  We present a class of curved-spacetime vacuum solutions which develope closed
timelike curves at some particular moment. We then use these vacuum solutions
to construct a time-machine model. The causality violation occurs inside an
empty torus, which constitutes the time-machine core. The matter field
surrounding this empty torus satisfies the weak, dominant, and strong energy
conditions. The model is regular, asymptotically-flat, and
topologically-trivial. Stability remains the main open question.
