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Time loops: the maths

Gödel's rotating universe (1949), an exact solution of Einstein's equations. The light cones along a circle are computed from the metric. Exact.

Equations

dt/dφ = −√2 sinh²r ± sinh r √(sinh²r + 1)

Light going round a circle of coordinate radius r: how the light cone tilts.

sinh r_c = 1

Beyond r_c = arsinh(1) ≈ 0.881 the circle itself lies inside the light cone: a closed timelike curve.

R_c = √2 · arsinh(1) · c / ω

That critical radius as a real distance, for a universe rotating at rate ω = 2π/P.

Constants

P

the rotation period (slider)

Gödel's universe; ours shows no measurable rotation (Saadeh et al. 2016)

Worked examples ✓ checked on every change

Loops begin, for a spin every 100 billion years	19.8 billion ly	√2 · arsinh(1) · P / 2π
At coordinate radius 1.06: a time loop?	yes	1.06 > arsinh(1)

The app's test suite puts the lab in each setup, reads the lab's own result, and fails if it strays from these values.

What's simplified

• The cone slices are drawn at a few radii; the circle walk is schematic in time.

Where it breaks

As a picture of our universe: it isn't rotating, and has matter that Gödel's solution doesn't.

Sources: K. Gödel, Rev. Mod. Phys. 21, 447 (1949); S. Hawking & G. Ellis, The Large Scale Structure of Space-Time (1973), §5.7.