Closed Timelike Curves in a Rotating Universe

Gödel Universe Parameters

Rotation scale a 1.00

Orbit radius R 1.20

Orbit speed 1.0×

Live State

spacelike loop — must climb in t

Critical radius R꜀—

g_φφ(R)—

Δt per revolution—

How it works

Gödel (1949) found an exact solution of Einstein's field equations for a universe filled with rotating dust. In cylindrical coordinates (t, r, φ, z), scaled by the rotation length a, the metric is:

ds² = 4a²[ -dt² - 2√2 sinh²ρ dφ dt
            + dρ² + dz² + (sinh⁴ρ - sinh²ρ) dφ² ]   ρ = R/(a√2)

The φφ metric component is g_φφ = 4a²·sinh²ρ(sinh²ρ - 1). It changes sign at sinh²ρ = 1, i.e. at the exact critical radius

R꜀ = a√2 · arcsinh(1) = a√2 · ln(1+√2) ≈ 1.246 a

For R < R꜀, g_φφ > 0 (spacelike): a circle at fixed t is a purely spatial loop, and a real worldline must also advance in coordinate time by at least Δt = √(g_φφ) per radian to stay timelike — it spirals forward, never closing. For R > R꜀, g_φφ < 0 (timelike): the circle itself, at constant t, is already a valid timelike curve. Since φ is periodic, that worldline returns to its own starting event — a genuine closed timelike curve (CTC), one of the most discussed causality violations in general relativity.

a — the Gödel rotation length scale (sets the overall size of the universe's vorticity).
R — the orbit radius of the test worldline (the sphere), in the same units as a.
Light cones — small tilted double-cones sampled at increasing radius: their tilt is driven by the exact frame-dragging term 2√2·sinh²ρ, and the ring of cones turns from green to orange right at R꜀.

This is not science fiction dressing: the Gödel metric is a standard textbook example (Hawking & Ellis, The Large Scale Structure of Space-Time) used to show that general relativity, as a purely local theory of curvature, does not by itself forbid closed timelike curves — global causality has to be imposed as an extra assumption or ruled out by other physical arguments (our own universe is not observed to rotate this way).

Test worldline Spacelike loop (R<R꜀) Closed timelike curve (R>R꜀)

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