Closed timelike curve

A closed timelike curve is a closed curve whose tangent is everywhere timelike, so an observer following it returns to the same spacetime event.

In the flat quotient $t\equiv t+P_t$ with noncompact space, choose any sufficiently large positive or negative temporal winding to make a lifted displacement timelike. Every event is in both the causal future and causal past of every other event. Constant-position curves are closed timelike curves. The quotient is a valid Lorentzian manifold, but it violates chronology and cannot support arbitrary globally consistent initial-value evolution.

On the equatorial plane of the Kerr metric, the displayed component is negative for sufficiently small negative $r$. A curve of constant $t,r,\theta =\pi /2$, with $\varphi$ increasing through $2\pi$, is therefore a regular closed timelike curve. It avoids the ring at $r=0$. Such circles occur on the negative-radius sheet of the exact analytic extension, not in the exterior region accessible from positive-radius infinity.