 Solution

On the equatorial plane, the Kerr metric coefficient simplifies toPut $r=-\epsilon$ on the negative-radius sheet. Then $g_{\varphi \varphi }={\epsilon }^2+a^2-2M{a}^2/\epsilon <0$ for all sufficiently small $\epsilon >0$. The curveis a closed timelike curve, since its tangent is ${\partial }_{\varphi }$ and the angular coordinate is periodic. It never meets the ring at $r=0$. These are the closed timelike circles in negative-radius Kerr spacetime; they need not be geodesics. Along them $\tilde{t}$ is also constant, because its difference from $t$ depends only on $r$.