In a rotating (Gödel-type) spacetime, a circular worldline at coordinate radius r becomes
a genuine closed timelike curve once r exceeds a critical radius r_c = arcsinh(1) / (Ω√2)
set by the rotation rate Ω. Past that threshold the local time-flow rate dτ/dt = 1 − (r/r_c)²
turns negative, and the particle's worldline closes on itself each circuit instead of climbing forward.
Sets the critical radius r_c — faster rotation shrinks the loop threshold.
Above r_c the worldline closes into a loop; below it, time flows forward.
Drives the self-consistency return map rn+1 = rn + δ·sin(2πrn/r_c) applied once per completed circuit — watch whether it converges (self-consistent) or wanders (unstable).