Black Hole Aperture: A Constructed Analogy

Correction

The original version of this post said the simulation reproduced black-hole time dilation without general relativity. It did not. The model defines an observer's clock from aperture-weighted state change and then closes that aperture near a horizon analogue. Its clock slows because the slowing is built into the definition.

That construction can still illustrate observer-dependent readout. It is not a derivation, independent test, or physical model of a black hole. The stronger claims about Landauer cost, LIGO ringdown, and an external observer's dimension falling from three to two are withdrawn.

This page now documents exactly what the simulation does.

The model

The toy substrate contains coupled oscillators with state

$\mathbf{x}\in\mathbb{R}^{2N}.$

Two simulated observers receive different filtered versions of that same state:

$\mathbf{x}_{\mathrm{obs}}(r)=\sqrt{\mathbf{w}(r)}\odot\mathbf{x}.$

The “infalling” channel keeps its weights open. The “external” channel progressively suppresses selected modes as a control parameter called radius approaches a chosen horizon value. Here $w_i$ acts as a non-negative metric or transmission-power weight; writing the amplitude factor as $\sqrt{w_i}$ makes the displayed quadratic quantities consistent with the channel definition.

The labels are analogical. There is no spacetime metric, stress-energy tensor, field equation, causal horizon, null geodesic, or gravitational redshift in the oscillator model.

The constructed clock

The simulation defines a clock rate from weighted state-space motion:

$\dot T_{\mathrm{channel}}=\frac{\sqrt{\sum_i w_i(r)\dot x_i^2}}{\sqrt{\sum_i \dot x_i^2}}.$

The denominator is the rate of the fully open channel. Closing weights removes positive terms from the numerator, so the normalized $\dot T_{\mathrm{channel}}$ falls. This is a valid clock definition for the toy model. It does not show that physical proper time is caused by lost dimensional access.

The inference is circular if presented as a test:

1. define elapsed channel time as aperture-weighted change;
2. make the aperture weights vanish near a chosen boundary;
3. observe that elapsed channel time stops accumulating.

The output confirms the implementation. It does not discriminate the aperture hypothesis from general relativity or from another theory of time.

What general relativity says

For a stationary observer outside a Schwarzschild black hole,

$\frac{d\tau_{\mathrm{proper}}}{dt}=\sqrt{1-\frac{r_s}{r}}.$

The ratio tends to zero as a stationary worldline approaches the horizon. No physical observer can remain stationary at the horizon. A freely falling observer crosses it in finite proper time, while signals sent outward become increasingly delayed and redshifted for a distant receiver.

These results follow from Lorentzian spacetime geometry. The horizon's area law does not mean that an external observer's state-space participation ratio becomes two. Spatial dimension, horizon area, Hilbert-space dimension, and effective statistical dimension are different quantities.

What the simulation can teach

The useful lesson concerns access rather than proper time. Two channels observing the same dynamics can differ in:

• which modes they retain;
• which source states they can distinguish;
• how estimation error changes with noise;
• which records remain readable after filtering;
• how a chosen channel clock accumulates.

Those are genuine observer-relative properties. They can matter in sensing, control, and memory without implying that the underlying dynamics slow down.

Auditing the four plotted quantities

Effective dimension. A participation ratio such as

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

summarizes how evenly the selected weights are distributed. It is a channel statistic, not a spatial dimension.

Accessible entropy. A log-determinant of a weighted covariance can summarize dispersion after choices about coordinates, units, regularization, and sampling. It is not automatically thermodynamic entropy.

Correlation clock. The aperture-weighted speed above is an engineered observable. Calling it $\tau$ does not make it relativistic proper time.

Erasure cost. A decrease in a covariance or accessible-information proxy is not a Landauer heat cost. Landauer's bound applies when a physical device implements a logically irreversible operation such as memory reset. The model contains no bath, reset protocol, temperature, or heat flow.

The retired comparisons

The earlier post described a qualitative merger and ringdown as a LIGO connection. Coupled oscillators can chirp and decay, but visual resemblance does not reproduce the Einstein equations, gravitational-wave propagation, or black-hole quasinormal modes. No LIGO prediction follows from this simulation.

It also compared a hand-chosen aperture curve with the Schwarzschild redshift factor. Tracking a target curve after choosing the weights is curve construction, not derivation. A meaningful comparison would first derive the weights from independent dynamics and then test out-of-sample observables.

Relation to meta-time

The corrected time essay uses meta-time for an epoch or traversal coordinate. It answers “which visit?” rather than “how much proper time elapsed?” A physical record can carry such an epoch, and a noisy channel can progressively lose access to it.

That is the defensible bridge to this toy model: filtering can make history harder to read. It does not freeze the history that the source system is generating.

A useful next version

A stronger observer-channel experiment would avoid black-hole claims and measure quantities the model actually contains:

1. Specify a source process and two observation kernels.
2. Test global identifiability, local Fisher information, and finite-noise decoding separately.
3. Add a physical memory with an explicit reset cycle before estimating dissipation.
4. Derive any channel weights from the model rather than choosing them to match a target clock.
5. Treat general-relativistic predictions as external constraints, not labels applied to the controls.

The current simulation remains useful as an interactive diagram of filtered access. Its evidence ends there.