= Solution
The periodic axial orbits have norm $m^2=g_{\varphi\varphi}$. On the equator,
$$
m^2=r^2+a^2+\frac{2Ma^2}{r}\longrightarrow-\infty\quad(r\to0^-).
$$
Thus a neighborhood on the negative sheet has $m^2<0$. A fixed-$(t,r,\theta)$ curve running once around periodic $\varphi$ 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. \b[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>.
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