Closed timelike circles in negative-radius Kerr spacetime ($g_{\varphi \varphi }=r^2+a^2+2M{a}^2/r<0(r<0\text{ near }0)$)

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.