 Solution

The periodic axial orbits have norm $m^2=g_{\phi \phi }$. On the equator,Thus a neighborhood on the negative sheet has $m^2<0$. A fixed-$(t,r,\theta )$ curve running once around periodic $\phi$ is a closed timelike curve: its timelike tangent returns to the same event.

No horizon in the over-rotating solution prevents passage from positive-$r$ infinity through the nonsingular disk to this region. Timelike periodic axial orbits provide the time-machine mechanism. This property of the analytic solution does not establish that such a naked geometry can be manufactured stably from generic regular collapse, as envisaged by the weak cosmic censorship conjecture.