= Solution
On the equatorial plane, the <Kerr metric> coefficient simplifies to
$$
g_{\phi\phi}=\frac{(r^2+a^2)^2-a^2\Delta}{r^2}
=r^2+a^2+\frac{2Ma^2}{r}.
$$
Put $r=-\varepsilon$ on the negative-radius sheet. Then $g_{\phi\phi}=\varepsilon^2+a^2-2Ma^2/\varepsilon<0$ for all sufficiently small $\varepsilon>0$. The curve
$$
\boxed{t=t_0,\quad r=-\varepsilon,\quad\theta=\pi/2,\quad
\phi\in[0,2\pi],}
$$
is a <closed timelike curve>, since its tangent is $\partial_\phi$ 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 $\widetilde t$ is also constant, because its difference from $t$ depends only on $r$.
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