Validated demo

Gödel CTC Visualizer

A visualization page for the causal transition in Gödel spacetime and the emergence of azimuthal closed timelike curves.

g_{\phi\phi} < 0

What the tool does

The visualizer will present the critical-radius idea, show where circular paths change causal character, and connect plotted regions to the sign of the angular metric component.

Why it matters

Visual representations help separate coordinate facts from physical interpretation, especially when studying causality violations in exact solutions.

Connection to CTC paper

The CTC paper uses Gödel spacetime as a central example for chronology violation. This tool turns the paper's causal-radius discussion into a reproducible visual reference.

Research role

The initial version should make the chronology-violating region visible without implying that the visualization itself establishes a new theorem or physical result.

Near-term build

A static diagram can come first, followed by parameter controls and source-linked derivations once the equations and units are locked down.

Validated Demo

Gödel Chronology Threshold

This first validation artifact checks the visualizer against the standard Gödel chronology threshold. In the normalized radial coordinate used here, the azimuthal metric component changes sign at the known critical radius, so circular azimuthal curves become timelike beyond that boundary. The result is an established feature of Gödel spacetime and is used here only to validate the site pipeline against known literature, not to claim new physics.

Angular metric component

g_{\phi\phi} = \sinh^2(r) - \sinh^4(r)

Critical radius

r_c = \ln(1 + \sqrt{2})

Static threshold diagram

g_{\\phi\\phi} \\ge 0

before

r_c

r_c

g_{\\phi\\phi} < 0

beyond

r_c

The diagram is a sign-check aid for the known Gödel result. It is not a simulation and does not replace derivation or source comparison.

Radius	$g_{\\phi\\phi}$	Sign	Interpretation
$0$	0.000	zero	Axis limit; no negative angular term.
$0.50$	0.198	positive	Outside the CTC region for this circular test.
$r_c \approx 0.881$	0.000	zero	Chronology threshold in this normalization.
$1.00$	-0.527	negative	Beyond the threshold; azimuthal circles are timelike.
$1.25$	-4.019	negative	Deeper into the chronology-violating region.

Reproducibility artifact

The public artifact records the formula, critical radius, sample values around the threshold, and a minimal reproduction snippet for this known Gödel result.

Open reproducibility artifact

Research caution

Validated demo, limited to the standard Gödel threshold check shown below. It is not a general-purpose relativity engine and does not establish new physical results.