No, it is literally impossible to enter or exit a CTC, at least in classical GR. All CTCs are shrouded by a Cauchy horizon, meaning that you cannot “push” the evolution of a spatial slice into a CTC region. Or out of it, if you apply the same logic in reverse. The issue most people forget is that the spacetime in GR is not a background on which matter exists, the spacetime metric is unique (up to diffeomorphisms) and it uniquely determines the stress energy tensor at every point in spacetime, which means it determines its matter content almost completely. Thus one cannot have one loop of a CTC without something being there, and another with it. Anyway, it’s getting a bit technical. My original comment was that the OP setup breaks physics as we know it, and so it’s kind of pointless to discuss.
No, it is literally impossible to enter or exit a CTC, at least in classical GR. All CTCs are shrouded by a Cauchy horizon, meaning that you cannot “push” the evolution of a spatial slice into a CTC region. Or out of it, if you apply the same logic in reverse. The issue most people forget is that the spacetime in GR is not a background on which matter exists, the spacetime metric is unique (up to diffeomorphisms) and it uniquely determines the stress energy tensor at every point in spacetime, which means it determines its matter content almost completely. Thus one cannot have one loop of a CTC without something being there, and another with it. Anyway, it’s getting a bit technical. My original comment was that the OP setup breaks physics as we know it, and so it’s kind of pointless to discuss.