Time flow ∝ distinguishability rate ∝ Fisher volume.

In a toy model where updates follow natural-gradient dynamics, the rate of distinguishable state change scales with the Fisher metric. Shrinking Fisher volume → fewer distinguishable updates per unit parameter-time → time slows.

When the Fisher volume contracts (dimensional collapse), geodesic motion slows. Time emerges from motion through state space; when the space itself contracts, motion slows.

This connects to the minimal embedding dimension paper—the minimum dimensionality needed to represent dynamics faithfully.

Thermodynamic angle

This is where time, aperture, and thermodynamics converge.

Define the accessible entropy for an observer with covariance $C$ over their observable subspace:

$S_{acc} = \frac{1}{2} \log \det C + \text{const}$

When the aperture squeezes, eigenvalues of $C$ collapse, $\det C$ shrinks, and $S_{acc}$ drops.

But here's the key: squeezing the aperture is information erasure. The degrees of freedom you lose access to don't vanish—they get traced out. And Landauer's principle says erasure has a minimum thermodynamic cost:

$Q \geq k_B T \cdot \Delta S_{erased}$

So the thermodynamic story becomes:

• Time = correlation accumulation rate through aperture
• Aperture squeeze = dimensional reduction = information erasure
• Information erasure has thermodynamic cost (Landauer)
• Time dilation is the operational symptom of a computational bottleneck

This is why Part 2 connected to Jacobson's derivation. Horizons are where apertures close. The thermodynamic cost of maintaining a decohered description at a horizon becomes extreme. Einstein's equations emerge from requiring thermodynamic consistency across observers with different apertures.

Time dilation isn't just about geometry. It's about the thermodynamic cost of compression.

This connects to the [dimensional Landauer bound](in preparation)—the formal treatment of thermodynamic costs under dimensional reduction.

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The Punchline

Time is not a backdrop on which physics happens. Time is a measure of correlation accumulation through dimensional apertures.

When the aperture squeezes—whether from gravity, velocity, or neural synchronization—correlation rate drops. From the inside, time slows.

This reframes several classic puzzles:

• Twin paradox: The traveling twin's dimensional aperture is contracted; they accumulate fewer correlations; they age less.
• Black hole information: The external observer's aperture closes at the horizon; they can't see further correlations; time freezes. (See the simulation.)
• Subjective time distortion: Neural aperture varies with brain state; subjective time varies accordingly.

The math is not complete. But the pattern is clear:

Squeeze k, slow time.

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One-sentence synthesis:

Time emerges from correlations through dimensional apertures; squeeze the aperture and time dilates—whether the squeezing comes from gravity, velocity, or synchronization.

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This is Part 3 of a series. See Part 1: Quantum Mechanics Without the Math and Part 2: Quantum Gravity Without the Paradox.

The formal connections to information geometry appear in our paper on minimal embedding dimension. The thermodynamic angle will appear in the forthcoming paper on the dimensional Landauer bound.

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Epilogue: Where This Goes

This trilogy makes a structural claim: projection through a dimensional aperture produces QM weirdness, dissolves the QG paradox, and reframes time as correlation rate. The math is information-geometric. The physics interpretation is... optional.

Here's the quiet truth: physics journals aren't the right adoption path for this. Not because the ideas are wrong, but because they're upstream of where physics knows how to referee. The question isn't "does this contradict GR/QM?"—it doesn't. The question is "what literature bucket does this go in?" And there isn't one yet.

So the research program runs differently:

The key insight: the math is substrate-agnostic. A variable-dimensional system with an observer-dependent metric produces the same clock behavior whether it's a robot, a cell, a brain, or spacetime. If the framework survives scrutiny in information geometry and improves engineered systems, the physics interpretation becomes retrospectively obvious.

This isn't avoidance—it's sequencing. You don't need to prove quantum gravity to show that aperture contraction slows distinguishability rate. You just need to define the manifold, the metric, and the clock. Then run it.

The next step is simulation. Black holes are the obvious target: same underlying dynamics, different observer apertures, different effective clocks. If even a toy model shows the external observer's time freezing while the infalling observer's flows—without invoking GR—then the conceptual bridge is built.

Feed this into the right simulation and it stops being philosophy overnight.

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Update: We built that simulation. See Black Hole as Aperture for the demo and the GitHub repo for the full code.